diff --git "a/Sky-T1_10k.json" "b/Sky-T1_10k.json" new file mode 100644--- /dev/null +++ "b/Sky-T1_10k.json" @@ -0,0 +1,146876 @@ +[ + { + "question": "Return your final response within \\boxed{}. The operation $\\otimes$ is defined for all nonzero numbers by $a\\otimes b =\\frac{a^{2}}{b}$. Determine $[(1\\otimes 2)\\otimes 3]-[1\\otimes (2\\otimes 3)]$.\n$\\text{(A)}\\ -\\frac{2}{3}\\qquad\\text{(B)}\\ -\\frac{1}{4}\\qquad\\text{(C)}\\ 0\\qquad\\text{(D)}\\ \\frac{1}{4}\\qquad\\text{(E)}\\ \\frac{2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_0", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The operation $\\otimes$ is defined for all nonzero numbers by $a\\otimes b =\\frac{a^{2}}{b}$. Determine $[(1\\otimes 2)\\otimes 3]-[1\\otimes (2\\otimes 3)]$.\n$\\text{(A)}\\ -\\frac{2}{3}\\qquad\\text{(B)}\\ -\\frac{1}{4}\\qquad\\text{(C)}\\ 0\\qquad\\text{(D)}\\ \\frac{1}{4}\\qquad\\text{(E)}\\ \\frac{2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?\n[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill((6,0)--(8,0)--(8,26)--(6,26)--cycle,gray); fill((12,0)--(14,0)--(14,26)--(12,26)--cycle,gray); fill((18,0)--(20,0)--(20,26)--(18,26)--cycle,gray); fill((24,0)--(26,0)--(26,26)--(24,26)--cycle,gray); fill((0,0)--(26,0)--(26,2)--(0,2)--cycle,gray); fill((0,12)--(26,12)--(26,14)--(0,14)--cycle,gray); fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); [/asy]\n$\\textbf{(A)}\\ 26\\qquad\\textbf{(B)}\\ 28\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 32\\qquad\\textbf{(E)}\\ 34$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?\n[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill((6,0)--(8,0)--(8,26)--(6,26)--cycle,gray); fill((12,0)--(14,0)--(14,26)--(12,26)--cycle,gray); fill((18,0)--(20,0)--(20,26)--(18,26)--cycle,gray); fill((24,0)--(26,0)--(26,26)--(24,26)--cycle,gray); fill((0,0)--(26,0)--(26,2)--(0,2)--cycle,gray); fill((0,12)--(26,12)--(26,14)--(0,14)--cycle,gray); fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); [/asy]\n$\\textbf{(A)}\\ 26\\qquad\\textbf{(B)}\\ 28\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 32\\qquad\\textbf{(E)}\\ 34$" + } + }, + { + "question": "Return your final response within \\boxed{}. $P(x)$ is a polynomial of degree $3n$ such that\n\\begin{eqnarray*} P(0) = P(3) = \\cdots &=& P(3n) = 2, \\\\ P(1) = P(4) = \\cdots &=& P(3n-2) = 1, \\\\ P(2) = P(5) = \\cdots &=& P(3n-1) = 0, \\quad\\text{ and }\\\\ && P(3n+1) = 730.\\end{eqnarray*}\nDetermine $n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "4", + "index": "Sky-T1_10k_2", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $P(x)$ is a polynomial of degree $3n$ such that\n\\begin{eqnarray*} P(0) = P(3) = \\cdots &=& P(3n) = 2, \\\\ P(1) = P(4) = \\cdots &=& P(3n-2) = 1, \\\\ P(2) = P(5) = \\cdots &=& P(3n-1) = 0, \\quad\\text{ and }\\\\ && P(3n+1) = 730.\\end{eqnarray*}\nDetermine $n$." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f$ be the function defined by $f(x)=ax^2-\\sqrt{2}$ for some positive $a$. If $f(f(\\sqrt{2}))=-\\sqrt{2}$ then $a=$\n$\\text{(A) } \\frac{2-\\sqrt{2}}{2}\\quad \\text{(B) } \\frac{1}{2}\\quad \\text{(C) } 2-\\sqrt{2}\\quad \\text{(D) } \\frac{\\sqrt{2}}{2}\\quad \\text{(E) } \\frac{2+\\sqrt{2}}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_3", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f$ be the function defined by $f(x)=ax^2-\\sqrt{2}$ for some positive $a$. If $f(f(\\sqrt{2}))=-\\sqrt{2}$ then $a=$\n$\\text{(A) } \\frac{2-\\sqrt{2}}{2}\\quad \\text{(B) } \\frac{1}{2}\\quad \\text{(C) } 2-\\sqrt{2}\\quad \\text{(D) } \\frac{\\sqrt{2}}{2}\\quad \\text{(E) } \\frac{2+\\sqrt{2}}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The total in-store price for an appliance is $\\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\\textdollar 29.98$ and a one-time shipping and handling charge of $\\textdollar 9.98$. How many cents are saved by buying the appliance from the television advertiser?\n$\\mathrm{(A) \\ 6 } \\qquad \\mathrm{(B) \\ 7 } \\qquad \\mathrm{(C) \\ 8 } \\qquad \\mathrm{(D) \\ 9 } \\qquad \\mathrm{(E) \\ 10 }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_4", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The total in-store price for an appliance is $\\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\\textdollar 29.98$ and a one-time shipping and handling charge of $\\textdollar 9.98$. How many cents are saved by buying the appliance from the television advertiser?\n$\\mathrm{(A) \\ 6 } \\qquad \\mathrm{(B) \\ 7 } \\qquad \\mathrm{(C) \\ 8 } \\qquad \\mathrm{(D) \\ 9 } \\qquad \\mathrm{(E) \\ 10 }$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ and $\\overline{AG}$ are parallel. Find $HC/JE$.\n$\\text{(A)}\\ 5/4 \\qquad \\text{(B)}\\ 4/3 \\qquad \\text{(C)}\\ 3/2 \\qquad \\text{(D)}\\ 5/3 \\qquad \\text{(E)}\\ 2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_5", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ and $\\overline{AG}$ are parallel. Find $HC/JE$.\n$\\text{(A)}\\ 5/4 \\qquad \\text{(B)}\\ 4/3 \\qquad \\text{(C)}\\ 3/2 \\qquad \\text{(D)}\\ 5/3 \\qquad \\text{(E)}\\ 2$" + } + }, + { + "question": "Return your final response within \\boxed{}. During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?\n$\\text{(A)}\\ 28\\% \\qquad \\text{(B)}\\ 35\\% \\qquad \\text{(C)}\\ 70\\% \\qquad \\text{(D)}\\ 75\\% \\qquad \\text{(E)}\\ 80\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_6", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?\n$\\text{(A)}\\ 28\\% \\qquad \\text{(B)}\\ 35\\% \\qquad \\text{(C)}\\ 70\\% \\qquad \\text{(D)}\\ 75\\% \\qquad \\text{(E)}\\ 80\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph, $G$ of $y=\\log_{10}x$ is rotated $90^{\\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?\n(A) $y=\\log_{10}\\left(\\frac{x+90}{9}\\right)$ (B) $y=\\log_{x}10$ (C) $y=\\frac{1}{x+1}$ (D) $y=10^{-x}$ (E) $y=10^x$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_7", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph, $G$ of $y=\\log_{10}x$ is rotated $90^{\\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?\n(A) $y=\\log_{10}\\left(\\frac{x+90}{9}\\right)$ (B) $y=\\log_{x}10$ (C) $y=\\frac{1}{x+1}$ (D) $y=10^{-x}$ (E) $y=10^x$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his number, and then doubles the result. Who gets the largest final answer?\n$\\text{(A)}\\ \\text{Jose} \\qquad \\text{(B)}\\ \\text{Thuy} \\qquad \\text{(C)}\\ \\text{Kareem} \\qquad \\text{(D)}\\ \\text{Jose and Thuy} \\qquad \\text{(E)}\\ \\text{Thuy and Kareem}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_8", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his number, and then doubles the result. Who gets the largest final answer?\n$\\text{(A)}\\ \\text{Jose} \\qquad \\text{(B)}\\ \\text{Thuy} \\qquad \\text{(C)}\\ \\text{Kareem} \\qquad \\text{(D)}\\ \\text{Jose and Thuy} \\qquad \\text{(E)}\\ \\text{Thuy and Kareem}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased? \n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 7 \\qquad \\textbf{(C)}\\ 8 \\qquad \\textbf{(D)}\\ 9\\qquad \\textbf{(E)}\\ \\text{cannot be determined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_9", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased? \n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 7 \\qquad \\textbf{(C)}\\ 8 \\qquad \\textbf{(D)}\\ 9\\qquad \\textbf{(E)}\\ \\text{cannot be determined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?\n\n[asy] unitsize(2mm); defaultpen(linewidth(.8pt)); fill((0,0)--(0,5)--(5,5)--cycle,gray); fill((25,0)--(25,5)--(20,5)--cycle,gray); draw((0,0)--(0,5)--(25,5)--(25,0)--cycle); draw((0,0)--(5,5)); draw((20,5)--(25,0)); [/asy]\n$\\mathrm{(A)}\\frac {1}{8}\\qquad \\mathrm{(B)}\\frac {1}{6}\\qquad \\mathrm{(C)}\\frac {1}{5}\\qquad \\mathrm{(D)}\\frac {1}{4}\\qquad \\mathrm{(E)}\\frac {1}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_10", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?\n\n[asy] unitsize(2mm); defaultpen(linewidth(.8pt)); fill((0,0)--(0,5)--(5,5)--cycle,gray); fill((25,0)--(25,5)--(20,5)--cycle,gray); draw((0,0)--(0,5)--(25,5)--(25,0)--cycle); draw((0,0)--(5,5)); draw((20,5)--(25,0)); [/asy]\n$\\mathrm{(A)}\\frac {1}{8}\\qquad \\mathrm{(B)}\\frac {1}{6}\\qquad \\mathrm{(C)}\\frac {1}{5}\\qquad \\mathrm{(D)}\\frac {1}{4}\\qquad \\mathrm{(E)}\\frac {1}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \\[(\\cos 40^\\circ,\\sin 40^\\circ), (\\cos 60^\\circ,\\sin 60^\\circ), \\text{ and } (\\cos t^\\circ,\\sin t^\\circ)\\]\nis isosceles? \n$\\textbf{(A)} \\: 100 \\qquad\\textbf{(B)} \\: 150 \\qquad\\textbf{(C)} \\: 330 \\qquad\\textbf{(D)} \\: 360 \\qquad\\textbf{(E)} \\: 380$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_11", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \\[(\\cos 40^\\circ,\\sin 40^\\circ), (\\cos 60^\\circ,\\sin 60^\\circ), \\text{ and } (\\cos t^\\circ,\\sin t^\\circ)\\]\nis isosceles? \n$\\textbf{(A)} \\: 100 \\qquad\\textbf{(B)} \\: 150 \\qquad\\textbf{(C)} \\: 330 \\qquad\\textbf{(D)} \\: 360 \\qquad\\textbf{(E)} \\: 380$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not always correct is:\n$\\textbf{(A)}\\ x + z > y + z \\qquad\\textbf{(B)}\\ x - z > y - z \\qquad\\textbf{(C)}\\ xz > yz$\n$\\textbf{(D)}\\ \\frac {x}{z^2} > \\frac {y}{z^2} \\qquad\\textbf{(E)}\\ xz^2 > yz^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_12", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not always correct is:\n$\\textbf{(A)}\\ x + z > y + z \\qquad\\textbf{(B)}\\ x - z > y - z \\qquad\\textbf{(C)}\\ xz > yz$\n$\\textbf{(D)}\\ \\frac {x}{z^2} > \\frac {y}{z^2} \\qquad\\textbf{(E)}\\ xz^2 > yz^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 45 \\qquad \\textbf{(C)}\\ 504 \\qquad \\textbf{(D)}\\ 1008 \\qquad \\textbf{(E)}\\ 2015$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_13", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 45 \\qquad \\textbf{(C)}\\ 504 \\qquad \\textbf{(D)}\\ 1008 \\qquad \\textbf{(E)}\\ 2015$" + } + }, + { + "question": "Return your final response within \\boxed{}. The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: \n$\\textbf{(A)}\\ -2\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ \\text{undetermined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_14", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: \n$\\textbf{(A)}\\ -2\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ \\text{undetermined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let a geometric progression with n terms have first term one, common ratio $r$ and sum $s$, where $r$ and $s$ are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is\n$\\textbf{(A) }\\frac{1}{s}\\qquad \\textbf{(B) }\\frac{1}{r^ns}\\qquad \\textbf{(C) }\\frac{s}{r^{n-1}}\\qquad \\textbf{(D) }\\frac{r^n}{s}\\qquad \\textbf{(E) } \\frac{r^{n-1}}{s}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_15", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let a geometric progression with n terms have first term one, common ratio $r$ and sum $s$, where $r$ and $s$ are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is\n$\\textbf{(A) }\\frac{1}{s}\\qquad \\textbf{(B) }\\frac{1}{r^ns}\\qquad \\textbf{(C) }\\frac{s}{r^{n-1}}\\qquad \\textbf{(D) }\\frac{r^n}{s}\\qquad \\textbf{(E) } \\frac{r^{n-1}}{s}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alicia had two containers. The first was $\\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?\n$\\textbf{(A) } \\frac{5}{8} \\qquad \\textbf{(B) } \\frac{4}{5} \\qquad \\textbf{(C) } \\frac{7}{8} \\qquad \\textbf{(D) } \\frac{9}{10} \\qquad \\textbf{(E) } \\frac{11}{12}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_16", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alicia had two containers. The first was $\\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?\n$\\textbf{(A) } \\frac{5}{8} \\qquad \\textbf{(B) } \\frac{4}{5} \\qquad \\textbf{(C) } \\frac{7}{8} \\qquad \\textbf{(D) } \\frac{9}{10} \\qquad \\textbf{(E) } \\frac{11}{12}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "647", + "index": "Sky-T1_10k_17", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$?\n$\\textbf{(A) }9 \\qquad\\textbf{(B) } 6\\sqrt{3} \\qquad\\textbf{(C) } 8\\sqrt{3} \\qquad\\textbf{(D) } 17 \\qquad\\textbf{(E) }12\\sqrt{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_18", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$?\n$\\textbf{(A) }9 \\qquad\\textbf{(B) } 6\\sqrt{3} \\qquad\\textbf{(C) } 8\\sqrt{3} \\qquad\\textbf{(D) } 17 \\qquad\\textbf{(E) }12\\sqrt{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "65", + "index": "Sky-T1_10k_19", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$." + } + }, + { + "question": "Return your final response within \\boxed{}. Find the smallest positive number from the numbers below.\n$\\textbf{(A)} \\ 10-3\\sqrt{11} \\qquad \\textbf{(B)} \\ 3\\sqrt{11}-10 \\qquad \\textbf{(C)}\\ 18-5\\sqrt{13}\\qquad \\textbf{(D)}\\ 51-10\\sqrt{26}\\qquad \\textbf{(E)}\\ 10\\sqrt{26}-51$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_20", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the smallest positive number from the numbers below.\n$\\textbf{(A)} \\ 10-3\\sqrt{11} \\qquad \\textbf{(B)} \\ 3\\sqrt{11}-10 \\qquad \\textbf{(C)}\\ 18-5\\sqrt{13}\\qquad \\textbf{(D)}\\ 51-10\\sqrt{26}\\qquad \\textbf{(E)}\\ 10\\sqrt{26}-51$" + } + }, + { + "question": "Return your final response within \\boxed{}. The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):\n$\\textbf{(A)}\\ 12\\qquad \\textbf{(B)}\\ -12\\qquad \\textbf{(C)}\\ \\pm 12\\qquad \\textbf{(D)}\\ {12}\\text{ or }{6}\\qquad \\textbf{(E)}\\ {6}\\text{ or }{6\\frac{2}{3}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_21", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):\n$\\textbf{(A)}\\ 12\\qquad \\textbf{(B)}\\ -12\\qquad \\textbf{(C)}\\ \\pm 12\\qquad \\textbf{(D)}\\ {12}\\text{ or }{6}\\qquad \\textbf{(E)}\\ {6}\\text{ or }{6\\frac{2}{3}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose $q_1,q_2,...$ is an infinite sequence of integers satisfying the following two conditions:\n(a) $m - n$ divides $q_m - q_n$ for $m>n \\geq 0$\n(b) There is a polynomial $P$ such that $|q_n|n \\geq 0$\n(b) There is a polynomial $P$ such that $|q_n|0$?\n$\\textbf{(A)}\\ \\text{For all x}, x^2 < 0\\qquad \\textbf{(B)}\\ \\text{For all x}, x^2 \\le 0\\qquad \\textbf{(C)}\\ \\text{For no x}, x^2>0\\qquad \\\\ \\textbf{(D)}\\ \\text{For some x}, x^2>0\\qquad \\textbf{(E)}\\ \\text{For some x}, x^2 \\le 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_28", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?\n$\\textbf{(A)}\\ \\text{For all x}, x^2 < 0\\qquad \\textbf{(B)}\\ \\text{For all x}, x^2 \\le 0\\qquad \\textbf{(C)}\\ \\text{For no x}, x^2>0\\qquad \\\\ \\textbf{(D)}\\ \\text{For some x}, x^2>0\\qquad \\textbf{(E)}\\ \\text{For some x}, x^2 \\le 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?\n$\\textbf{(A) } 2^{1002} \\qquad\\textbf{(B) } 2^{1003} \\qquad\\textbf{(C) } 2^{1004} \\qquad\\textbf{(D) } 2^{1005} \\qquad\\textbf{(E) }2^{1006}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_29", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?\n$\\textbf{(A) } 2^{1002} \\qquad\\textbf{(B) } 2^{1003} \\qquad\\textbf{(C) } 2^{1004} \\qquad\\textbf{(D) } 2^{1005} \\qquad\\textbf{(E) }2^{1006}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider all 1000-element subsets of the set $\\{1, 2, 3, ... , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "431", + "index": "Sky-T1_10k_30", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider all 1000-element subsets of the set $\\{1, 2, 3, ... , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$." + } + }, + { + "question": "Return your final response within \\boxed{}. A man on his way to dinner short after $6: 00$ p.m. observes that the hands of his watch form an angle of $110^{\\circ}$. Returning before $7: 00$ p.m. he notices that again the hands of his watch form an angle of $110^{\\circ}$. The number of minutes that he has been away is: \n$\\textbf{(A)}\\ 36\\frac{2}3\\qquad\\textbf{(B)}\\ 40\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 42.4\\qquad\\textbf{(E)}\\ 45$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_31", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A man on his way to dinner short after $6: 00$ p.m. observes that the hands of his watch form an angle of $110^{\\circ}$. Returning before $7: 00$ p.m. he notices that again the hands of his watch form an angle of $110^{\\circ}$. The number of minutes that he has been away is: \n$\\textbf{(A)}\\ 36\\frac{2}3\\qquad\\textbf{(B)}\\ 40\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 42.4\\qquad\\textbf{(E)}\\ 45$" + } + }, + { + "question": "Return your final response within \\boxed{}. A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?\n$\\text{(A)}\\ 60 \\qquad \\text{(B)}\\ 72 \\qquad \\text{(C)}\\ 84 \\qquad \\text{(D)}\\ 90 \\qquad \\text{(E)} 96$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_32", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?\n$\\text{(A)}\\ 60 \\qquad \\text{(B)}\\ 72 \\qquad \\text{(C)}\\ 84 \\qquad \\text{(D)}\\ 90 \\qquad \\text{(E)} 96$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$? \n\\[\\begin{array}{cccc}& A & B & A\\\\ \\times & & C & D\\\\ \\hline C & D & C & D\\\\ \\end{array}\\]\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_33", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$? \n\\[\\begin{array}{cccc}& A & B & A\\\\ \\times & & C & D\\\\ \\hline C & D & C & D\\\\ \\end{array}\\]\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \\,$ where $\\,m\\,$ and $\\,n\\,$ are positive integers. Prove that $\\,a^m + a^n \\geq m^m + n^n$.\n[You may wish to analyze the ratio $\\,(a^N - N^N)/(a-N),$ for real $\\, a \\geq 0 \\,$ and integer $\\, N \\geq 1$.]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "a^m + a^n \\geq m^m + n^n", + "index": "Sky-T1_10k_34", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \\,$ where $\\,m\\,$ and $\\,n\\,$ are positive integers. Prove that $\\,a^m + a^n \\geq m^m + n^n$.\n[You may wish to analyze the ratio $\\,(a^N - N^N)/(a-N),$ for real $\\, a \\geq 0 \\,$ and integer $\\, N \\geq 1$.]" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?\n$\\textbf{(A)} ~10{,}272\\qquad\\textbf{(B)} ~11{,}700\\qquad\\textbf{(C)} ~13{,}362\\qquad\\textbf{(D)} ~14{,}238\\qquad\\textbf{(E)} ~15{,}426$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_35", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?\n$\\textbf{(A)} ~10{,}272\\qquad\\textbf{(B)} ~11{,}700\\qquad\\textbf{(C)} ~13{,}362\\qquad\\textbf{(D)} ~14{,}238\\qquad\\textbf{(E)} ~15{,}426$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of \\[2^{\\left(0^{\\left(1^9\\right)}\\right)}+\\left(\\left(2^0\\right)^1\\right)^9?\\]\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_36", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of \\[2^{\\left(0^{\\left(1^9\\right)}\\right)}+\\left(\\left(2^0\\right)^1\\right)^9?\\]\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?\n$\\textbf{(A) } \\text{the\\,empty\\,set}\\qquad \\textbf{(B) } \\textrm{one\\,point}\\qquad \\textbf{(C) } \\textrm{two\\,lines} \\qquad \\textbf{(D) } \\textrm{a\\,circle} \\qquad \\textbf{(E) } \\textrm{the\\,entire\\,plane}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_37", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?\n$\\textbf{(A) } \\text{the\\,empty\\,set}\\qquad \\textbf{(B) } \\textrm{one\\,point}\\qquad \\textbf{(C) } \\textrm{two\\,lines} \\qquad \\textbf{(D) } \\textrm{a\\,circle} \\qquad \\textbf{(E) } \\textrm{the\\,entire\\,plane}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\texttt{a}$ and $\\texttt{b}$ are digits for which\n$\\begin{array}{ccc}& 2 & a\\\\ \\times & b & 3\\\\ \\hline & 6 & 9\\\\ 9 & 2\\\\ \\hline 9 & 8 & 9\\end{array}$\nthen $\\texttt{a+b =}$\n$\\mathrm{(A)\\ } 3 \\qquad \\mathrm{(B) \\ }4 \\qquad \\mathrm{(C) \\ } 7 \\qquad \\mathrm{(D) \\ } 9 \\qquad \\mathrm{(E) \\ }12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_38", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\texttt{a}$ and $\\texttt{b}$ are digits for which\n$\\begin{array}{ccc}& 2 & a\\\\ \\times & b & 3\\\\ \\hline & 6 & 9\\\\ 9 & 2\\\\ \\hline 9 & 8 & 9\\end{array}$\nthen $\\texttt{a+b =}$\n$\\mathrm{(A)\\ } 3 \\qquad \\mathrm{(B) \\ }4 \\qquad \\mathrm{(C) \\ } 7 \\qquad \\mathrm{(D) \\ } 9 \\qquad \\mathrm{(E) \\ }12$" + } + }, + { + "question": "Return your final response within \\boxed{}. At $2: 15$ o'clock, the hour and minute hands of a clock form an angle of:\n$\\textbf{(A)}\\ 30^{\\circ} \\qquad\\textbf{(B)}\\ 5^{\\circ} \\qquad\\textbf{(C)}\\ 22\\frac {1}{2}^{\\circ} \\qquad\\textbf{(D)}\\ 7\\frac {1}{2} ^{\\circ} \\qquad\\textbf{(E)}\\ 28^{\\circ}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_39", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. At $2: 15$ o'clock, the hour and minute hands of a clock form an angle of:\n$\\textbf{(A)}\\ 30^{\\circ} \\qquad\\textbf{(B)}\\ 5^{\\circ} \\qquad\\textbf{(C)}\\ 22\\frac {1}{2}^{\\circ} \\qquad\\textbf{(D)}\\ 7\\frac {1}{2} ^{\\circ} \\qquad\\textbf{(E)}\\ 28^{\\circ}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://artofproblemsolving.com/wiki/index.php/Point) $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two [circles](https://artofproblemsolving.com/wiki/index.php/Circle) with equal [radii](https://artofproblemsolving.com/wiki/index.php/Radius). The circle with center $O_1$ is tangent to the [hypotenuse](https://artofproblemsolving.com/wiki/index.php/Hypotenuse) and to the extension of leg $CA$, the circle with center $O_2$ is [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to the hypotenuse and to the extension of [leg](https://artofproblemsolving.com/wiki/index.php/Leg) $CB$, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as $p/q$, where $p$ and $q$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [integers](https://artofproblemsolving.com/wiki/index.php/Integer). Find $p+q$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "737", + "index": "Sky-T1_10k_40", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://artofproblemsolving.com/wiki/index.php/Point) $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two [circles](https://artofproblemsolving.com/wiki/index.php/Circle) with equal [radii](https://artofproblemsolving.com/wiki/index.php/Radius). The circle with center $O_1$ is tangent to the [hypotenuse](https://artofproblemsolving.com/wiki/index.php/Hypotenuse) and to the extension of leg $CA$, the circle with center $O_2$ is [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to the hypotenuse and to the extension of [leg](https://artofproblemsolving.com/wiki/index.php/Leg) $CB$, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as $p/q$, where $p$ and $q$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [integers](https://artofproblemsolving.com/wiki/index.php/Integer). Find $p+q$." + } + }, + { + "question": "Return your final response within \\boxed{}. A finite [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest [prime](https://artofproblemsolving.com/wiki/index.php/Prime) [factor](https://artofproblemsolving.com/wiki/index.php/Factor) that always divides $S$? \n$\\mathrm{(A)}\\ 3\\qquad \\mathrm{(B)}\\ 7\\qquad \\mathrm{(C)}\\ 13\\qquad \\mathrm{(D)}\\ 37\\qquad \\mathrm{(E)}\\ 43$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_41", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A finite [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest [prime](https://artofproblemsolving.com/wiki/index.php/Prime) [factor](https://artofproblemsolving.com/wiki/index.php/Factor) that always divides $S$? \n$\\mathrm{(A)}\\ 3\\qquad \\mathrm{(B)}\\ 7\\qquad \\mathrm{(C)}\\ 13\\qquad \\mathrm{(D)}\\ 37\\qquad \\mathrm{(E)}\\ 43$" + } + }, + { + "question": "Return your final response within \\boxed{}. What time was it $2011$ minutes after midnight on January 1, 2011? \n$\\textbf{(A)}\\ \\text{January 1 at 9:31PM}$\n$\\textbf{(B)}\\ \\text{January 1 at 11:51PM}$\n$\\textbf{(C)}\\ \\text{January 2 at 3:11AM}$\n$\\textbf{(D)}\\ \\text{January 2 at 9:31AM}$\n$\\textbf{(E)}\\ \\text{January 2 at 6:01PM}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_42", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What time was it $2011$ minutes after midnight on January 1, 2011? \n$\\textbf{(A)}\\ \\text{January 1 at 9:31PM}$\n$\\textbf{(B)}\\ \\text{January 1 at 11:51PM}$\n$\\textbf{(C)}\\ \\text{January 2 at 3:11AM}$\n$\\textbf{(D)}\\ \\text{January 2 at 9:31AM}$\n$\\textbf{(E)}\\ \\text{January 2 at 6:01PM}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_43", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?\n$\\text{(A)}\\,\\frac{49}{64} \\qquad\\text{(B)}\\,\\frac{25}{32} \\qquad\\text{(C)}\\,\\frac78 \\qquad\\text{(D)}\\,\\frac{5\\sqrt{2}}{8} \\qquad\\text{(E)}\\,\\frac{\\sqrt{14}}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_44", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?\n$\\text{(A)}\\,\\frac{49}{64} \\qquad\\text{(B)}\\,\\frac{25}{32} \\qquad\\text{(C)}\\,\\frac78 \\qquad\\text{(D)}\\,\\frac{5\\sqrt{2}}{8} \\qquad\\text{(E)}\\,\\frac{\\sqrt{14}}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A house worth $ $9000$ is sold by Mr. A to Mr. B at a $10$ % loss. Mr. B sells the house back to Mr. A at a $10$ % gain. \nThe result of the two transactions is: \n$\\textbf{(A)}\\ \\text{Mr. A breaks even} \\qquad \\textbf{(B)}\\ \\text{Mr. B gains }$900 \\qquad \\textbf{(C)}\\ \\text{Mr. A loses }$900\\\\ \\textbf{(D)}\\ \\text{Mr. A loses }$810\\qquad \\textbf{(E)}\\ \\text{Mr. B gains }$1710$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_45", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A house worth $ $9000$ is sold by Mr. A to Mr. B at a $10$ % loss. Mr. B sells the house back to Mr. A at a $10$ % gain. \nThe result of the two transactions is: \n$\\textbf{(A)}\\ \\text{Mr. A breaks even} \\qquad \\textbf{(B)}\\ \\text{Mr. B gains }$900 \\qquad \\textbf{(C)}\\ \\text{Mr. A loses }$900\\\\ \\textbf{(D)}\\ \\text{Mr. A loses }$810\\qquad \\textbf{(E)}\\ \\text{Mr. B gains }$1710$" + } + }, + { + "question": "Return your final response within \\boxed{}. Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents\nsend him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns\n$16$ dollars per week for his paper route. He will use all of his birthday money and all\nof the money he earns from his paper route. In how many weeks will he be able\nto buy the mountain bike?\n$\\mathrm{(A)}\\ 24 \\qquad\\mathrm{(B)}\\ 25 \\qquad\\mathrm{(C)}\\ 26 \\qquad\\mathrm{(D)}\\ 27 \\qquad\\mathrm{(E)}\\ 28$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_46", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents\nsend him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns\n$16$ dollars per week for his paper route. He will use all of his birthday money and all\nof the money he earns from his paper route. In how many weeks will he be able\nto buy the mountain bike?\n$\\mathrm{(A)}\\ 24 \\qquad\\mathrm{(B)}\\ 25 \\qquad\\mathrm{(C)}\\ 26 \\qquad\\mathrm{(D)}\\ 27 \\qquad\\mathrm{(E)}\\ 28$" + } + }, + { + "question": "Return your final response within \\boxed{}. The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?\n$\\textbf{(A) }\\frac{3}{7}\\qquad\\textbf{(B) }\\frac{7}{12}\\qquad\\textbf{(C) }\\frac{12}{7}\\qquad\\textbf{(D) }\\frac{7}{4}\\qquad \\textbf{(E) }\\frac{7}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_47", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?\n$\\textbf{(A) }\\frac{3}{7}\\qquad\\textbf{(B) }\\frac{7}{12}\\qquad\\textbf{(C) }\\frac{12}{7}\\qquad\\textbf{(D) }\\frac{7}{4}\\qquad \\textbf{(E) }\\frac{7}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?\n\n$\\text{(A)}\\ 2250 \\qquad \\text{(B)}\\ 3000 \\qquad \\text{(C)}\\ 4200 \\qquad \\text{(D)}\\ 4350 \\qquad \\text{(E)}\\ 5650$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_48", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?\n\n$\\text{(A)}\\ 2250 \\qquad \\text{(B)}\\ 3000 \\qquad \\text{(C)}\\ 4200 \\qquad \\text{(D)}\\ 4350 \\qquad \\text{(E)}\\ 5650$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the following instructions are carried out by a computer, which value of $X$ will be printed because of instruction $5$?\n\n1. START $X$ AT $3$ AND $S$ AT $0$. \n2. INCREASE THE VALUE OF $X$ BY $2$. \n3. INCREASE THE VALUE OF $S$ BY THE VALUE OF $X$. \n4. IF $S$ IS AT LEAST $10000$, \n THEN GO TO INSTRUCTION $5$; \n OTHERWISE, GO TO INSTRUCTION $2$. \n AND PROCEED FROM THERE. \n5. PRINT THE VALUE OF $X$. \n6. STOP. \n\n$\\text{(A) } 19\\quad \\text{(B) } 21\\quad \\text{(C) } 23\\quad \\text{(D) } 199\\quad \\text{(E) } 201$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_49", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the following instructions are carried out by a computer, which value of $X$ will be printed because of instruction $5$?\n\n1. START $X$ AT $3$ AND $S$ AT $0$. \n2. INCREASE THE VALUE OF $X$ BY $2$. \n3. INCREASE THE VALUE OF $S$ BY THE VALUE OF $X$. \n4. IF $S$ IS AT LEAST $10000$, \n THEN GO TO INSTRUCTION $5$; \n OTHERWISE, GO TO INSTRUCTION $2$. \n AND PROCEED FROM THERE. \n5. PRINT THE VALUE OF $X$. \n6. STOP. \n\n$\\text{(A) } 19\\quad \\text{(B) } 21\\quad \\text{(C) } 23\\quad \\text{(D) } 199\\quad \\text{(E) } 201$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is equivalent to \"If P is true, then Q is false.\"?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "If Q is true, then P is false.", + "index": "Sky-T1_10k_50", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is equivalent to \"If P is true, then Q is false.\"?" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "476", + "index": "Sky-T1_10k_51", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$." + } + }, + { + "question": "Return your final response within \\boxed{}. A shopper plans to purchase an item that has a listed price greater than $\\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\\%$ off the listed price, Coupon B gives $\\textdollar 30$ off the listed price, and Coupon C gives $25\\%$ off the amount by which the listed price exceeds\n$\\textdollar 100$. \nLet $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x$?\n$\\textbf{(A)}\\ 50 \\qquad \\textbf{(B)}\\ 60 \\qquad \\textbf{(C)}\\ 75 \\qquad \\textbf{(D)}\\ 80 \\qquad \\textbf{(E)}\\ 100$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_52", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A shopper plans to purchase an item that has a listed price greater than $\\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\\%$ off the listed price, Coupon B gives $\\textdollar 30$ off the listed price, and Coupon C gives $25\\%$ off the amount by which the listed price exceeds\n$\\textdollar 100$. \nLet $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x$?\n$\\textbf{(A)}\\ 50 \\qquad \\textbf{(B)}\\ 60 \\qquad \\textbf{(C)}\\ 75 \\qquad \\textbf{(D)}\\ 80 \\qquad \\textbf{(E)}\\ 100$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$, $\\angle ABC=45^\\circ$. Point $D$ is on $\\overline{BC}$ so that $2\\cdot BD=CD$ and $\\angle DAB=15^\\circ$. Find $\\angle ACB.$\n$\\text{(A) }54^\\circ \\qquad \\text{(B) }60^\\circ \\qquad \\text{(C) }72^\\circ \\qquad \\text{(D) }75^\\circ \\qquad \\text{(E) }90^\\circ$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_53", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$, $\\angle ABC=45^\\circ$. Point $D$ is on $\\overline{BC}$ so that $2\\cdot BD=CD$ and $\\angle DAB=15^\\circ$. Find $\\angle ACB.$\n$\\text{(A) }54^\\circ \\qquad \\text{(B) }60^\\circ \\qquad \\text{(C) }72^\\circ \\qquad \\text{(D) }75^\\circ \\qquad \\text{(E) }90^\\circ$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters?\n$\\mathrm{(A)}\\ 1 \\qquad \\mathrm{(B)}\\ 2 \\qquad \\mathrm{(C)}\\ 3 \\qquad \\mathrm{(D)}\\ 4 \\qquad \\mathrm{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_54", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters?\n$\\mathrm{(A)}\\ 1 \\qquad \\mathrm{(B)}\\ 2 \\qquad \\mathrm{(C)}\\ 3 \\qquad \\mathrm{(D)}\\ 4 \\qquad \\mathrm{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The equation $2^{2x}-8\\cdot 2^x+12=0$ is satisfied by:\n$\\text{(A) } log(3)\\quad \\text{(B) } \\tfrac{1}{2}log(6)\\quad \\text{(C) } 1+log(\\tfrac{3}{2})\\quad \\text{(D) } 1+\\frac{log(3)}{log(2)}\\quad \\text{(E) none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_55", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The equation $2^{2x}-8\\cdot 2^x+12=0$ is satisfied by:\n$\\text{(A) } log(3)\\quad \\text{(B) } \\tfrac{1}{2}log(6)\\quad \\text{(C) } 1+log(\\tfrac{3}{2})\\quad \\text{(D) } 1+\\frac{log(3)}{log(2)}\\quad \\text{(E) none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is\n$\\textbf{(A)}\\ \\Big\\{\\frac{5}{2}\\Big\\}\\qquad \\textbf{(B)}\\ \\big\\{x\\ |\\ 2 \\le x \\le 3\\big\\}\\qquad \\textbf{(C)}\\ \\big\\{x\\ |\\ 2\\le x < 3\\big\\}\\qquad\\\\ \\textbf{(D)}\\ \\Big\\{x\\ |\\ 2 < x\\le 3\\Big\\}\\qquad \\textbf{(E)}\\ \\Big\\{x\\ |\\ 2 < x < 3\\Big\\}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_56", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is\n$\\textbf{(A)}\\ \\Big\\{\\frac{5}{2}\\Big\\}\\qquad \\textbf{(B)}\\ \\big\\{x\\ |\\ 2 \\le x \\le 3\\big\\}\\qquad \\textbf{(C)}\\ \\big\\{x\\ |\\ 2\\le x < 3\\big\\}\\qquad\\\\ \\textbf{(D)}\\ \\Big\\{x\\ |\\ 2 < x\\le 3\\Big\\}\\qquad \\textbf{(E)}\\ \\Big\\{x\\ |\\ 2 < x < 3\\Big\\}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately:\n$\\textbf{(A)}\\ \\text{200 yd.}\\qquad\\textbf{(B)}\\ \\text{352 yd.}\\qquad\\textbf{(C)}\\ \\text{300 yd.}\\qquad\\textbf{(D)}\\ \\text{245 yd.}\\qquad\\textbf{(E)}\\ \\text{512 yd.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_57", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately:\n$\\textbf{(A)}\\ \\text{200 yd.}\\qquad\\textbf{(B)}\\ \\text{352 yd.}\\qquad\\textbf{(C)}\\ \\text{300 yd.}\\qquad\\textbf{(D)}\\ \\text{245 yd.}\\qquad\\textbf{(E)}\\ \\text{512 yd.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?\n\n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 15\\qquad\\textbf{(D)}\\ 18\\qquad\\textbf{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_58", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?\n\n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 15\\qquad\\textbf{(D)}\\ 18\\qquad\\textbf{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)} \\ 4 \\qquad\\textbf{(C)} \\ 5 \\qquad\\textbf{(D)} \\ 6 \\qquad\\textbf{(E)} \\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_59", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)} \\ 4 \\qquad\\textbf{(C)} \\ 5 \\qquad\\textbf{(D)} \\ 6 \\qquad\\textbf{(E)} \\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\theta$ is a constant such that $0 < \\theta < \\pi$ and $x + \\dfrac{1}{x} = 2\\cos{\\theta}$, then for each positive integer $n$, $x^n + \\dfrac{1}{x^n}$ equals\n$\\textbf{(A)}\\ 2\\cos\\theta\\qquad \\textbf{(B)}\\ 2^n\\cos\\theta\\qquad \\textbf{(C)}\\ 2\\cos^n\\theta\\qquad \\textbf{(D)}\\ 2\\cos n\\theta\\qquad \\textbf{(E)}\\ 2^n\\cos^n\\theta$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_60", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\theta$ is a constant such that $0 < \\theta < \\pi$ and $x + \\dfrac{1}{x} = 2\\cos{\\theta}$, then for each positive integer $n$, $x^n + \\dfrac{1}{x^n}$ equals\n$\\textbf{(A)}\\ 2\\cos\\theta\\qquad \\textbf{(B)}\\ 2^n\\cos\\theta\\qquad \\textbf{(C)}\\ 2\\cos^n\\theta\\qquad \\textbf{(D)}\\ 2\\cos n\\theta\\qquad \\textbf{(E)}\\ 2^n\\cos^n\\theta$" + } + }, + { + "question": "Return your final response within \\boxed{}. Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?\n$\\textbf{(A)}\\ {-20}\\qquad\\textbf{(B)}\\ {-18}\\qquad\\textbf{(C)}\\ {-15}\\qquad\\textbf{(D)}\\ {-12}\\qquad\\textbf{(E)}\\ {-8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_61", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?\n$\\textbf{(A)}\\ {-20}\\qquad\\textbf{(B)}\\ {-18}\\qquad\\textbf{(C)}\\ {-15}\\qquad\\textbf{(D)}\\ {-12}\\qquad\\textbf{(E)}\\ {-8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Titu Andreescu)\nProve that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "For every positive integer n , such an n-digit number exists.", + "index": "Sky-T1_10k_62", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Titu Andreescu)\nProve that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd." + } + }, + { + "question": "Return your final response within \\boxed{}. Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\tfrac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\\tfrac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\\overline{PQ}$. What is $\\overline{AR}+\\overline{BR}+\\overline{CR}+\\overline{DR}$ ?\n$\\textbf{(A)}\\; 180 \\qquad\\textbf{(B)}\\; 184 \\qquad\\textbf{(C)}\\; 188 \\qquad\\textbf{(D)}\\; 192\\qquad\\textbf{(E)}\\; 196$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_63", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\tfrac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\\tfrac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\\overline{PQ}$. What is $\\overline{AR}+\\overline{BR}+\\overline{CR}+\\overline{DR}$ ?\n$\\textbf{(A)}\\; 180 \\qquad\\textbf{(B)}\\; 184 \\qquad\\textbf{(C)}\\; 188 \\qquad\\textbf{(D)}\\; 192\\qquad\\textbf{(E)}\\; 196$" + } + }, + { + "question": "Return your final response within \\boxed{}. A figure is an equiangular parallelogram if and only if it is a\n$\\textbf{(A)}\\ \\text{rectangle}\\qquad \\textbf{(B)}\\ \\text{regular polygon}\\qquad \\textbf{(C)}\\ \\text{rhombus}\\qquad \\textbf{(D)}\\ \\text{square}\\qquad \\textbf{(E)}\\ \\text{trapezoid}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_64", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A figure is an equiangular parallelogram if and only if it is a\n$\\textbf{(A)}\\ \\text{rectangle}\\qquad \\textbf{(B)}\\ \\text{regular polygon}\\qquad \\textbf{(C)}\\ \\text{rhombus}\\qquad \\textbf{(D)}\\ \\text{square}\\qquad \\textbf{(E)}\\ \\text{trapezoid}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the minimum value of $\\sqrt{x^2+y^2}$ if $5x+12y=60$. \n$\\textbf{(A)}\\ \\frac{60}{13}\\qquad \\textbf{(B)}\\ \\frac{13}{5}\\qquad \\textbf{(C)}\\ \\frac{13}{12}\\qquad \\textbf{(D)}\\ 1\\qquad \\textbf{(E)}\\ 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_65", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the minimum value of $\\sqrt{x^2+y^2}$ if $5x+12y=60$. \n$\\textbf{(A)}\\ \\frac{60}{13}\\qquad \\textbf{(B)}\\ \\frac{13}{5}\\qquad \\textbf{(C)}\\ \\frac{13}{12}\\qquad \\textbf{(D)}\\ 1\\qquad \\textbf{(E)}\\ 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? \n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 32\\qquad\\textbf{(C)}\\ 35\\qquad\\textbf{(D)}\\ 49\\qquad\\textbf{(E)}\\ 112$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_66", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? \n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 32\\qquad\\textbf{(C)}\\ 35\\qquad\\textbf{(D)}\\ 49\\qquad\\textbf{(E)}\\ 112$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\\frac{1}{12}$. What is the least possible number of faces on the two dice combined?\n$\\textbf{(A) }16 \\qquad \\textbf{(B) }17 \\qquad \\textbf{(C) }18\\qquad \\textbf{(D) }19 \\qquad \\textbf{(E) }20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_67", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\\frac{1}{12}$. What is the least possible number of faces on the two dice combined?\n$\\textbf{(A) }16 \\qquad \\textbf{(B) }17 \\qquad \\textbf{(C) }18\\qquad \\textbf{(D) }19 \\qquad \\textbf{(E) }20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "544", + "index": "Sky-T1_10k_68", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way)." + } + }, + { + "question": "Return your final response within \\boxed{}. The number $2.5252525\\ldots$ can be written as a fraction. \nWhen reduced to lowest terms the sum of the numerator and denominator of this fraction is:\n$\\textbf{(A) }7\\qquad \\textbf{(B) }29\\qquad \\textbf{(C) }141\\qquad \\textbf{(D) }349\\qquad \\textbf{(E) }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_69", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number $2.5252525\\ldots$ can be written as a fraction. \nWhen reduced to lowest terms the sum of the numerator and denominator of this fraction is:\n$\\textbf{(A) }7\\qquad \\textbf{(B) }29\\qquad \\textbf{(C) }141\\qquad \\textbf{(D) }349\\qquad \\textbf{(E) }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\\left(x-\\frac{1}{x}\\right)\\left(y+\\frac{1}{y}\\right)$ equals\n$\\textbf{(A) }2x^2\\qquad \\textbf{(B) }2y^2\\qquad \\textbf{(C) }x^2+y^2\\qquad \\textbf{(D) }x^2-y^2\\qquad \\textbf{(E) }y^2-x^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_70", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\\left(x-\\frac{1}{x}\\right)\\left(y+\\frac{1}{y}\\right)$ equals\n$\\textbf{(A) }2x^2\\qquad \\textbf{(B) }2y^2\\qquad \\textbf{(C) }x^2+y^2\\qquad \\textbf{(D) }x^2-y^2\\qquad \\textbf{(E) }y^2-x^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: \n$\\textbf{(A)}\\ 9\\text{ and }-7\\qquad\\textbf{(B)}\\ \\text{only }-7\\qquad\\textbf{(C)}\\ \\text{9 and 7}\\\\ \\textbf{(D)}\\ -9\\text{ and }-7\\qquad\\textbf{(E)}\\ \\text{only 9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_71", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: \n$\\textbf{(A)}\\ 9\\text{ and }-7\\qquad\\textbf{(B)}\\ \\text{only }-7\\qquad\\textbf{(C)}\\ \\text{9 and 7}\\\\ \\textbf{(D)}\\ -9\\text{ and }-7\\qquad\\textbf{(E)}\\ \\text{only 9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }9\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }57\\qquad \\textbf{(E) }58$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_72", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }9\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }57\\qquad \\textbf{(E) }58$" + } + }, + { + "question": "Return your final response within \\boxed{}. A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \\frac{\\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?\n$\\textbf{(A)}\\ 2\\sqrt{6} \\qquad\\textbf{(B)} \\ 2+2\\sqrt{3} \\qquad\\textbf{(C)} \\ 6 \\qquad\\textbf{(D)} \\ 3 + 2\\sqrt{3} \\qquad\\textbf{(E)} \\ 6 + \\frac{\\sqrt{3}}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_73", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \\frac{\\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?\n$\\textbf{(A)}\\ 2\\sqrt{6} \\qquad\\textbf{(B)} \\ 2+2\\sqrt{3} \\qquad\\textbf{(C)} \\ 6 \\qquad\\textbf{(D)} \\ 3 + 2\\sqrt{3} \\qquad\\textbf{(E)} \\ 6 + \\frac{\\sqrt{3}}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alicia earns 20 dollars per hour, of which $1.45\\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?\n$\\mathrm{(A) \\ } 0.0029 \\qquad \\mathrm{(B) \\ } 0.029 \\qquad \\mathrm{(C) \\ } 0.29 \\qquad \\mathrm{(D) \\ } 2.9 \\qquad \\mathrm{(E) \\ } 29$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_74", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alicia earns 20 dollars per hour, of which $1.45\\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?\n$\\mathrm{(A) \\ } 0.0029 \\qquad \\mathrm{(B) \\ } 0.029 \\qquad \\mathrm{(C) \\ } 0.29 \\qquad \\mathrm{(D) \\ } 2.9 \\qquad \\mathrm{(E) \\ } 29$" + } + }, + { + "question": "Return your final response within \\boxed{}. The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:\n\n 1. The largest isn't first, but it is in one of the first three places. \n 2. The smallest isn't last, but it is in one of the last three places. \n 3. The median isn't first or last.\n\nWhat is the average of the first and last numbers?\n$\\textbf{(A)}\\ 3.5 \\qquad \\textbf{(B)}\\ 5 \\qquad \\textbf{(C)}\\ 6.5 \\qquad \\textbf{(D)}\\ 7.5 \\qquad \\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_75", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:\n\n 1. The largest isn't first, but it is in one of the first three places. \n 2. The smallest isn't last, but it is in one of the last three places. \n 3. The median isn't first or last.\n\nWhat is the average of the first and last numbers?\n$\\textbf{(A)}\\ 3.5 \\qquad \\textbf{(B)}\\ 5 \\qquad \\textbf{(C)}\\ 6.5 \\qquad \\textbf{(D)}\\ 7.5 \\qquad \\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? \n$\\textbf{(A)}\\ 11\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 13\\qquad\\textbf{(D)}\\ 14\\qquad\\textbf{(E)}\\ 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_76", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? \n$\\textbf{(A)}\\ 11\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 13\\qquad\\textbf{(D)}\\ 14\\qquad\\textbf{(E)}\\ 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. Trapezoid $ABCD$ has $\\overline{AB}\\parallel\\overline{CD},BC=CD=43$, and $\\overline{AD}\\perp\\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?\n$\\textbf{(A) }65 \\qquad \\textbf{(B) }132 \\qquad \\textbf{(C) }157 \\qquad \\textbf{(D) }194\\qquad \\textbf{(E) }215$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_77", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Trapezoid $ABCD$ has $\\overline{AB}\\parallel\\overline{CD},BC=CD=43$, and $\\overline{AD}\\perp\\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?\n$\\textbf{(A) }65 \\qquad \\textbf{(B) }132 \\qquad \\textbf{(C) }157 \\qquad \\textbf{(D) }194\\qquad \\textbf{(E) }215$" + } + }, + { + "question": "Return your final response within \\boxed{}. The expression $\\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to: \n$\\textbf{(A)}\\ \\frac{4y-1}{8}\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ \\frac{15}{2}\\qquad\\textbf{(D)}\\ \\frac{15}{8}\\qquad\\textbf{(E)}\\ \\frac{1}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_78", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The expression $\\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to: \n$\\textbf{(A)}\\ \\frac{4y-1}{8}\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ \\frac{15}{2}\\qquad\\textbf{(D)}\\ \\frac{15}{8}\\qquad\\textbf{(E)}\\ \\frac{1}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest?\n$\\textbf{(A) }560 \\qquad \\textbf{(B) }960 \\qquad \\textbf{(C) }1120 \\qquad \\textbf{(D) }1920 \\qquad \\textbf{(E) }3840$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_79", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest?\n$\\textbf{(A) }560 \\qquad \\textbf{(B) }960 \\qquad \\textbf{(C) }1120 \\qquad \\textbf{(D) }1920 \\qquad \\textbf{(E) }3840$" + } + }, + { + "question": "Return your final response within \\boxed{}. When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then:\n$\\text{(A) } r>r' \\text{ always}\\quad \\text{(B) } rr' \\text{ sometimes and } rr' \\text{ sometimes and } r=r' \\text{ sometimes}\\quad\\\\ \\text{(E) } r=r' \\text{ always}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_80", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then:\n$\\text{(A) } r>r' \\text{ always}\\quad \\text{(B) } rr' \\text{ sometimes and } rr' \\text{ sometimes and } r=r' \\text{ sometimes}\\quad\\\\ \\text{(E) } r=r' \\text{ always}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?\n\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_81", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?\n\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing $4.5$ ounces was\n$\\text{(A)}\\ \\text{96 cents} \\qquad \\text{(B)}\\ \\text{1.07 dollars} \\qquad \\text{(C)}\\ \\text{1.18 dollars} \\qquad \\text{(D)}\\ \\text{1.20 dollars} \\qquad \\text{(E)}\\ \\text{1.40 dollars}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_82", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing $4.5$ ounces was\n$\\text{(A)}\\ \\text{96 cents} \\qquad \\text{(B)}\\ \\text{1.07 dollars} \\qquad \\text{(C)}\\ \\text{1.18 dollars} \\qquad \\text{(D)}\\ \\text{1.20 dollars} \\qquad \\text{(E)}\\ \\text{1.40 dollars}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that\n\\[ (2 + i)^n = a_n + b_ni \\]for all integers $n\\geq 0$, where $i = \\sqrt{-1}$. What is\\[\\sum_{n=0}^\\infty\\frac{a_nb_n}{7^n}\\,?\\]\n$\\textbf{(A) }\\frac 38\\qquad\\textbf{(B) }\\frac7{16}\\qquad\\textbf{(C) }\\frac12\\qquad\\textbf{(D) }\\frac9{16}\\qquad\\textbf{(E) }\\frac47$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_83", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that\n\\[ (2 + i)^n = a_n + b_ni \\]for all integers $n\\geq 0$, where $i = \\sqrt{-1}$. What is\\[\\sum_{n=0}^\\infty\\frac{a_nb_n}{7^n}\\,?\\]\n$\\textbf{(A) }\\frac 38\\qquad\\textbf{(B) }\\frac7{16}\\qquad\\textbf{(C) }\\frac12\\qquad\\textbf{(D) }\\frac9{16}\\qquad\\textbf{(E) }\\frac47$" + } + }, + { + "question": "Return your final response within \\boxed{}. An $11\\times 11\\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?\n$\\text{(A) 328} \\quad \\text{(B) 329} \\quad \\text{(C) 330} \\quad \\text{(D) 331} \\quad \\text{(E) 332}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_84", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An $11\\times 11\\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?\n$\\text{(A) 328} \\quad \\text{(B) 329} \\quad \\text{(C) 330} \\quad \\text{(D) 331} \\quad \\text{(E) 332}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, \nthen the possible number of points of intersection with the hyperbola is: \n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 2\\text{ or }3\\qquad \\textbf{(C)}\\ 2\\text{ or }4\\qquad \\textbf{(D)}\\ 3\\text{ or }4\\qquad \\textbf{(E)}\\ 2,3,\\text{ or }4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_85", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, \nthen the possible number of points of intersection with the hyperbola is: \n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 2\\text{ or }3\\qquad \\textbf{(C)}\\ 2\\text{ or }4\\qquad \\textbf{(D)}\\ 3\\text{ or }4\\qquad \\textbf{(E)}\\ 2,3,\\text{ or }4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "No", + "index": "Sky-T1_10k_86", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\frac {1}{x} - \\frac {1}{y} = \\frac {1}{z}$, then $z$ equals:\n$\\textbf{(A)}\\ y - x\\qquad \\textbf{(B)}\\ x - y\\qquad \\textbf{(C)}\\ \\frac {y - x}{xy}\\qquad \\textbf{(D)}\\ \\frac {xy}{y - x}\\qquad \\textbf{(E)}\\ \\frac {xy}{x - y}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_87", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\frac {1}{x} - \\frac {1}{y} = \\frac {1}{z}$, then $z$ equals:\n$\\textbf{(A)}\\ y - x\\qquad \\textbf{(B)}\\ x - y\\qquad \\textbf{(C)}\\ \\frac {y - x}{xy}\\qquad \\textbf{(D)}\\ \\frac {xy}{y - x}\\qquad \\textbf{(E)}\\ \\frac {xy}{x - y}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty and Carlos had was $7$ dollars, between Carlos and Dick was $5$ dollars, between Dick and Elgin was $4$ dollars, and between Elgin and Ashley was $11$ dollars. How many dollars did Elgin have?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_88", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty and Carlos had was $7$ dollars, between Carlos and Dick was $5$ dollars, between Dick and Elgin was $4$ dollars, and between Elgin and Ashley was $11$ dollars. How many dollars did Elgin have?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?\n$\\textbf{(A)}\\ \\dfrac{1}{10} \\qquad \\textbf{(B)}\\ \\dfrac{1}{9} \\qquad \\textbf{(C)}\\ \\dfrac{1}{7} \\qquad \\textbf{(D)}\\ \\dfrac{1}{6} \\qquad \\textbf{(E)}\\ \\dfrac{1}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_89", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?\n$\\textbf{(A)}\\ \\dfrac{1}{10} \\qquad \\textbf{(B)}\\ \\dfrac{1}{9} \\qquad \\textbf{(C)}\\ \\dfrac{1}{7} \\qquad \\textbf{(D)}\\ \\dfrac{1}{6} \\qquad \\textbf{(E)}\\ \\dfrac{1}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P$ be an interior point of circle $K$ other than the center of $K$. Form all chords of $K$ which pass through $P$, and determine their midpoints. The locus of these midpoints is \n$\\textbf{(A)} \\text{ a circle with one point deleted} \\qquad \\\\ \\textbf{(B)} \\text{ a circle if the distance from } P \\text{ to the center of } K \\text{ is less than one half the radius of } K; \\\\ \\text{otherwise a circular arc of less than } 360^{\\circ} \\qquad \\\\ \\textbf{(C)} \\text{ a semicircle with one point deleted} \\qquad \\\\ \\textbf{(D)} \\text{ a semicircle} \\qquad \\textbf{(E)} \\text{ a circle}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_90", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P$ be an interior point of circle $K$ other than the center of $K$. Form all chords of $K$ which pass through $P$, and determine their midpoints. The locus of these midpoints is \n$\\textbf{(A)} \\text{ a circle with one point deleted} \\qquad \\\\ \\textbf{(B)} \\text{ a circle if the distance from } P \\text{ to the center of } K \\text{ is less than one half the radius of } K; \\\\ \\text{otherwise a circular arc of less than } 360^{\\circ} \\qquad \\\\ \\textbf{(C)} \\text{ a semicircle with one point deleted} \\qquad \\\\ \\textbf{(D)} \\text{ a semicircle} \\qquad \\textbf{(E)} \\text{ a circle}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\\underline{not}$ a divisor of the product of the first $n$ positive integers?\n$\\textbf{(A) } 995 \\qquad\\textbf{(B) } 996 \\qquad\\textbf{(C) } 997 \\qquad\\textbf{(D) } 998 \\qquad\\textbf{(E) } 999$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_91", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\\underline{not}$ a divisor of the product of the first $n$ positive integers?\n$\\textbf{(A) } 995 \\qquad\\textbf{(B) } 996 \\qquad\\textbf{(C) } 997 \\qquad\\textbf{(D) } 998 \\qquad\\textbf{(E) } 999$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\\triangle CHM$ is $K$, then the area of $\\triangle ABC$ is \n $\\textbf{(A)}\\ 6K\\qquad\\textbf{(B)}\\ 4\\sqrt3\\ K\\qquad\\textbf{(C)}\\ 3\\sqrt3\\ K\\qquad\\textbf{(D)}\\ 3K\\qquad\\textbf{(E)}\\ 4K$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_92", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\\triangle CHM$ is $K$, then the area of $\\triangle ABC$ is \n $\\textbf{(A)}\\ 6K\\qquad\\textbf{(B)}\\ 4\\sqrt3\\ K\\qquad\\textbf{(C)}\\ 3\\sqrt3\\ K\\qquad\\textbf{(D)}\\ 3K\\qquad\\textbf{(E)}\\ 4K$" + } + }, + { + "question": "Return your final response within \\boxed{}. Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?\n$\\textbf{(A)}\\ 37\\qquad\\textbf{(B)}\\ 63\\qquad\\textbf{(C)}\\ 117\\qquad\\textbf{(D)}\\ 143\\qquad\\textbf{(E)}\\ 163$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_93", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?\n$\\textbf{(A)}\\ 37\\qquad\\textbf{(B)}\\ 63\\qquad\\textbf{(C)}\\ 117\\qquad\\textbf{(D)}\\ 143\\qquad\\textbf{(E)}\\ 163$" + } + }, + { + "question": "Return your final response within \\boxed{}. Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?\n$\\textbf{(A) }6 \\qquad \\textbf{(B) }8 \\qquad \\textbf{(C) }12 \\qquad \\textbf{(D) }18 \\qquad \\textbf{(E) }24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_94", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?\n$\\textbf{(A) }6 \\qquad \\textbf{(B) }8 \\qquad \\textbf{(C) }12 \\qquad \\textbf{(D) }18 \\qquad \\textbf{(E) }24$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ray's car averages $40$ miles per gallon of gasoline, and Tom's car averages $10$ miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?\n$\\textbf{(A)}\\ 10 \\qquad \\textbf{(B)}\\ 16 \\qquad \\textbf{(C)}\\ 25 \\qquad \\textbf{(D)}\\ 30 \\qquad \\textbf{(E)}\\ 40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_95", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ray's car averages $40$ miles per gallon of gasoline, and Tom's car averages $10$ miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?\n$\\textbf{(A)}\\ 10 \\qquad \\textbf{(B)}\\ 16 \\qquad \\textbf{(C)}\\ 25 \\qquad \\textbf{(D)}\\ 30 \\qquad \\textbf{(E)}\\ 40$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\\triangle ABC$?\n\n$\\mathrm{(A)}\\ 6\\qquad \\mathrm{(B)}\\ 8\\qquad \\mathrm{(C)}\\ 10\\qquad \\mathrm{(D)}\\ 12\\qquad \\mathrm{(E)}\\ 14$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_96", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\\triangle ABC$?\n\n$\\mathrm{(A)}\\ 6\\qquad \\mathrm{(B)}\\ 8\\qquad \\mathrm{(C)}\\ 10\\qquad \\mathrm{(D)}\\ 12\\qquad \\mathrm{(E)}\\ 14$" + } + }, + { + "question": "Return your final response within \\boxed{}. A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \\%$ of its games for the season?\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 23 \\qquad \\text{(C)}\\ 28 \\qquad \\text{(D)}\\ 30 \\qquad \\text{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_97", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \\%$ of its games for the season?\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 23 \\qquad \\text{(C)}\\ 28 \\qquad \\text{(D)}\\ 30 \\qquad \\text{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. For each real number $x$, let $\\textbf{[}x\\textbf{]}$ be the largest integer not exceeding $x$ \n(i.e., the integer $n$ such that $n\\le x0$. For how many $x_0$ is it true that $x_0=x_5$?\n$\\text{(A) 0} \\quad \\text{(B) 1} \\quad \\text{(C) 5} \\quad \\text{(D) 31} \\quad \\text{(E) }\\infty$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_105", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given $0\\le x_0<1$, let \n\\[x_n=\\left\\{ \\begin{array}{ll} 2x_{n-1} &\\text{ if }2x_{n-1}<1 \\\\ 2x_{n-1}-1 &\\text{ if }2x_{n-1}\\ge 1 \\end{array}\\right.\\]\nfor all integers $n>0$. For how many $x_0$ is it true that $x_0=x_5$?\n$\\text{(A) 0} \\quad \\text{(B) 1} \\quad \\text{(C) 5} \\quad \\text{(D) 31} \\quad \\text{(E) }\\infty$" + } + }, + { + "question": "Return your final response within \\boxed{}. Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?\n$\\textbf{(A)}\\ 0.04 \\qquad \\textbf{(B)}\\ \\frac{0.4}{\\pi} \\qquad \\textbf{(C)}\\ 0.4 \\qquad \\textbf{(D)}\\ \\frac{4}{\\pi} \\qquad \\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_106", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?\n$\\textbf{(A)}\\ 0.04 \\qquad \\textbf{(B)}\\ \\frac{0.4}{\\pi} \\qquad \\textbf{(C)}\\ 0.4 \\qquad \\textbf{(D)}\\ \\frac{4}{\\pi} \\qquad \\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is: \n$\\textbf{(A)}\\ 20\\% \\qquad \\textbf{(B)}\\ 25\\% \\qquad \\textbf{(C)}\\ 30\\% \\qquad \\textbf{(D)}\\ 33\\frac {1}{3}\\% \\qquad \\textbf{(E)}\\ 35\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_107", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is: \n$\\textbf{(A)}\\ 20\\% \\qquad \\textbf{(B)}\\ 25\\% \\qquad \\textbf{(C)}\\ 30\\% \\qquad \\textbf{(D)}\\ 33\\frac {1}{3}\\% \\qquad \\textbf{(E)}\\ 35\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the tens digit of $7^{2011}$?\n$\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }4\\qquad\\textbf{(E) }7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_108", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the tens digit of $7^{2011}$?\n$\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }4\\qquad\\textbf{(E) }7$" + } + }, + { + "question": "Return your final response within \\boxed{}. For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?\n$\\textbf{(A) } 14 \\qquad \\textbf{(B) } 16 \\qquad \\textbf{(C) } 18 \\qquad \\textbf{(D) } 19 \\qquad \\textbf{(E) } 21$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_109", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?\n$\\textbf{(A) } 14 \\qquad \\textbf{(B) } 16 \\qquad \\textbf{(C) } 18 \\qquad \\textbf{(D) } 19 \\qquad \\textbf{(E) } 21$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f(t)=\\frac{t}{1-t}$, $t \\not= 1$. If $y=f(x)$, then $x$ can be expressed as \n$\\textbf{(A)}\\ f\\left(\\frac{1}{y}\\right)\\qquad \\textbf{(B)}\\ -f(y)\\qquad \\textbf{(C)}\\ -f(-y)\\qquad \\textbf{(D)}\\ f(-y)\\qquad \\textbf{(E)}\\ f(y)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_110", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f(t)=\\frac{t}{1-t}$, $t \\not= 1$. If $y=f(x)$, then $x$ can be expressed as \n$\\textbf{(A)}\\ f\\left(\\frac{1}{y}\\right)\\qquad \\textbf{(B)}\\ -f(y)\\qquad \\textbf{(C)}\\ -f(-y)\\qquad \\textbf{(D)}\\ f(-y)\\qquad \\textbf{(E)}\\ f(y)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parenthese but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?\n$\\textbf{(A)}\\ -5 \\qquad \\textbf{(B)}\\ -3 \\qquad \\textbf{(C)}\\ 0 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_111", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parenthese but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?\n$\\textbf{(A)}\\ -5 \\qquad \\textbf{(B)}\\ -3 \\qquad \\textbf{(C)}\\ 0 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x\\geq 0$, then $\\sqrt{x\\sqrt{x\\sqrt{x}}}=$\n$\\textbf{(A) } x\\sqrt{x}\\qquad \\textbf{(B) } x\\sqrt[4]{x}\\qquad \\textbf{(C) } \\sqrt[8]{x}\\qquad \\textbf{(D) } \\sqrt[8]{x^3}\\qquad \\textbf{(E) } \\sqrt[8]{x^7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_112", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x\\geq 0$, then $\\sqrt{x\\sqrt{x\\sqrt{x}}}=$\n$\\textbf{(A) } x\\sqrt{x}\\qquad \\textbf{(B) } x\\sqrt[4]{x}\\qquad \\textbf{(C) } \\sqrt[8]{x}\\qquad \\textbf{(D) } \\sqrt[8]{x^3}\\qquad \\textbf{(E) } \\sqrt[8]{x^7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A cylindrical oil tank, lying horizontally, has an interior length of $10$ feet and an interior diameter of $6$ feet. \nIf the rectangular surface of the oil has an area of $40$ square feet, the depth of the oil is: \n$\\textbf{(A)}\\ \\sqrt{5}\\qquad\\textbf{(B)}\\ 2\\sqrt{5}\\qquad\\textbf{(C)}\\ 3-\\sqrt{5}\\qquad\\textbf{(D)}\\ 3+\\sqrt{5}\\\\ \\textbf{(E)}\\ \\text{either }3-\\sqrt{5}\\text{ or }3+\\sqrt{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_113", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A cylindrical oil tank, lying horizontally, has an interior length of $10$ feet and an interior diameter of $6$ feet. \nIf the rectangular surface of the oil has an area of $40$ square feet, the depth of the oil is: \n$\\textbf{(A)}\\ \\sqrt{5}\\qquad\\textbf{(B)}\\ 2\\sqrt{5}\\qquad\\textbf{(C)}\\ 3-\\sqrt{5}\\qquad\\textbf{(D)}\\ 3+\\sqrt{5}\\\\ \\textbf{(E)}\\ \\text{either }3-\\sqrt{5}\\text{ or }3+\\sqrt{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Through the use of theorems on logarithms \n\\[\\log{\\frac{a}{b}} + \\log{\\frac{b}{c}} + \\log{\\frac{c}{d}} - \\log{\\frac{ay}{dx}}\\]\ncan be reduced to: \n$\\textbf{(A)}\\ \\log{\\frac{y}{x}}\\qquad \\textbf{(B)}\\ \\log{\\frac{x}{y}}\\qquad \\textbf{(C)}\\ 1\\qquad \\\\ \\textbf{(D)}\\ 140x-24x^2+x^3\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_114", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Through the use of theorems on logarithms \n\\[\\log{\\frac{a}{b}} + \\log{\\frac{b}{c}} + \\log{\\frac{c}{d}} - \\log{\\frac{ay}{dx}}\\]\ncan be reduced to: \n$\\textbf{(A)}\\ \\log{\\frac{y}{x}}\\qquad \\textbf{(B)}\\ \\log{\\frac{x}{y}}\\qquad \\textbf{(C)}\\ 1\\qquad \\\\ \\textbf{(D)}\\ 140x-24x^2+x^3\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?\n$\\text{(A)}\\ 30 \\qquad \\text{(B)}\\ 40 \\qquad \\text{(C)}\\ 41 \\qquad \\text{(D)}\\ 60 \\qquad \\text{(E)}\\ 119$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_115", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?\n$\\text{(A)}\\ 30 \\qquad \\text{(B)}\\ 40 \\qquad \\text{(C)}\\ 41 \\qquad \\text{(D)}\\ 60 \\qquad \\text{(E)}\\ 119$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jose is $4$ years younger than Zack. Zack is $3$ years older than Inez. Inez is $15$ years old. How old is Jose?\n$\\text{(A)}\\ 8 \\qquad \\text{(B)}\\ 11 \\qquad \\text{(C)}\\ 14 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 22$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_116", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jose is $4$ years younger than Zack. Zack is $3$ years older than Inez. Inez is $15$ years old. How old is Jose?\n$\\text{(A)}\\ 8 \\qquad \\text{(B)}\\ 11 \\qquad \\text{(C)}\\ 14 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 22$" + } + }, + { + "question": "Return your final response within \\boxed{}. Of the following statements, the only one that is incorrect is:\n$\\textbf{(A)}\\ \\text{An inequality will remain true after each side is increased,}$ $\\text{ decreased, multiplied or divided zero excluded by the same positive quantity.}$\n$\\textbf{(B)}\\ \\text{The arithmetic mean of two unequal positive quantities is greater than their geometric mean.}$\n$\\textbf{(C)}\\ \\text{If the sum of two positive quantities is given, ther product is largest when they are equal.}$\n$\\textbf{(D)}\\ \\text{If }a\\text{ and }b\\text{ are positive and unequal, }\\frac{1}{2}(a^{2}+b^{2})\\text{ is greater than }[\\frac{1}{2}(a+b)]^{2}.$\n$\\textbf{(E)}\\ \\text{If the product of two positive quantities is given, their sum is greatest when they are equal.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_117", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Of the following statements, the only one that is incorrect is:\n$\\textbf{(A)}\\ \\text{An inequality will remain true after each side is increased,}$ $\\text{ decreased, multiplied or divided zero excluded by the same positive quantity.}$\n$\\textbf{(B)}\\ \\text{The arithmetic mean of two unequal positive quantities is greater than their geometric mean.}$\n$\\textbf{(C)}\\ \\text{If the sum of two positive quantities is given, ther product is largest when they are equal.}$\n$\\textbf{(D)}\\ \\text{If }a\\text{ and }b\\text{ are positive and unequal, }\\frac{1}{2}(a^{2}+b^{2})\\text{ is greater than }[\\frac{1}{2}(a+b)]^{2}.$\n$\\textbf{(E)}\\ \\text{If the product of two positive quantities is given, their sum is greatest when they are equal.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the expression $xy^2$, the values of $x$ and $y$ are each decreased $25$ %; the value of the expression is: \n$\\textbf{(A)}\\ \\text{decreased } 50\\% \\qquad \\textbf{(B)}\\ \\text{decreased }75\\%\\\\ \\textbf{(C)}\\ \\text{decreased }\\frac{37}{64}\\text{ of its value}\\qquad \\textbf{(D)}\\ \\text{decreased }\\frac{27}{64}\\text{ of its value}\\\\ \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_118", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the expression $xy^2$, the values of $x$ and $y$ are each decreased $25$ %; the value of the expression is: \n$\\textbf{(A)}\\ \\text{decreased } 50\\% \\qquad \\textbf{(B)}\\ \\text{decreased }75\\%\\\\ \\textbf{(C)}\\ \\text{decreased }\\frac{37}{64}\\text{ of its value}\\qquad \\textbf{(D)}\\ \\text{decreased }\\frac{27}{64}\\text{ of its value}\\\\ \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of all the roots of $4x^3-8x^2-63x-9=0$ is: \n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ -8 \\qquad \\textbf{(D)}\\ -2 \\qquad \\textbf{(E)}\\ 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_120", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of all the roots of $4x^3-8x^2-63x-9=0$ is: \n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ -8 \\qquad \\textbf{(D)}\\ -2 \\qquad \\textbf{(E)}\\ 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?\n$\\textbf{(I)}\\ x+y < a+b\\qquad$\n$\\textbf{(II)}\\ x-y < a-b\\qquad$\n$\\textbf{(III)}\\ xy < ab\\qquad$\n$\\textbf{(IV)}\\ \\frac{x}{y} < \\frac{a}{b}$\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_121", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?\n$\\textbf{(I)}\\ x+y < a+b\\qquad$\n$\\textbf{(II)}\\ x-y < a-b\\qquad$\n$\\textbf{(III)}\\ xy < ab\\qquad$\n$\\textbf{(IV)}\\ \\frac{x}{y} < \\frac{a}{b}$\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x>0$, then the correct relationship is: \n$\\textbf{(A)}\\ \\log (1+x) = \\frac{x}{1+x} \\qquad \\textbf{(B)}\\ \\log (1+x) < \\frac{x}{1+x} \\\\ \\textbf{(C)}\\ \\log(1+x) > x\\qquad \\textbf{(D)}\\ \\log (1+x) < x\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_122", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x>0$, then the correct relationship is: \n$\\textbf{(A)}\\ \\log (1+x) = \\frac{x}{1+x} \\qquad \\textbf{(B)}\\ \\log (1+x) < \\frac{x}{1+x} \\\\ \\textbf{(C)}\\ \\log(1+x) > x\\qquad \\textbf{(D)}\\ \\log (1+x) < x\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the \naverage of these $36$ numbers. The ratio of the second average to the true average was\n$\\textbf{(A) }1:1\\qquad \\textbf{(B) }35:36\\qquad \\textbf{(C) }36:35\\qquad \\textbf{(D) }2:1\\qquad \\textbf{(E) }\\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_123", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the \naverage of these $36$ numbers. The ratio of the second average to the true average was\n$\\textbf{(A) }1:1\\qquad \\textbf{(B) }35:36\\qquad \\textbf{(C) }36:35\\qquad \\textbf{(D) }2:1\\qquad \\textbf{(E) }\\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?\n$\\textbf{(A)}\\ \\frac{bde}{ac}\\qquad\\textbf{(B)}\\ \\frac{ac}{bde}\\qquad\\textbf{(C)}\\ \\frac{abde}{c}\\qquad\\textbf{(D)}\\ \\frac{bcde}{a}\\qquad\\textbf{(E)}\\ \\frac{abc}{de}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_124", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?\n$\\textbf{(A)}\\ \\frac{bde}{ac}\\qquad\\textbf{(B)}\\ \\frac{ac}{bde}\\qquad\\textbf{(C)}\\ \\frac{abde}{c}\\qquad\\textbf{(D)}\\ \\frac{bcde}{a}\\qquad\\textbf{(E)}\\ \\frac{abc}{de}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?\n$\\textbf{(A) } \\frac{1}{10} \\qquad \\textbf{(B) } \\frac{1}{9} \\qquad \\textbf{(C) } \\frac{1}{3} \\qquad \\textbf{(D) } \\frac{4}{9} \\qquad \\textbf{(E) } \\frac{5}{9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_125", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?\n$\\textbf{(A) } \\frac{1}{10} \\qquad \\textbf{(B) } \\frac{1}{9} \\qquad \\textbf{(C) } \\frac{1}{3} \\qquad \\textbf{(D) } \\frac{4}{9} \\qquad \\textbf{(E) } \\frac{5}{9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of the expression $\\sqrt{16\\sqrt{8\\sqrt{4}}}$?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }4\\sqrt{2}\\qquad\\textbf{(C) }8\\qquad\\textbf{(D) }8\\sqrt{2}\\qquad\\textbf{(E) }16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_126", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of the expression $\\sqrt{16\\sqrt{8\\sqrt{4}}}$?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }4\\sqrt{2}\\qquad\\textbf{(C) }8\\qquad\\textbf{(D) }8\\sqrt{2}\\qquad\\textbf{(E) }16$" + } + }, + { + "question": "Return your final response within \\boxed{}. A jacket and a shirt originally sold for $80$ dollars and $40$ dollars, respectively. During a sale Chris bought the $80$ dollar jacket at a $40\\%$ discount and the $40$ dollar shirt at a $55\\%$ discount. The total amount saved was what percent of the total of the original prices?\n$\\text{(A)}\\ 45\\% \\qquad \\text{(B)}\\ 47\\dfrac{1}{2}\\% \\qquad \\text{(C)}\\ 50\\% \\qquad \\text{(D)}\\ 79\\dfrac{1}{6}\\% \\qquad \\text{(E)}\\ 95\\%$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_127", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A jacket and a shirt originally sold for $80$ dollars and $40$ dollars, respectively. During a sale Chris bought the $80$ dollar jacket at a $40\\%$ discount and the $40$ dollar shirt at a $55\\%$ discount. The total amount saved was what percent of the total of the original prices?\n$\\text{(A)}\\ 45\\% \\qquad \\text{(B)}\\ 47\\dfrac{1}{2}\\% \\qquad \\text{(C)}\\ 50\\% \\qquad \\text{(D)}\\ 79\\dfrac{1}{6}\\% \\qquad \\text{(E)}\\ 95\\%$." + } + }, + { + "question": "Return your final response within \\boxed{}. Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels,dimes, and quarters, whose total value is $8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?\n$\\text {(A) } 0 \\qquad \\text {(B) } 13 \\qquad \\text {(C) } 37 \\qquad \\text {(D) } 64 \\qquad \\text {(E) } 83$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_128", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels,dimes, and quarters, whose total value is $8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?\n$\\text {(A) } 0 \\qquad \\text {(B) } 13 \\qquad \\text {(C) } 37 \\qquad \\text {(D) } 64 \\qquad \\text {(E) } 83$" + } + }, + { + "question": "Return your final response within \\boxed{}. Six distinct positive integers are randomly chosen between $1$ and $2006$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$? \n$\\textbf{(A) } \\frac{1}{2}\\qquad\\textbf{(B) } \\frac{3}{5}\\qquad\\textbf{(C) } \\frac{2}{3}\\qquad\\textbf{(D) } \\frac{4}{5}\\qquad\\textbf{(E) } 1\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_129", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Six distinct positive integers are randomly chosen between $1$ and $2006$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$? \n$\\textbf{(A) } \\frac{1}{2}\\qquad\\textbf{(B) } \\frac{3}{5}\\qquad\\textbf{(C) } \\frac{2}{3}\\qquad\\textbf{(D) } \\frac{4}{5}\\qquad\\textbf{(E) } 1\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?\n$\\textbf{(A)}\\ \\frac{\\pi}{3} \\qquad\\textbf{(B)}\\ \\frac{2\\pi}{3} \\qquad\\textbf{(C)}\\ \\pi \\qquad\\textbf{(D)}\\ \\frac{4\\pi}{3} \\qquad\\textbf{(E)}\\ \\frac{5\\pi}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_130", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?\n$\\textbf{(A)}\\ \\frac{\\pi}{3} \\qquad\\textbf{(B)}\\ \\frac{2\\pi}{3} \\qquad\\textbf{(C)}\\ \\pi \\qquad\\textbf{(D)}\\ \\frac{4\\pi}{3} \\qquad\\textbf{(E)}\\ \\frac{5\\pi}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? \n\n$\\textbf{(A)}\\ \\frac{\\sqrt 2}{4}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ \\frac{2}{\\pi}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{\\sqrt 2}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_131", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? \n\n$\\textbf{(A)}\\ \\frac{\\sqrt 2}{4}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ \\frac{2}{\\pi}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{\\sqrt 2}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes?\n$\\textbf{(A)}\\ \\frac{a}{1080r}\\qquad \\textbf{(B)}\\ \\frac{30r}{a}\\qquad \\textbf{(C)}\\ \\frac{30a}{r}\\qquad \\textbf{(D)}\\ \\frac{10r}{a}\\qquad \\textbf{(E)}\\ \\frac{10a}{r}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_132", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes?\n$\\textbf{(A)}\\ \\frac{a}{1080r}\\qquad \\textbf{(B)}\\ \\frac{30r}{a}\\qquad \\textbf{(C)}\\ \\frac{30a}{r}\\qquad \\textbf{(D)}\\ \\frac{10r}{a}\\qquad \\textbf{(E)}\\ \\frac{10a}{r}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ratio of the radii of two concentric circles is $1:3$. If $\\overline{AC}$ is a diameter of the larger circle, $\\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$, then the radius of the larger circle is\n$\\text{(A) } 13\\quad \\text{(B) } 18\\quad \\text{(C) } 21\\quad \\text{(D) } 24\\quad \\text{(E) } 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_133", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ratio of the radii of two concentric circles is $1:3$. If $\\overline{AC}$ is a diameter of the larger circle, $\\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$, then the radius of the larger circle is\n$\\text{(A) } 13\\quad \\text{(B) } 18\\quad \\text{(C) } 21\\quad \\text{(D) } 24\\quad \\text{(E) } 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. For real numbers $a$ and $b$, define $a \\diamond b = \\sqrt{a^2 + b^2}$. What is the value of\n$(5 \\diamond 12) \\diamond ((-12) \\diamond (-5))$?\n$\\textbf{(A) } 0 \\qquad \\textbf{(B) } \\frac{17}{2} \\qquad \\textbf{(C) } 13 \\qquad \\textbf{(D) } 13\\sqrt{2} \\qquad \\textbf{(E) } 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_134", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For real numbers $a$ and $b$, define $a \\diamond b = \\sqrt{a^2 + b^2}$. What is the value of\n$(5 \\diamond 12) \\diamond ((-12) \\diamond (-5))$?\n$\\textbf{(A) } 0 \\qquad \\textbf{(B) } \\frac{17}{2} \\qquad \\textbf{(C) } 13 \\qquad \\textbf{(D) } 13\\sqrt{2} \\qquad \\textbf{(E) } 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.\n[1988 AIME-12.png](https://artofproblemsolving.com/wiki/index.php/File:1988_AIME-12.png)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "441", + "index": "Sky-T1_10k_135", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.\n[1988 AIME-12.png](https://artofproblemsolving.com/wiki/index.php/File:1988_AIME-12.png)" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\sqrt{8}+\\sqrt{18}=$\n\\[\\text{(A)}\\ \\sqrt{20}\\qquad\\text{(B)}\\ 2(\\sqrt{2}+\\sqrt{3})\\qquad\\text{(C)}\\ 7\\qquad\\text{(D)}\\ 5\\sqrt{2}\\qquad\\text{(E)}\\ 2\\sqrt{13}\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_136", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\sqrt{8}+\\sqrt{18}=$\n\\[\\text{(A)}\\ \\sqrt{20}\\qquad\\text{(B)}\\ 2(\\sqrt{2}+\\sqrt{3})\\qquad\\text{(C)}\\ 7\\qquad\\text{(D)}\\ 5\\sqrt{2}\\qquad\\text{(E)}\\ 2\\sqrt{13}\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. How many positive integer factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1,2,3,4,6,$ and $12.$)\n$\\textbf{(A) }6 \\qquad \\textbf{(B) }7 \\qquad \\textbf{(C) }8 \\qquad \\textbf{(D) }9 \\qquad \\textbf{(E) }10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_137", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many positive integer factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1,2,3,4,6,$ and $12.$)\n$\\textbf{(A) }6 \\qquad \\textbf{(B) }7 \\qquad \\textbf{(C) }8 \\qquad \\textbf{(D) }9 \\qquad \\textbf{(E) }10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?\n$\\textbf{(A) } 9\\%\\qquad \\textbf{(B) } 19\\%\\qquad \\textbf{(C) } 22\\%\\qquad \\textbf{(D) } 23\\%\\qquad \\textbf{(E) } 25\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_138", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?\n$\\textbf{(A) } 9\\%\\qquad \\textbf{(B) } 19\\%\\qquad \\textbf{(C) } 22\\%\\qquad \\textbf{(D) } 23\\%\\qquad \\textbf{(E) } 25\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Before the district play, the Unicorns had won $45$% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 50\\qquad\\textbf{(C)}\\ 52\\qquad\\textbf{(D)}\\ 54\\qquad\\textbf{(E)}\\ 60$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_139", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Before the district play, the Unicorns had won $45$% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 50\\qquad\\textbf{(C)}\\ 52\\qquad\\textbf{(D)}\\ 54\\qquad\\textbf{(E)}\\ 60$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, \"If I get an A, then Beth will get an A.\" Beth said, \"If I get an A, then Carlos will get an A.\" Carlos said, \"If I get an A, then Diana will get an A.\" All of these statements were true, but only two of the students received an A. Which two received A's?\n$\\text{(A)}\\ \\text{Alan, Beth} \\qquad \\text{(B)}\\ \\text{Beth, Carlos} \\qquad \\text{(C)}\\ \\text{Carlos, Diana}$\n$\\text{(D)}\\ \\text{Alan, Diana} \\qquad \\text{(E)}\\ \\text{Beth, Diana}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_140", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, \"If I get an A, then Beth will get an A.\" Beth said, \"If I get an A, then Carlos will get an A.\" Carlos said, \"If I get an A, then Diana will get an A.\" All of these statements were true, but only two of the students received an A. Which two received A's?\n$\\text{(A)}\\ \\text{Alan, Beth} \\qquad \\text{(B)}\\ \\text{Beth, Carlos} \\qquad \\text{(C)}\\ \\text{Carlos, Diana}$\n$\\text{(D)}\\ \\text{Alan, Diana} \\qquad \\text{(E)}\\ \\text{Beth, Diana}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) of cents. What is the total cost, in cents, of one pencil and one eraser?\n$\\mathrm{(A)}\\ 10\\qquad\\mathrm{(B)}\\ 12\\qquad\\mathrm{(C)}\\ 15\\qquad\\mathrm{(D)}\\ 18\\qquad\\mathrm{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_141", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) of cents. What is the total cost, in cents, of one pencil and one eraser?\n$\\mathrm{(A)}\\ 10\\qquad\\mathrm{(B)}\\ 12\\qquad\\mathrm{(C)}\\ 15\\qquad\\mathrm{(D)}\\ 18\\qquad\\mathrm{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $m$ men can do a job in $d$ days, then $m+r$ men can do the job in:\n$\\textbf{(A)}\\ d+r \\text{ days}\\qquad\\textbf{(B)}\\ d-r\\text{ days}\\qquad\\textbf{(C)}\\ \\frac{md}{m+r}\\text{ days}\\qquad\\\\ \\textbf{(D)}\\ \\frac{d}{m+r}\\text{ days}\\qquad\\textbf{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_142", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $m$ men can do a job in $d$ days, then $m+r$ men can do the job in:\n$\\textbf{(A)}\\ d+r \\text{ days}\\qquad\\textbf{(B)}\\ d-r\\text{ days}\\qquad\\textbf{(C)}\\ \\frac{md}{m+r}\\text{ days}\\qquad\\\\ \\textbf{(D)}\\ \\frac{d}{m+r}\\text{ days}\\qquad\\textbf{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $B$ and $C$ lie on $\\overline{AD}$. The length of $\\overline{AB}$ is $4$ times the length of $\\overline{BD}$, and the length of $\\overline{AC}$ is $9$ times the length of $\\overline{CD}$. The length of $\\overline{BC}$ is what fraction of the length of $\\overline{AD}$?\n$\\textbf{(A)}\\ \\frac {1}{36} \\qquad \\textbf{(B)}\\ \\frac {1}{13} \\qquad \\textbf{(C)}\\ \\frac {1}{10} \\qquad \\textbf{(D)}\\ \\frac {5}{36} \\qquad \\textbf{(E)}\\ \\frac {1}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_143", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $B$ and $C$ lie on $\\overline{AD}$. The length of $\\overline{AB}$ is $4$ times the length of $\\overline{BD}$, and the length of $\\overline{AC}$ is $9$ times the length of $\\overline{CD}$. The length of $\\overline{BC}$ is what fraction of the length of $\\overline{AD}$?\n$\\textbf{(A)}\\ \\frac {1}{36} \\qquad \\textbf{(B)}\\ \\frac {1}{13} \\qquad \\textbf{(C)}\\ \\frac {1}{10} \\qquad \\textbf{(D)}\\ \\frac {5}{36} \\qquad \\textbf{(E)}\\ \\frac {1}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the greatest number of consecutive integers whose sum is $45?$\n$\\textbf{(A) } 9 \\qquad\\textbf{(B) } 25 \\qquad\\textbf{(C) } 45 \\qquad\\textbf{(D) } 90 \\qquad\\textbf{(E) } 120$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_144", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the greatest number of consecutive integers whose sum is $45?$\n$\\textbf{(A) } 9 \\qquad\\textbf{(B) } 25 \\qquad\\textbf{(C) } 45 \\qquad\\textbf{(D) } 90 \\qquad\\textbf{(E) } 120$" + } + }, + { + "question": "Return your final response within \\boxed{}. A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?\n$\\textbf{(A)}\\ \\frac{1}{36} \\qquad \\textbf{(B)}\\ \\frac{1}{12} \\qquad \\textbf{(C)}\\ \\frac{1}{6} \\qquad \\textbf{(D)}\\ \\frac{1}{4} \\qquad \\textbf{(E)}\\ \\frac{5}{18}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_145", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?\n$\\textbf{(A)}\\ \\frac{1}{36} \\qquad \\textbf{(B)}\\ \\frac{1}{12} \\qquad \\textbf{(C)}\\ \\frac{1}{6} \\qquad \\textbf{(D)}\\ \\frac{1}{4} \\qquad \\textbf{(E)}\\ \\frac{5}{18}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is: \n$\\textbf{(A)}\\ \\frac{1}{4} \\text{ the area of the original square}\\\\ \\textbf{(B)}\\ \\frac{1}{2}\\text{ the area of the original square}\\\\ \\textbf{(C)}\\ \\frac{1}{2}\\text{ the area of the circular piece}\\\\ \\textbf{(D)}\\ \\frac{1}{4}\\text{ the area of the circular piece}\\\\ \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_146", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is: \n$\\textbf{(A)}\\ \\frac{1}{4} \\text{ the area of the original square}\\\\ \\textbf{(B)}\\ \\frac{1}{2}\\text{ the area of the original square}\\\\ \\textbf{(C)}\\ \\frac{1}{2}\\text{ the area of the circular piece}\\\\ \\textbf{(D)}\\ \\frac{1}{4}\\text{ the area of the circular piece}\\\\ \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The fraction $\\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to: \n$\\textbf{(A)}\\ a^{-6}-b^{-6}\\qquad\\textbf{(B)}\\ a^{-2}-b^{-2}\\qquad\\textbf{(C)}\\ a^{-2}+b^{-2}\\\\ \\textbf{(D)}\\ a^2+b^2\\qquad\\textbf{(E)}\\ a^2-b^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_147", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The fraction $\\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to: \n$\\textbf{(A)}\\ a^{-6}-b^{-6}\\qquad\\textbf{(B)}\\ a^{-2}-b^{-2}\\qquad\\textbf{(C)}\\ a^{-2}+b^{-2}\\\\ \\textbf{(D)}\\ a^2+b^2\\qquad\\textbf{(E)}\\ a^2-b^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper?\n[asy] size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label(\"$A$\",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype(\"2.5 2.5\")+linewidth(.5)); draw((3,0)--(-3,0),linetype(\"2.5 2.5\")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label(\"$A$\",(6.5,0),NW); dot((6.5,0)); [/asy]\n$\\textbf{(A) } 2(w+h)^2 \\qquad \\textbf{(B) } \\frac{(w+h)^2}2 \\qquad \\textbf{(C) } 2w^2+4wh \\qquad \\textbf{(D) } 2w^2 \\qquad \\textbf{(E) } w^2h$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_148", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper?\n[asy] size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label(\"$A$\",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype(\"2.5 2.5\")+linewidth(.5)); draw((3,0)--(-3,0),linetype(\"2.5 2.5\")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label(\"$A$\",(6.5,0),NW); dot((6.5,0)); [/asy]\n$\\textbf{(A) } 2(w+h)^2 \\qquad \\textbf{(B) } \\frac{(w+h)^2}2 \\qquad \\textbf{(C) } 2w^2+4wh \\qquad \\textbf{(D) } 2w^2 \\qquad \\textbf{(E) } w^2h$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a_n$ be the number of permutations $(x_1, x_2, \\dots, x_n)$ of the numbers $(1,2,\\dots, n)$ such that the $n$ ratios $\\frac{x_k}{k}$ for $1\\le k\\le n$ are all distinct. Prove that $a_n$ is odd for all $n\\ge 1.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "a_n is odd for all n \\geq 1", + "index": "Sky-T1_10k_149", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a_n$ be the number of permutations $(x_1, x_2, \\dots, x_n)$ of the numbers $(1,2,\\dots, n)$ such that the $n$ ratios $\\frac{x_k}{k}$ for $1\\le k\\le n$ are all distinct. Prove that $a_n$ is odd for all $n\\ge 1.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Of the following expressions the one equal to $\\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}}$ is:\n$\\textbf{(A)}\\ \\frac{a^2b^2}{b^2 - a^2}\\qquad \\textbf{(B)}\\ \\frac{a^2b^2}{b^3 - a^3}\\qquad \\textbf{(C)}\\ \\frac{ab}{b^3 - a^3}\\qquad \\textbf{(D)}\\ \\frac{a^3 - b^3}{ab}\\qquad \\textbf{(E)}\\ \\frac{a^2b^2}{a - b}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_150", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Of the following expressions the one equal to $\\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}}$ is:\n$\\textbf{(A)}\\ \\frac{a^2b^2}{b^2 - a^2}\\qquad \\textbf{(B)}\\ \\frac{a^2b^2}{b^3 - a^3}\\qquad \\textbf{(C)}\\ \\frac{ab}{b^3 - a^3}\\qquad \\textbf{(D)}\\ \\frac{a^3 - b^3}{ab}\\qquad \\textbf{(E)}\\ \\frac{a^2b^2}{a - b}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.\n[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "732", + "index": "Sky-T1_10k_151", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.\n[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $\\dfrac{11!-10!}{9!}$?\n$\\textbf{(A)}\\ 99\\qquad\\textbf{(B)}\\ 100\\qquad\\textbf{(C)}\\ 110\\qquad\\textbf{(D)}\\ 121\\qquad\\textbf{(E)}\\ 132$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_152", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $\\dfrac{11!-10!}{9!}$?\n$\\textbf{(A)}\\ 99\\qquad\\textbf{(B)}\\ 100\\qquad\\textbf{(C)}\\ 110\\qquad\\textbf{(D)}\\ 121\\qquad\\textbf{(E)}\\ 132$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "42", + "index": "Sky-T1_10k_153", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$." + } + }, + { + "question": "Return your final response within \\boxed{}. Frieda the frog begins a sequence of hops on a $3 \\times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she \"wraps around\" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops \"up\", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?\n$\\textbf{(A)} ~\\frac{9}{16}\\qquad\\textbf{(B)} ~\\frac{5}{8}\\qquad\\textbf{(C)} ~\\frac{3}{4}\\qquad\\textbf{(D)} ~\\frac{25}{32}\\qquad\\textbf{(E)} ~\\frac{13}{16}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_154", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Frieda the frog begins a sequence of hops on a $3 \\times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she \"wraps around\" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops \"up\", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?\n$\\textbf{(A)} ~\\frac{9}{16}\\qquad\\textbf{(B)} ~\\frac{5}{8}\\qquad\\textbf{(C)} ~\\frac{3}{4}\\qquad\\textbf{(D)} ~\\frac{25}{32}\\qquad\\textbf{(E)} ~\\frac{13}{16}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be: \n$\\textbf{(A)}\\ \\text{equal to the second}\\qquad\\textbf{(B)}\\ \\frac{4}{3}\\text{ times the second}\\qquad\\textbf{(C)}\\ \\frac{2}{\\sqrt{3}}\\text{ times the second}\\\\ \\textbf{(D)}\\ \\frac{\\sqrt{2}}{\\sqrt{3}}\\text{ times the second}\\qquad\\textbf{(E)}\\ \\text{indeterminately related to the second}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_155", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be: \n$\\textbf{(A)}\\ \\text{equal to the second}\\qquad\\textbf{(B)}\\ \\frac{4}{3}\\text{ times the second}\\qquad\\textbf{(C)}\\ \\frac{2}{\\sqrt{3}}\\text{ times the second}\\\\ \\textbf{(D)}\\ \\frac{\\sqrt{2}}{\\sqrt{3}}\\text{ times the second}\\qquad\\textbf{(E)}\\ \\text{indeterminately related to the second}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $A_1A_2A_3\\ldots A_{12}$ be a dodecagon ($12$-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "19", + "index": "Sky-T1_10k_156", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $A_1A_2A_3\\ldots A_{12}$ be a dodecagon ($12$-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. $\\dfrac{2+4+6+\\cdots + 34}{3+6+9+\\cdots+51}=$\n$\\text{(A)}\\ \\dfrac{1}{3} \\qquad \\text{(B)}\\ \\dfrac{2}{3} \\qquad \\text{(C)}\\ \\dfrac{3}{2} \\qquad \\text{(D)}\\ \\dfrac{17}{3} \\qquad \\text{(E)}\\ \\dfrac{34}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_157", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\dfrac{2+4+6+\\cdots + 34}{3+6+9+\\cdots+51}=$\n$\\text{(A)}\\ \\dfrac{1}{3} \\qquad \\text{(B)}\\ \\dfrac{2}{3} \\qquad \\text{(C)}\\ \\dfrac{3}{2} \\qquad \\text{(D)}\\ \\dfrac{17}{3} \\qquad \\text{(E)}\\ \\dfrac{34}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of [digits](https://artofproblemsolving.com/wiki/index.php/Digit) in $4^{16}5^{25}$ (when written in the usual [base](https://artofproblemsolving.com/wiki/index.php/Base_number) $10$ form) is\n$\\mathrm{(A) \\ }31 \\qquad \\mathrm{(B) \\ }30 \\qquad \\mathrm{(C) \\ } 29 \\qquad \\mathrm{(D) \\ }28 \\qquad \\mathrm{(E) \\ } 27$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_158", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of [digits](https://artofproblemsolving.com/wiki/index.php/Digit) in $4^{16}5^{25}$ (when written in the usual [base](https://artofproblemsolving.com/wiki/index.php/Base_number) $10$ form) is\n$\\mathrm{(A) \\ }31 \\qquad \\mathrm{(B) \\ }30 \\qquad \\mathrm{(C) \\ } 29 \\qquad \\mathrm{(D) \\ }28 \\qquad \\mathrm{(E) \\ } 27$" + } + }, + { + "question": "Return your final response within \\boxed{}. The population of the United States in $1980$ was $226,504,825$. The area of the country is $3,615,122$ square miles. There are $(5280)^{2}$ \nsquare feet in one square mile. Which number below best approximates the average number of square feet per person?\n$\\textbf{(A)}\\ 5,000\\qquad \\textbf{(B)}\\ 10,000\\qquad \\textbf{(C)}\\ 50,000\\qquad \\textbf{(D)}\\ 100,000\\qquad \\textbf{(E)}\\ 500,000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_159", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The population of the United States in $1980$ was $226,504,825$. The area of the country is $3,615,122$ square miles. There are $(5280)^{2}$ \nsquare feet in one square mile. Which number below best approximates the average number of square feet per person?\n$\\textbf{(A)}\\ 5,000\\qquad \\textbf{(B)}\\ 10,000\\qquad \\textbf{(C)}\\ 50,000\\qquad \\textbf{(D)}\\ 100,000\\qquad \\textbf{(E)}\\ 500,000$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many integers $n \\geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that\n\\[|z_1| = |z_2| = ... = |z_n| = 1 \\text{ and } z_1 + z_2 + ... + z_n = 0,\\]\nthen the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2 \\qquad\\textbf{(C)}\\ 3 \\qquad\\textbf{(D)}\\ 4 \\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_160", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many integers $n \\geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that\n\\[|z_1| = |z_2| = ... = |z_n| = 1 \\text{ and } z_1 + z_2 + ... + z_n = 0,\\]\nthen the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2 \\qquad\\textbf{(C)}\\ 3 \\qquad\\textbf{(D)}\\ 4 \\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk?\n$\\textbf{(A) }\\frac{x(x+2)(x+5)}{(x+1)(x+3)}\\qquad \\textbf{(B) }\\frac{x(x+1)(x+5)}{(x+2)(x+3)}\\qquad\\\\ \\textbf{(C) }\\frac{(x+1)(x+3)(x+5)}{x(x+2)}\\qquad \\textbf{(D) }\\frac{(x+1)(x+3)}{x(x+2)(x+5)}\\qquad \\\\ \\textbf{(E) }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_161", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk?\n$\\textbf{(A) }\\frac{x(x+2)(x+5)}{(x+1)(x+3)}\\qquad \\textbf{(B) }\\frac{x(x+1)(x+5)}{(x+2)(x+3)}\\qquad\\\\ \\textbf{(C) }\\frac{(x+1)(x+3)(x+5)}{x(x+2)}\\qquad \\textbf{(D) }\\frac{(x+1)(x+3)}{x(x+2)(x+5)}\\qquad \\\\ \\textbf{(E) }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A $6$-inch and $18$-inch diameter poles are placed together and bound together with wire. \nThe length of the shortest wire that will go around them is: \n$\\textbf{(A)}\\ 12\\sqrt{3}+16\\pi\\qquad\\textbf{(B)}\\ 12\\sqrt{3}+7\\pi\\qquad\\textbf{(C)}\\ 12\\sqrt{3}+14\\pi\\\\ \\textbf{(D)}\\ 12+15\\pi\\qquad\\textbf{(E)}\\ 24\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_162", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A $6$-inch and $18$-inch diameter poles are placed together and bound together with wire. \nThe length of the shortest wire that will go around them is: \n$\\textbf{(A)}\\ 12\\sqrt{3}+16\\pi\\qquad\\textbf{(B)}\\ 12\\sqrt{3}+7\\pi\\qquad\\textbf{(C)}\\ 12\\sqrt{3}+14\\pi\\\\ \\textbf{(D)}\\ 12+15\\pi\\qquad\\textbf{(E)}\\ 24\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?\n\n$\\textbf{(A) }\\text{2 o' clock} \\qquad\\textbf{(B) }\\text{3 o' clock} \\qquad\\textbf{(C) }\\text{4 o' clock} \\qquad\\textbf{(D) }\\text{6 o' clock} \\qquad\\textbf{(E) }\\text{8 o' clock}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_163", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?\n\n$\\textbf{(A) }\\text{2 o' clock} \\qquad\\textbf{(B) }\\text{3 o' clock} \\qquad\\textbf{(C) }\\text{4 o' clock} \\qquad\\textbf{(D) }\\text{6 o' clock} \\qquad\\textbf{(E) }\\text{8 o' clock}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $650$ students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?\n\n$\\mathrm{(A)} \\frac{2}{5} \\qquad \\mathrm{(B)} \\frac{1}{2} \\qquad \\mathrm{(C)} \\frac{5}{4} \\qquad \\mathrm{(D)} \\frac{5}{3} \\qquad \\mathrm{(E)} \\frac{5}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_164", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $650$ students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?\n\n$\\mathrm{(A)} \\frac{2}{5} \\qquad \\mathrm{(B)} \\frac{1}{2} \\qquad \\mathrm{(C)} \\frac{5}{4} \\qquad \\mathrm{(D)} \\frac{5}{3} \\qquad \\mathrm{(E)} \\frac{5}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $A_1,A_2,...,A_{n+1}$ be distinct subsets of $[n]$ with $|A_1|=|A_2|=\\cdots =|A_{n+1}|=3$. Prove that $|A_i\\cap A_j|=1$ for some pair $\\{i,j\\}$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "|A_i \\cap A_j| = 1 for some pair \\{i, j\\}", + "index": "Sky-T1_10k_165", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $A_1,A_2,...,A_{n+1}$ be distinct subsets of $[n]$ with $|A_1|=|A_2|=\\cdots =|A_{n+1}|=3$. Prove that $|A_i\\cap A_j|=1$ for some pair $\\{i,j\\}$." + } + }, + { + "question": "Return your final response within \\boxed{}. In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red. \nThereafter, $7$ out of every $8$ counted were red. If, in all, $90$ % or more of the balls counted were red, the maximum value of $n$ is:\n$\\textbf{(A)}\\ 225 \\qquad \\textbf{(B)}\\ 210 \\qquad \\textbf{(C)}\\ 200 \\qquad \\textbf{(D)}\\ 180 \\qquad \\textbf{(E)}\\ 175$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_166", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red. \nThereafter, $7$ out of every $8$ counted were red. If, in all, $90$ % or more of the balls counted were red, the maximum value of $n$ is:\n$\\textbf{(A)}\\ 225 \\qquad \\textbf{(B)}\\ 210 \\qquad \\textbf{(C)}\\ 200 \\qquad \\textbf{(D)}\\ 180 \\qquad \\textbf{(E)}\\ 175$" + } + }, + { + "question": "Return your final response within \\boxed{}. The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 3 \\qquad \\text{(D)}\\ 4 \\qquad \\text{(E)}\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_167", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 3 \\qquad \\text{(D)}\\ 4 \\qquad \\text{(E)}\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a magic triangle, each of the six [whole numbers](https://artofproblemsolving.com/wiki/index.php/Whole_number) $10-15$ is placed in one of the [circles](https://artofproblemsolving.com/wiki/index.php/Circle) so that the sum, $S$, of the three numbers on each side of the [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) is the same. The largest possible value for $S$ is\n[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(60)); draw(circle(14*dir(60),1)); draw((1,0)--(6,0)); draw(circle((7,0),1)); draw((8,0)--(13,0)); draw(circle((14,0),1)); draw(circle((10.5,6.0621778264910705273460621952706),1)); draw((13.5,0.86602540378443864676372317075294)--(11,5.1961524227066318805823390245176)); draw((10,6.9282032302755091741097853660235)--(7.5,11.258330249197702407928401219788)); [/asy]\n$\\text{(A)}\\ 36 \\qquad \\text{(B)}\\ 37 \\qquad \\text{(C)}\\ 38 \\qquad \\text{(D)}\\ 39 \\qquad \\text{(E)}\\ 40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_168", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a magic triangle, each of the six [whole numbers](https://artofproblemsolving.com/wiki/index.php/Whole_number) $10-15$ is placed in one of the [circles](https://artofproblemsolving.com/wiki/index.php/Circle) so that the sum, $S$, of the three numbers on each side of the [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) is the same. The largest possible value for $S$ is\n[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(60)); draw(circle(14*dir(60),1)); draw((1,0)--(6,0)); draw(circle((7,0),1)); draw((8,0)--(13,0)); draw(circle((14,0),1)); draw(circle((10.5,6.0621778264910705273460621952706),1)); draw((13.5,0.86602540378443864676372317075294)--(11,5.1961524227066318805823390245176)); draw((10,6.9282032302755091741097853660235)--(7.5,11.258330249197702407928401219788)); [/asy]\n$\\text{(A)}\\ 36 \\qquad \\text{(B)}\\ 37 \\qquad \\text{(C)}\\ 38 \\qquad \\text{(D)}\\ 39 \\qquad \\text{(E)}\\ 40$" + } + }, + { + "question": "Return your final response within \\boxed{}. Supposed that $x$ and $y$ are nonzero real numbers such that $\\frac{3x+y}{x-3y}=-2$. What is the value of $\\frac{x+3y}{3x-y}$?\n$\\textbf{(A)}\\ -3\\qquad\\textbf{(B)}\\ -1\\qquad\\textbf{(C)}\\ 1\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_169", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Supposed that $x$ and $y$ are nonzero real numbers such that $\\frac{3x+y}{x-3y}=-2$. What is the value of $\\frac{x+3y}{3x-y}$?\n$\\textbf{(A)}\\ -3\\qquad\\textbf{(B)}\\ -1\\qquad\\textbf{(C)}\\ 1\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. The [product](https://artofproblemsolving.com/wiki/index.php/Product) $8\\times .25\\times 2\\times .125 =$\n$\\text{(A)}\\ \\frac18 \\qquad \\text{(B)}\\ \\frac14 \\qquad \\text{(C)}\\ \\frac12 \\qquad \\text{(D)}\\ 1 \\qquad \\text{(E)}\\ 2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_170", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The [product](https://artofproblemsolving.com/wiki/index.php/Product) $8\\times .25\\times 2\\times .125 =$\n$\\text{(A)}\\ \\frac18 \\qquad \\text{(B)}\\ \\frac14 \\qquad \\text{(C)}\\ \\frac12 \\qquad \\text{(D)}\\ 1 \\qquad \\text{(E)}\\ 2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 16 \\qquad\\textbf{(C)}\\ 28 \\qquad\\textbf{(D)}\\ 32 \\qquad\\textbf{(E)}\\ 40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_171", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 16 \\qquad\\textbf{(C)}\\ 28 \\qquad\\textbf{(D)}\\ 32 \\qquad\\textbf{(E)}\\ 40$" + } + }, + { + "question": "Return your final response within \\boxed{}. If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is\n$\\textbf{(A) }8\\qquad \\textbf{(B) }9\\qquad \\textbf{(C) }12\\qquad \\textbf{(D) }18\\qquad \\textbf{(E) }24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_172", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is\n$\\textbf{(A) }8\\qquad \\textbf{(B) }9\\qquad \\textbf{(C) }12\\qquad \\textbf{(D) }18\\qquad \\textbf{(E) }24$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice, Bob, and Carol play a game in which each of them chooses a real number between $0$ and $1.$ The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between $0$ and $1,$ and Bob announces that he will choose his number uniformly at random from all the numbers between $\\tfrac{1}{2}$ and $\\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning?\n$\\textbf{(A) }\\frac{1}{2}\\qquad \\textbf{(B) }\\frac{13}{24} \\qquad \\textbf{(C) }\\frac{7}{12} \\qquad \\textbf{(D) }\\frac{5}{8} \\qquad \\textbf{(E) }\\frac{2}{3}\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_173", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice, Bob, and Carol play a game in which each of them chooses a real number between $0$ and $1.$ The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between $0$ and $1,$ and Bob announces that he will choose his number uniformly at random from all the numbers between $\\tfrac{1}{2}$ and $\\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning?\n$\\textbf{(A) }\\frac{1}{2}\\qquad \\textbf{(B) }\\frac{13}{24} \\qquad \\textbf{(C) }\\frac{7}{12} \\qquad \\textbf{(D) }\\frac{5}{8} \\qquad \\textbf{(E) }\\frac{2}{3}\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is:\n$\\text{(A) } \\frac{1}{3}\\quad \\text{(B) } \\frac{2}{3}\\quad \\text{(C) } 6\\quad \\text{(D) } 6\\tfrac{1}{3}\\quad \\text{(E) } 6\\tfrac{2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_174", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is:\n$\\text{(A) } \\frac{1}{3}\\quad \\text{(B) } \\frac{2}{3}\\quad \\text{(C) } 6\\quad \\text{(D) } 6\\tfrac{1}{3}\\quad \\text{(E) } 6\\tfrac{2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be an isosceles trapezoid with $\\overline{BC}\\parallel \\overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\\angle AXD = \\angle BYC = 90^\\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD?$\n\n$\\textbf{(A)}\\: 15\\qquad\\textbf{(B)} \\: 5\\sqrt{11}\\qquad\\textbf{(C)} \\: 3\\sqrt{35}\\qquad\\textbf{(D)} \\: 18\\qquad\\textbf{(E)} \\: 7\\sqrt{7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_175", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be an isosceles trapezoid with $\\overline{BC}\\parallel \\overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\\angle AXD = \\angle BYC = 90^\\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD?$\n\n$\\textbf{(A)}\\: 15\\qquad\\textbf{(B)} \\: 5\\sqrt{11}\\qquad\\textbf{(C)} \\: 3\\sqrt{35}\\qquad\\textbf{(D)} \\: 18\\qquad\\textbf{(E)} \\: 7\\sqrt{7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. ([Sam Vandervelde](https://artofproblemsolving.com/wiki/index.php/Sam_Vandervelde)) At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $2^k$ for some positive integer $k$).", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "2^k", + "index": "Sky-T1_10k_176", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. ([Sam Vandervelde](https://artofproblemsolving.com/wiki/index.php/Sam_Vandervelde)) At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $2^k$ for some positive integer $k$)." + } + }, + { + "question": "Return your final response within \\boxed{}. The number of revolutions of a wheel, with fixed center and with an outside diameter of $6$ feet, required to cause a point on the rim to go one mile is: \n$\\textbf{(A)}\\ 880 \\qquad\\textbf{(B)}\\ \\frac{440}{\\pi} \\qquad\\textbf{(C)}\\ \\frac{880}{\\pi} \\qquad\\textbf{(D)}\\ 440\\pi\\qquad\\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_177", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of revolutions of a wheel, with fixed center and with an outside diameter of $6$ feet, required to cause a point on the rim to go one mile is: \n$\\textbf{(A)}\\ 880 \\qquad\\textbf{(B)}\\ \\frac{440}{\\pi} \\qquad\\textbf{(C)}\\ \\frac{880}{\\pi} \\qquad\\textbf{(D)}\\ 440\\pi\\qquad\\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{(.2)^3}{(.02)^2} =$\n$\\text{(A)}\\ .2 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 10 \\qquad \\text{(D)}\\ 15 \\qquad \\text{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_178", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{(.2)^3}{(.02)^2} =$\n$\\text{(A)}\\ .2 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 10 \\qquad \\text{(D)}\\ 15 \\qquad \\text{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?\n$\\textbf{(A) } 1\\qquad\\textbf{(B) } 3\\qquad\\textbf{(C) }6 \\qquad\\textbf{(D) }12 \\qquad\\textbf{(E) }24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_179", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?\n$\\textbf{(A) } 1\\qquad\\textbf{(B) } 3\\qquad\\textbf{(C) }6 \\qquad\\textbf{(D) }12 \\qquad\\textbf{(E) }24$" + } + }, + { + "question": "Return your final response within \\boxed{}. A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx +2$ passes through no lattice point with $0 < x \\le 100$ for all $m$ such that $\\frac{1}{2} < m < a$. What is the maximum possible value of $a$?\n$\\textbf{(A)}\\ \\frac{51}{101} \\qquad\\textbf{(B)}\\ \\frac{50}{99} \\qquad\\textbf{(C)}\\ \\frac{51}{100} \\qquad\\textbf{(D)}\\ \\frac{52}{101} \\qquad\\textbf{(E)}\\ \\frac{13}{25}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_180", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx +2$ passes through no lattice point with $0 < x \\le 100$ for all $m$ such that $\\frac{1}{2} < m < a$. What is the maximum possible value of $a$?\n$\\textbf{(A)}\\ \\frac{51}{101} \\qquad\\textbf{(B)}\\ \\frac{50}{99} \\qquad\\textbf{(C)}\\ \\frac{51}{100} \\qquad\\textbf{(D)}\\ \\frac{52}{101} \\qquad\\textbf{(E)}\\ \\frac{13}{25}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $P$ and $Q$ are on line segment $AB$, and both points are on the same side of the midpoint of $AB$. Point $P$ divides $AB$ in the ratio $2:3$, and $Q$ divides $AB$ in the ratio $3:4$. If $PQ$=2, then the length of segment $AB$ is\n$\\text{(A) } 12\\quad \\text{(B) } 28\\quad \\text{(C) } 70\\quad \\text{(D) } 75\\quad \\text{(E) } 105$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_181", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $P$ and $Q$ are on line segment $AB$, and both points are on the same side of the midpoint of $AB$. Point $P$ divides $AB$ in the ratio $2:3$, and $Q$ divides $AB$ in the ratio $3:4$. If $PQ$=2, then the length of segment $AB$ is\n$\\text{(A) } 12\\quad \\text{(B) } 28\\quad \\text{(C) } 70\\quad \\text{(D) } 75\\quad \\text{(E) } 105$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $f(x)=\\frac{x^4+x^2}{x+1}$, then $f(i)$, where $i=\\sqrt{-1}$, is equal to\n$\\text{(A) } 1+i\\quad \\text{(B) } 1\\quad \\text{(C) } -1\\quad \\text{(D) } 0\\quad \\text{(E) } -1-i$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_182", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $f(x)=\\frac{x^4+x^2}{x+1}$, then $f(i)$, where $i=\\sqrt{-1}$, is equal to\n$\\text{(A) } 1+i\\quad \\text{(B) } 1\\quad \\text{(C) } -1\\quad \\text{(D) } 0\\quad \\text{(E) } -1-i$" + } + }, + { + "question": "Return your final response within \\boxed{}. Joy has $30$ thin rods, one each of every integer length from $1 \\text{ cm}$ through $30 \\text{ cm}$. She places the rods with lengths $3 \\text{ cm}$, $7 \\text{ cm}$, and $15 \\text{cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?\n$\\textbf{(A)}\\ 16 \\qquad\\textbf{(B)}\\ 17 \\qquad\\textbf{(C)}\\ 18 \\qquad\\textbf{(D)}\\ 19 \\qquad\\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_183", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Joy has $30$ thin rods, one each of every integer length from $1 \\text{ cm}$ through $30 \\text{ cm}$. She places the rods with lengths $3 \\text{ cm}$, $7 \\text{ cm}$, and $15 \\text{cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?\n$\\textbf{(A)}\\ 16 \\qquad\\textbf{(B)}\\ 17 \\qquad\\textbf{(C)}\\ 18 \\qquad\\textbf{(D)}\\ 19 \\qquad\\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, \"My house number has two digits, and exactly three of the following four statements about it are true.\"\n(1) It is prime.\n(2) It is even.\n(3) It is divisible by 7.\n(4) One of its digits is 9.\nThis information allows Malcolm to determine Isabella's house number. What is its units digit?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }6\\qquad\\textbf{(C) }7\\qquad\\textbf{(D) }8\\qquad\\textbf{(E) }9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_184", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, \"My house number has two digits, and exactly three of the following four statements about it are true.\"\n(1) It is prime.\n(2) It is even.\n(3) It is divisible by 7.\n(4) One of its digits is 9.\nThis information allows Malcolm to determine Isabella's house number. What is its units digit?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }6\\qquad\\textbf{(C) }7\\qquad\\textbf{(D) }8\\qquad\\textbf{(E) }9$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?\n$\\textbf{(A)}\\ 32 \\qquad\\textbf{(B)}\\ 34\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 38\\qquad\\textbf{(E)}\\ 40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_185", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?\n$\\textbf{(A)}\\ 32 \\qquad\\textbf{(B)}\\ 34\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 38\\qquad\\textbf{(E)}\\ 40$" + } + }, + { + "question": "Return your final response within \\boxed{}. The manager of a company planned to distribute a $$50$ bonus to each employee from the company fund, but the fund contained $$5$ less than what was needed. Instead the manager gave each employee a $$45$ bonus and kept the remaining $$95$ in the company fund. The amount of money in the company fund before any bonuses were paid was\n$\\text{(A)}\\ 945\\text{ dollars} \\qquad \\text{(B)}\\ 950\\text{ dollars} \\qquad \\text{(C)}\\ 955\\text{ dollars} \\qquad \\text{(D)}\\ 990\\text{ dollars} \\qquad \\text{(E)}\\ 995\\text{ dollars}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_186", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The manager of a company planned to distribute a $$50$ bonus to each employee from the company fund, but the fund contained $$5$ less than what was needed. Instead the manager gave each employee a $$45$ bonus and kept the remaining $$95$ in the company fund. The amount of money in the company fund before any bonuses were paid was\n$\\text{(A)}\\ 945\\text{ dollars} \\qquad \\text{(B)}\\ 950\\text{ dollars} \\qquad \\text{(C)}\\ 955\\text{ dollars} \\qquad \\text{(D)}\\ 990\\text{ dollars} \\qquad \\text{(E)}\\ 995\\text{ dollars}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $6^6+6^6+6^6+6^6+6^6+6^6=$\n$\\text{(A) } 6^6 \\quad \\text{(B) } 6^7\\quad \\text{(C) } 36^6\\quad \\text{(D) } 6^{36}\\quad \\text{(E) } 36^{36}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_187", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $6^6+6^6+6^6+6^6+6^6+6^6=$\n$\\text{(A) } 6^6 \\quad \\text{(B) } 6^7\\quad \\text{(C) } 36^6\\quad \\text{(D) } 6^{36}\\quad \\text{(E) } 36^{36}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $\\frac{(2112-2021)^2}{169}$?\n$\\textbf{(A) } 7 \\qquad\\textbf{(B) } 21 \\qquad\\textbf{(C) } 49 \\qquad\\textbf{(D) } 64 \\qquad\\textbf{(E) } 91$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_188", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $\\frac{(2112-2021)^2}{169}$?\n$\\textbf{(A) } 7 \\qquad\\textbf{(B) } 21 \\qquad\\textbf{(C) } 49 \\qquad\\textbf{(D) } 64 \\qquad\\textbf{(E) } 91$" + } + }, + { + "question": "Return your final response within \\boxed{}. Rectangle $ABCD$ has $AB=4$ and $BC=3$. Segment $EF$ is constructed through $B$ so that $EF$ is perpendicular to $DB$, and $A$ and $C$ lie on $DE$ and $DF$, respectively. What is $EF$?\n$\\mathrm{(A)}\\ 9 \\qquad \\mathrm{(B)}\\ 10 \\qquad \\mathrm{(C)}\\ \\frac {125}{12} \\qquad \\mathrm{(D)}\\ \\frac {103}{9} \\qquad \\mathrm{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_189", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Rectangle $ABCD$ has $AB=4$ and $BC=3$. Segment $EF$ is constructed through $B$ so that $EF$ is perpendicular to $DB$, and $A$ and $C$ lie on $DE$ and $DF$, respectively. What is $EF$?\n$\\mathrm{(A)}\\ 9 \\qquad \\mathrm{(B)}\\ 10 \\qquad \\mathrm{(C)}\\ \\frac {125}{12} \\qquad \\mathrm{(D)}\\ \\frac {103}{9} \\qquad \\mathrm{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the set of numbers $\\{1, 10, 10^2, 10^3, \\ldots, 10^{10}\\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 9 \\qquad\\textbf{(C)}\\ 10 \\qquad\\textbf{(D)}\\ 11 \\qquad\\textbf{(E)} 101$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_190", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the set of numbers $\\{1, 10, 10^2, 10^3, \\ldots, 10^{10}\\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 9 \\qquad\\textbf{(C)}\\ 10 \\qquad\\textbf{(D)}\\ 11 \\qquad\\textbf{(E)} 101$" + } + }, + { + "question": "Return your final response within \\boxed{}. Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:\n$\\quad\\bullet\\qquad$ $L,$ a rotation of $90^{\\circ}$ counterclockwise around the origin;\n$\\quad\\bullet\\qquad$ $R,$ a rotation of $90^{\\circ}$ clockwise around the origin;\n$\\quad\\bullet\\qquad$ $H,$ a reflection across the $x$-axis; and\n$\\quad\\bullet\\qquad$ $V,$ a reflection across the $y$-axis.\nEach of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\\{L, R, H, V\\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)\n$\\textbf{(A)}\\ 2^{37} \\qquad\\textbf{(B)}\\ 3\\cdot 2^{36} \\qquad\\textbf{(C)}\\ 2^{38} \\qquad\\textbf{(D)}\\ 3\\cdot 2^{37} \\qquad\\textbf{(E)}\\ 2^{39}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_191", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:\n$\\quad\\bullet\\qquad$ $L,$ a rotation of $90^{\\circ}$ counterclockwise around the origin;\n$\\quad\\bullet\\qquad$ $R,$ a rotation of $90^{\\circ}$ clockwise around the origin;\n$\\quad\\bullet\\qquad$ $H,$ a reflection across the $x$-axis; and\n$\\quad\\bullet\\qquad$ $V,$ a reflection across the $y$-axis.\nEach of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\\{L, R, H, V\\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)\n$\\textbf{(A)}\\ 2^{37} \\qquad\\textbf{(B)}\\ 3\\cdot 2^{36} \\qquad\\textbf{(C)}\\ 2^{38} \\qquad\\textbf{(D)}\\ 3\\cdot 2^{37} \\qquad\\textbf{(E)}\\ 2^{39}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\\overline{AD}$. What is the area of $\\triangle AMC$?\n\n$\\text{(A) }12\\qquad\\text{(B) }15\\qquad\\text{(C) }18\\qquad\\text{(D) }20\\qquad \\text{(E) }24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_192", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\\overline{AD}$. What is the area of $\\triangle AMC$?\n\n$\\text{(A) }12\\qquad\\text{(B) }15\\qquad\\text{(C) }18\\qquad\\text{(D) }20\\qquad \\text{(E) }24$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $f(x)=3x+2$ for all real $x$, then the statement:\n\"$|f(x)+4|0$ and $b>0$\"\nis true when\n$\\mathrm{(A)}\\ b\\le a/3\\qquad\\mathrm{(B)}\\ b > a/3\\qquad\\mathrm{(C)}\\ a\\le b/3\\qquad\\mathrm{(D)}\\ a > b/3\\\\ \\qquad\\mathrm{(E)}\\ \\text{The statement is never true.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_193", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $f(x)=3x+2$ for all real $x$, then the statement:\n\"$|f(x)+4|0$ and $b>0$\"\nis true when\n$\\mathrm{(A)}\\ b\\le a/3\\qquad\\mathrm{(B)}\\ b > a/3\\qquad\\mathrm{(C)}\\ a\\le b/3\\qquad\\mathrm{(D)}\\ a > b/3\\\\ \\qquad\\mathrm{(E)}\\ \\text{The statement is never true.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the adjoining figure of a rectangular solid, $\\angle DHG=45^\\circ$ and $\\angle FHB=60^\\circ$. Find the cosine of $\\angle BHD$.\n\n$\\text {(A)} \\frac{\\sqrt{3}}{6} \\qquad \\text {(B)} \\frac{\\sqrt{2}}{6} \\qquad \\text {(C)} \\frac{\\sqrt{6}}{3} \\qquad \\text{(D)}\\frac{\\sqrt{6}}{4}\\qquad \\text{(E)}\\frac{\\sqrt{6}-\\sqrt{2}}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "(D) \\frac{\\sqrt{6}}{4}", + "index": "Sky-T1_10k_194", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the adjoining figure of a rectangular solid, $\\angle DHG=45^\\circ$ and $\\angle FHB=60^\\circ$. Find the cosine of $\\angle BHD$.\n\n$\\text {(A)} \\frac{\\sqrt{3}}{6} \\qquad \\text {(B)} \\frac{\\sqrt{2}}{6} \\qquad \\text {(C)} \\frac{\\sqrt{6}}{3} \\qquad \\text{(D)}\\frac{\\sqrt{6}}{4}\\qquad \\text{(E)}\\frac{\\sqrt{6}-\\sqrt{2}}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a,b,c$ are positive integers less than $10$, then $(10a + b)(10a + c) = 100a(a + 1) + bc$ if:\n$\\textbf{(A) }b+c=10$\n$\\qquad\\textbf{(B) }b=c$\n$\\qquad\\textbf{(C) }a+b=10$\n$\\qquad\\textbf {(D) }a=b$\n$\\qquad\\textbf{(E) }a+b+c=10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_195", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a,b,c$ are positive integers less than $10$, then $(10a + b)(10a + c) = 100a(a + 1) + bc$ if:\n$\\textbf{(A) }b+c=10$\n$\\qquad\\textbf{(B) }b=c$\n$\\qquad\\textbf{(C) }a+b=10$\n$\\qquad\\textbf {(D) }a=b$\n$\\qquad\\textbf{(E) }a+b+c=10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$\n\nArjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?\n$\\textbf{(A) }(6,1,1) \\qquad \\textbf{(B) }(6,2,1) \\qquad \\textbf{(C) }(6,2,2)\\qquad \\textbf{(D) }(6,3,1) \\qquad \\textbf{(E) }(6,3,2)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_196", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$\n\nArjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?\n$\\textbf{(A) }(6,1,1) \\qquad \\textbf{(B) }(6,2,1) \\qquad \\textbf{(C) }(6,2,2)\\qquad \\textbf{(D) }(6,3,1) \\qquad \\textbf{(E) }(6,3,2)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$? (Note: Both months have $31$ days.)\n$\\textrm{(A)}\\ \\text{Monday} \\qquad \\textrm{(B)}\\ \\text{Tuesday} \\qquad \\textrm{(C)}\\ \\text{Wednesday} \\qquad \\textrm{(D)}\\ \\text{Thursday} \\qquad \\textrm{(E)}\\ \\text{Friday}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_197", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$? (Note: Both months have $31$ days.)\n$\\textrm{(A)}\\ \\text{Monday} \\qquad \\textrm{(B)}\\ \\text{Tuesday} \\qquad \\textrm{(C)}\\ \\text{Wednesday} \\qquad \\textrm{(D)}\\ \\text{Thursday} \\qquad \\textrm{(E)}\\ \\text{Friday}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If an integer of two digits is $k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by\n$\\textbf{(A) \\ } 9-k \\qquad \\textbf{(B) \\ } 10-k \\qquad \\textbf{(C) \\ } 11-k \\qquad \\textbf{(D) \\ } k-1 \\qquad \\textbf{(E) \\ } k+1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_198", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If an integer of two digits is $k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by\n$\\textbf{(A) \\ } 9-k \\qquad \\textbf{(B) \\ } 10-k \\qquad \\textbf{(C) \\ } 11-k \\qquad \\textbf{(D) \\ } k-1 \\qquad \\textbf{(E) \\ } k+1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $A_1A_2A_3$ be a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) and let $\\omega_1$ be a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\\omega_2, \\omega_3, \\dots, \\omega_7$ such that for $k = 2, 3, \\dots, 7,$ $\\omega_k$ is externally [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent_(geometry)) to $\\omega_{k - 1}$ and passes through $A_k$ and $A_{k + 1},$ where $A_{n + 3} = A_{n}$ for all $n \\ge 1$. Prove that $\\omega_7 = \\omega_1.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\omega_7 = \\omega_1", + "index": "Sky-T1_10k_199", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $A_1A_2A_3$ be a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) and let $\\omega_1$ be a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\\omega_2, \\omega_3, \\dots, \\omega_7$ such that for $k = 2, 3, \\dots, 7,$ $\\omega_k$ is externally [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent_(geometry)) to $\\omega_{k - 1}$ and passes through $A_k$ and $A_{k + 1},$ where $A_{n + 3} = A_{n}$ for all $n \\ge 1$. Prove that $\\omega_7 = \\omega_1.$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $F(n+1)=\\frac{2F(n)+1}{2}$ for $n=1,2,\\cdots$ and $F(1)=2$, then $F(101)$ equals:\n$\\text{(A) } 49 \\quad \\text{(B) } 50 \\quad \\text{(C) } 51 \\quad \\text{(D) } 52 \\quad \\text{(E) } 53$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_200", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $F(n+1)=\\frac{2F(n)+1}{2}$ for $n=1,2,\\cdots$ and $F(1)=2$, then $F(101)$ equals:\n$\\text{(A) } 49 \\quad \\text{(B) } 50 \\quad \\text{(C) } 51 \\quad \\text{(D) } 52 \\quad \\text{(E) } 53$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following sets of $x$-values satisfy the inequality $2x^2 + x < 6$? \n$\\textbf{(A)}\\ -2 < x <\\frac{3}{2}\\qquad\\textbf{(B)}\\ x >\\frac{3}2\\text{ or }x <-2\\qquad\\textbf{(C)}\\ x <\\frac{3}2\\qquad$\n$\\textbf{(D)}\\ \\frac{3}2 < x < 2\\qquad\\textbf{(E)}\\ x <-2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_201", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following sets of $x$-values satisfy the inequality $2x^2 + x < 6$? \n$\\textbf{(A)}\\ -2 < x <\\frac{3}{2}\\qquad\\textbf{(B)}\\ x >\\frac{3}2\\text{ or }x <-2\\qquad\\textbf{(C)}\\ x <\\frac{3}2\\qquad$\n$\\textbf{(D)}\\ \\frac{3}2 < x < 2\\qquad\\textbf{(E)}\\ x <-2$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids?\n[asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy]\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_202", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids?\n[asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy]\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number $n$ can be written in base $14$ as $\\underline{a}\\text{ }\\underline{b}\\text{ }\\underline{c}$, can be written in base $15$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{b}$, and can be written in base $6$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{a}\\text{ }\\underline{c}\\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "925", + "index": "Sky-T1_10k_203", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number $n$ can be written in base $14$ as $\\underline{a}\\text{ }\\underline{b}\\text{ }\\underline{c}$, can be written in base $15$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{b}$, and can be written in base $6$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{a}\\text{ }\\underline{c}\\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$." + } + }, + { + "question": "Return your final response within \\boxed{}. A number $x$ is $2$ more than the product of its [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal) and its additive [inverse](https://artofproblemsolving.com/wiki/index.php/Inverse). In which [interval](https://artofproblemsolving.com/wiki/index.php/Interval) does the number lie?\n$\\textbf{(A) }\\ -4\\le x\\le -2\\qquad\\textbf{(B) }\\ -2 < x\\le 0\\qquad\\textbf{(C) }0$ $< x \\le 2 \\qquad \\textbf{(D) }\\ 2 < x\\le 4\\qquad\\textbf{(E) }\\ 4 < x\\le 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_204", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A number $x$ is $2$ more than the product of its [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal) and its additive [inverse](https://artofproblemsolving.com/wiki/index.php/Inverse). In which [interval](https://artofproblemsolving.com/wiki/index.php/Interval) does the number lie?\n$\\textbf{(A) }\\ -4\\le x\\le -2\\qquad\\textbf{(B) }\\ -2 < x\\le 0\\qquad\\textbf{(C) }0$ $< x \\le 2 \\qquad \\textbf{(D) }\\ 2 < x\\le 4\\qquad\\textbf{(E) }\\ 4 < x\\le 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. A carton contains milk that is $2$% fat, an amount that is $40$% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?\n$\\mathrm{(A)}\\ \\frac{12}{5} \\qquad \\mathrm{(B)}\\ 3 \\qquad \\mathrm{(C)}\\ \\frac{10}{3} \\qquad \\mathrm{(D)}\\ 38 \\qquad \\mathrm{(E)}\\ 42$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_205", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A carton contains milk that is $2$% fat, an amount that is $40$% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?\n$\\mathrm{(A)}\\ \\frac{12}{5} \\qquad \\mathrm{(B)}\\ 3 \\qquad \\mathrm{(C)}\\ \\frac{10}{3} \\qquad \\mathrm{(D)}\\ 38 \\qquad \\mathrm{(E)}\\ 42$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\\circ$, $60^\\circ$, and $60.001^\\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\\triangle A_nB_nC_n$ is obtuse?\n$\\textbf{(A) } 10 \\qquad \\textbf{(B) }11 \\qquad \\textbf{(C) } 13\\qquad \\textbf{(D) } 14 \\qquad \\textbf{(E) } 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_206", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\\circ$, $60^\\circ$, and $60.001^\\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\\triangle A_nB_nC_n$ is obtuse?\n$\\textbf{(A) } 10 \\qquad \\textbf{(B) }11 \\qquad \\textbf{(C) } 13\\qquad \\textbf{(D) } 14 \\qquad \\textbf{(E) } 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four \ntimes. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?\n$\\mathrm{(A)}\\ 2500\\qquad\\mathrm{(B)}\\ 3000\\qquad\\mathrm{(C)}\\ 3500\\qquad\\mathrm{(D)}\\ 4000\\qquad\\mathrm{(E)}\\ 4500$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_207", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four \ntimes. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?\n$\\mathrm{(A)}\\ 2500\\qquad\\mathrm{(B)}\\ 3000\\qquad\\mathrm{(C)}\\ 3500\\qquad\\mathrm{(D)}\\ 4000\\qquad\\mathrm{(E)}\\ 4500$" + } + }, + { + "question": "Return your final response within \\boxed{}. $3^3+3^3+3^3 =$\n$\\text{(A)}\\ 3^4 \\qquad \\text{(B)}\\ 9^3 \\qquad \\text{(C)}\\ 3^9 \\qquad \\text{(D)}\\ 27^3 \\qquad \\text{(E)}\\ 3^{27}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_208", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $3^3+3^3+3^3 =$\n$\\text{(A)}\\ 3^4 \\qquad \\text{(B)}\\ 9^3 \\qquad \\text{(C)}\\ 3^9 \\qquad \\text{(D)}\\ 27^3 \\qquad \\text{(E)}\\ 3^{27}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, \nforming a roll $10$ cm in diameter. Approximate the length of the paper in meters. \n(Pretend the paper forms $600$ concentric circles with diameters evenly spaced from $2$ cm to $10$ cm.)\n$\\textbf{(A)}\\ 36\\pi \\qquad \\textbf{(B)}\\ 45\\pi \\qquad \\textbf{(C)}\\ 60\\pi \\qquad \\textbf{(D)}\\ 72\\pi \\qquad \\textbf{(E)}\\ 90\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_209", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, \nforming a roll $10$ cm in diameter. Approximate the length of the paper in meters. \n(Pretend the paper forms $600$ concentric circles with diameters evenly spaced from $2$ cm to $10$ cm.)\n$\\textbf{(A)}\\ 36\\pi \\qquad \\textbf{(B)}\\ 45\\pi \\qquad \\textbf{(C)}\\ 60\\pi \\qquad \\textbf{(D)}\\ 72\\pi \\qquad \\textbf{(E)}\\ 90\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a high school with $500$ students, $40\\%$ of the seniors play a musical instrument, while $30\\%$ of the non-seniors do not play a musical instrument. In all, $46.8\\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?\n$\\textbf{(A) } 66 \\qquad\\textbf{(B) } 154 \\qquad\\textbf{(C) } 186 \\qquad\\textbf{(D) } 220 \\qquad\\textbf{(E) } 266$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_210", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a high school with $500$ students, $40\\%$ of the seniors play a musical instrument, while $30\\%$ of the non-seniors do not play a musical instrument. In all, $46.8\\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?\n$\\textbf{(A) } 66 \\qquad\\textbf{(B) } 154 \\qquad\\textbf{(C) } 186 \\qquad\\textbf{(D) } 220 \\qquad\\textbf{(E) } 266$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $f(x)=\\frac{x(x-1)}{2}$, then $f(x+2)$ equals: \n$\\textbf{(A)}\\ f(x)+f(2) \\qquad \\textbf{(B)}\\ (x+2)f(x) \\qquad \\textbf{(C)}\\ x(x+2)f(x) \\qquad \\textbf{(D)}\\ \\frac{xf(x)}{x+2}\\\\ \\textbf{(E)}\\ \\frac{(x+2)f(x+1)}{x}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_211", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $f(x)=\\frac{x(x-1)}{2}$, then $f(x+2)$ equals: \n$\\textbf{(A)}\\ f(x)+f(2) \\qquad \\textbf{(B)}\\ (x+2)f(x) \\qquad \\textbf{(C)}\\ x(x+2)f(x) \\qquad \\textbf{(D)}\\ \\frac{xf(x)}{x+2}\\\\ \\textbf{(E)}\\ \\frac{(x+2)f(x+1)}{x}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the table shown, the formula relating x and y is: \n\\[\\begin{array}{|c|c|c|c|c|c|}\\hline x & 1 & 2 & 3 & 4 & 5\\\\ \\hline y & 3 & 7 & 13 & 21 & 31\\\\ \\hline\\end{array}\\]\n$\\text{(A) } y = 4x - 1 \\qquad\\quad \\text{(B) } y = x^3 - x^2 + x + 2 \\qquad\\\\ \\text{(C) } y = x^2 + x + 1 \\qquad \\text{(D) } y = (x^2 + x + 1)(x - 1) \\qquad\\\\ \\text{(E) } \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_212", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the table shown, the formula relating x and y is: \n\\[\\begin{array}{|c|c|c|c|c|c|}\\hline x & 1 & 2 & 3 & 4 & 5\\\\ \\hline y & 3 & 7 & 13 & 21 & 31\\\\ \\hline\\end{array}\\]\n$\\text{(A) } y = 4x - 1 \\qquad\\quad \\text{(B) } y = x^3 - x^2 + x + 2 \\qquad\\\\ \\text{(C) } y = x^2 + x + 1 \\qquad \\text{(D) } y = (x^2 + x + 1)(x - 1) \\qquad\\\\ \\text{(E) } \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the median of the following list of $4040$ numbers$?$\n\\[1, 2, 3, \\ldots, 2020, 1^2, 2^2, 3^2, \\ldots, 2020^2\\]\n$\\textbf{(A)}\\ 1974.5\\qquad\\textbf{(B)}\\ 1975.5\\qquad\\textbf{(C)}\\ 1976.5\\qquad\\textbf{(D)}\\ 1977.5\\qquad\\textbf{(E)}\\ 1978.5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_213", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the median of the following list of $4040$ numbers$?$\n\\[1, 2, 3, \\ldots, 2020, 1^2, 2^2, 3^2, \\ldots, 2020^2\\]\n$\\textbf{(A)}\\ 1974.5\\qquad\\textbf{(B)}\\ 1975.5\\qquad\\textbf{(C)}\\ 1976.5\\qquad\\textbf{(D)}\\ 1977.5\\qquad\\textbf{(E)}\\ 1978.5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is: \n$\\textbf{(A)}\\ n^2\\qquad\\textbf{(B)}\\ n(n+1)\\qquad\\textbf{(C)}\\ n(n+2)\\qquad\\textbf{(D)}\\ (n+1)^2\\qquad\\textbf{(E)}\\ (n+1)(n+2)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_214", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is: \n$\\textbf{(A)}\\ n^2\\qquad\\textbf{(B)}\\ n(n+1)\\qquad\\textbf{(C)}\\ n(n+2)\\qquad\\textbf{(D)}\\ (n+1)^2\\qquad\\textbf{(E)}\\ (n+1)(n+2)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $s_1, s_2, s_3, \\ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \\cdots.$ Suppose that $t_1, t_2, t_3, \\ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "r", + "index": "Sky-T1_10k_215", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $s_1, s_2, s_3, \\ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \\cdots.$ Suppose that $t_1, t_2, t_3, \\ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$." + } + }, + { + "question": "Return your final response within \\boxed{}. A straight line joins the points $(-1,1)$ and $(3,9)$. Its $x$-intercept is:\n$\\textbf{(A)}\\ -\\frac{3}{2}\\qquad \\textbf{(B)}\\ -\\frac{2}{3}\\qquad \\textbf{(C)}\\ \\frac{2}{5}\\qquad \\textbf{(D)}\\ 2\\qquad \\textbf{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_216", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A straight line joins the points $(-1,1)$ and $(3,9)$. Its $x$-intercept is:\n$\\textbf{(A)}\\ -\\frac{3}{2}\\qquad \\textbf{(B)}\\ -\\frac{2}{3}\\qquad \\textbf{(C)}\\ \\frac{2}{5}\\qquad \\textbf{(D)}\\ 2\\qquad \\textbf{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$ and $c$ be fixed [positive numbers](https://artofproblemsolving.com/wiki/index.php/Natural_numbers). For each [real number](https://artofproblemsolving.com/wiki/index.php/Real_number) $t$ let $(x_t, y_t)$ be the [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) of the [parabola](https://artofproblemsolving.com/wiki/index.php/Parabola) $y=ax^2+bx+c$. If the set of the vertices $(x_t, y_t)$ for all real numbers of $t$ is graphed on the [plane](https://artofproblemsolving.com/wiki/index.php/Cartesian_plane), the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) is\n$\\mathrm{(A) \\ } \\text{a straight line} \\qquad \\mathrm{(B) \\ } \\text{a parabola} \\qquad \\mathrm{(C) \\ } \\text{part, but not all, of a parabola} \\qquad \\mathrm{(D) \\ } \\text{one branch of a hyperbola} \\qquad$ $\\mathrm{(E) \\ } \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_217", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$ and $c$ be fixed [positive numbers](https://artofproblemsolving.com/wiki/index.php/Natural_numbers). For each [real number](https://artofproblemsolving.com/wiki/index.php/Real_number) $t$ let $(x_t, y_t)$ be the [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) of the [parabola](https://artofproblemsolving.com/wiki/index.php/Parabola) $y=ax^2+bx+c$. If the set of the vertices $(x_t, y_t)$ for all real numbers of $t$ is graphed on the [plane](https://artofproblemsolving.com/wiki/index.php/Cartesian_plane), the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) is\n$\\mathrm{(A) \\ } \\text{a straight line} \\qquad \\mathrm{(B) \\ } \\text{a parabola} \\qquad \\mathrm{(C) \\ } \\text{part, but not all, of a parabola} \\qquad \\mathrm{(D) \\ } \\text{one branch of a hyperbola} \\qquad$ $\\mathrm{(E) \\ } \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of all integers between 50 and 350 which end in 1 is \n $\\textbf{(A)}\\ 5880\\qquad\\textbf{(B)}\\ 5539\\qquad\\textbf{(C)}\\ 5208\\qquad\\textbf{(D)}\\ 4877\\qquad\\textbf{(E)}\\ 4566$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_218", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of all integers between 50 and 350 which end in 1 is \n $\\textbf{(A)}\\ 5880\\qquad\\textbf{(B)}\\ 5539\\qquad\\textbf{(C)}\\ 5208\\qquad\\textbf{(D)}\\ 4877\\qquad\\textbf{(E)}\\ 4566$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\log_6 x=2.5$, the value of $x$ is: \n$\\textbf{(A)}\\ 90 \\qquad \\textbf{(B)}\\ 36 \\qquad \\textbf{(C)}\\ 36\\sqrt{6} \\qquad \\textbf{(D)}\\ 0.5 \\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_219", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\log_6 x=2.5$, the value of $x$ is: \n$\\textbf{(A)}\\ 90 \\qquad \\textbf{(B)}\\ 36 \\qquad \\textbf{(C)}\\ 36\\sqrt{6} \\qquad \\textbf{(D)}\\ 0.5 \\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. \nThe area of $ABCD$ is $256$ square inches and the area of $\\triangle CEF$ is $200$ square inches. Then the number of inches in $BE$ is:\n\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 14 \\qquad \\textbf{(C)}\\ 15 \\qquad \\textbf{(D)}\\ 16 \\qquad \\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_220", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. \nThe area of $ABCD$ is $256$ square inches and the area of $\\triangle CEF$ is $200$ square inches. Then the number of inches in $BE$ is:\n\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 14 \\qquad \\textbf{(C)}\\ 15 \\qquad \\textbf{(D)}\\ 16 \\qquad \\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. At the end of $1994$, Walter was half as old as his grandmother. The sum of the years in which they were born was $3838$. How old will Walter be at the end of $1999$?\n$\\textbf{(A)}\\ 48 \\qquad \\textbf{(B)}\\ 49\\qquad \\textbf{(C)}\\ 53\\qquad \\textbf{(D)}\\ 55\\qquad \\textbf{(E)}\\ 101$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_221", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. At the end of $1994$, Walter was half as old as his grandmother. The sum of the years in which they were born was $3838$. How old will Walter be at the end of $1999$?\n$\\textbf{(A)}\\ 48 \\qquad \\textbf{(B)}\\ 49\\qquad \\textbf{(C)}\\ 53\\qquad \\textbf{(D)}\\ 55\\qquad \\textbf{(E)}\\ 101$" + } + }, + { + "question": "Return your final response within \\boxed{}. A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $3\\%$ to $2\\%$ is approximately: \n$\\textbf{(A)}\\ 10000\\text{ ft.}\\qquad\\textbf{(B)}\\ 20000\\text{ ft.}\\qquad\\textbf{(C)}\\ 30000\\text{ ft.}\\qquad\\textbf{(D)}\\ 12000\\text{ ft.}\\qquad\\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_222", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $3\\%$ to $2\\%$ is approximately: \n$\\textbf{(A)}\\ 10000\\text{ ft.}\\qquad\\textbf{(B)}\\ 20000\\text{ ft.}\\qquad\\textbf{(C)}\\ 30000\\text{ ft.}\\qquad\\textbf{(D)}\\ 12000\\text{ ft.}\\qquad\\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $A$ be the area of the triangle with sides of length $25, 25$, and $30$. Let $B$ be the area of the triangle with sides of length $25, 25,$ and $40$. What is the relationship between $A$ and $B$?\n$\\textbf{(A) } A = \\dfrac9{16}B \\qquad\\textbf{(B) } A = \\dfrac34B \\qquad\\textbf{(C) } A=B \\qquad \\textbf{(D) } A = \\dfrac43B \\qquad \\textbf{(E) }A = \\dfrac{16}9B$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_223", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $A$ be the area of the triangle with sides of length $25, 25$, and $30$. Let $B$ be the area of the triangle with sides of length $25, 25,$ and $40$. What is the relationship between $A$ and $B$?\n$\\textbf{(A) } A = \\dfrac9{16}B \\qquad\\textbf{(B) } A = \\dfrac34B \\qquad\\textbf{(C) } A=B \\qquad \\textbf{(D) } A = \\dfrac43B \\qquad \\textbf{(E) }A = \\dfrac{16}9B$" + } + }, + { + "question": "Return your final response within \\boxed{}. The product of $\\sqrt[3]{4}$ and $\\sqrt[4]{8}$ equals\n$\\textbf{(A) }\\sqrt[7]{12}\\qquad \\textbf{(B) }2\\sqrt[7]{12}\\qquad \\textbf{(C) }\\sqrt[7]{32}\\qquad \\textbf{(D) }\\sqrt[12]{32}\\qquad \\textbf{(E) }2\\sqrt[12]{32}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_224", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The product of $\\sqrt[3]{4}$ and $\\sqrt[4]{8}$ equals\n$\\textbf{(A) }\\sqrt[7]{12}\\qquad \\textbf{(B) }2\\sqrt[7]{12}\\qquad \\textbf{(C) }\\sqrt[7]{32}\\qquad \\textbf{(D) }\\sqrt[12]{32}\\qquad \\textbf{(E) }2\\sqrt[12]{32}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)\n$\\mathrm{(A)}\\ {{{\\frac{3} {8}}}} \\qquad \\mathrm{(B)}\\ {{{\\frac{1} {2}}}} \\qquad \\mathrm{(C)}\\ {{{\\frac{43} {72}}}} \\qquad \\mathrm{(D)}\\ {{{\\frac{5} {8}}}} \\qquad \\mathrm{(E)}\\ {{{\\frac{2} {3}}}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_225", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)\n$\\mathrm{(A)}\\ {{{\\frac{3} {8}}}} \\qquad \\mathrm{(B)}\\ {{{\\frac{1} {2}}}} \\qquad \\mathrm{(C)}\\ {{{\\frac{43} {72}}}} \\qquad \\mathrm{(D)}\\ {{{\\frac{5} {8}}}} \\qquad \\mathrm{(E)}\\ {{{\\frac{2} {3}}}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the figure, it is given that angle $C = 90^{\\circ}$, $\\overline{AD} = \\overline{DB}$, $DE \\perp AB$, $\\overline{AB} = 20$, and $\\overline{AC} = 12$. The area of quadrilateral $ADEC$ is: \n\n$\\textbf{(A)}\\ 75\\qquad\\textbf{(B)}\\ 58\\frac{1}{2}\\qquad\\textbf{(C)}\\ 48\\qquad\\textbf{(D)}\\ 37\\frac{1}{2}\\qquad\\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_226", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the figure, it is given that angle $C = 90^{\\circ}$, $\\overline{AD} = \\overline{DB}$, $DE \\perp AB$, $\\overline{AB} = 20$, and $\\overline{AC} = 12$. The area of quadrilateral $ADEC$ is: \n\n$\\textbf{(A)}\\ 75\\qquad\\textbf{(B)}\\ 58\\frac{1}{2}\\qquad\\textbf{(C)}\\ 48\\qquad\\textbf{(D)}\\ 37\\frac{1}{2}\\qquad\\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\, p \\,$ be an odd prime. The sequence $(a_n)_{n \\geq 0}$ is defined as follows: $\\, a_0 = 0,$ $a_1 = 1, \\, \\ldots, \\, a_{p-2} = p-2 \\,$ and, for all $\\, n \\geq p-1, \\,$ $\\, a_n \\,$ is the least positive integer that does not form an arithmetic sequence of length $\\, p \\,$ with any of the preceding terms. Prove that, for all $\\, n, \\,$ $\\, a_n \\,$ is the number obtained by writing $\\, n \\,$ in base $\\, p-1 \\,$ and reading the result in base $\\, p$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "a_n is the number obtained by writing n in base p-1 and reading it in base p.", + "index": "Sky-T1_10k_227", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\, p \\,$ be an odd prime. The sequence $(a_n)_{n \\geq 0}$ is defined as follows: $\\, a_0 = 0,$ $a_1 = 1, \\, \\ldots, \\, a_{p-2} = p-2 \\,$ and, for all $\\, n \\geq p-1, \\,$ $\\, a_n \\,$ is the least positive integer that does not form an arithmetic sequence of length $\\, p \\,$ with any of the preceding terms. Prove that, for all $\\, n, \\,$ $\\, a_n \\,$ is the number obtained by writing $\\, n \\,$ in base $\\, p-1 \\,$ and reading the result in base $\\, p$." + } + }, + { + "question": "Return your final response within \\boxed{}. The polynomial $(x+y)^9$ is expanded in decreasing powers of $x$. The second and third terms have equal values \nwhen evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is one. What is the value of $p$? \n$\\textbf{(A)}\\ 1/5 \\qquad \\textbf{(B)}\\ 4/5 \\qquad \\textbf{(C)}\\ 1/4 \\qquad \\textbf{(D)}\\ 3/4 \\qquad \\textbf{(E)}\\ 8/9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_228", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The polynomial $(x+y)^9$ is expanded in decreasing powers of $x$. The second and third terms have equal values \nwhen evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is one. What is the value of $p$? \n$\\textbf{(A)}\\ 1/5 \\qquad \\textbf{(B)}\\ 4/5 \\qquad \\textbf{(C)}\\ 1/4 \\qquad \\textbf{(D)}\\ 3/4 \\qquad \\textbf{(E)}\\ 8/9$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$?\n$\\textbf{(A)}\\ 16\\qquad\\textbf{(B)}\\ 17\\qquad\\textbf{(C)}\\ 18\\qquad\\textbf{(D)}\\ 19\\qquad\\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_229", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$?\n$\\textbf{(A)}\\ 16\\qquad\\textbf{(B)}\\ 17\\qquad\\textbf{(C)}\\ 18\\qquad\\textbf{(D)}\\ 19\\qquad\\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer (1,000 meters)?\n$\\textbf{(A)}\\ 9^\\text{th} \\qquad \\textbf{(B)}\\ 10^\\text{th} \\qquad \\textbf{(C)}\\ 11^\\text{th} \\qquad \\textbf{(D)}\\ 12^\\text{th} \\qquad \\textbf{(E)}\\ 13^\\text{th}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_230", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer (1,000 meters)?\n$\\textbf{(A)}\\ 9^\\text{th} \\qquad \\textbf{(B)}\\ 10^\\text{th} \\qquad \\textbf{(C)}\\ 11^\\text{th} \\qquad \\textbf{(D)}\\ 12^\\text{th} \\qquad \\textbf{(E)}\\ 13^\\text{th}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following numbers is the largest?\n$\\text{(A)}\\ .99 \\qquad \\text{(B)}\\ .9099 \\qquad \\text{(C)}\\ .9 \\qquad \\text{(D)}\\ .909 \\qquad \\text{(E)}\\ .9009$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_231", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following numbers is the largest?\n$\\text{(A)}\\ .99 \\qquad \\text{(B)}\\ .9099 \\qquad \\text{(C)}\\ .9 \\qquad \\text{(D)}\\ .909 \\qquad \\text{(E)}\\ .9009$" + } + }, + { + "question": "Return your final response within \\boxed{}. When three different numbers from the set $\\{ -3, -2, -1, 4, 5 \\}$ are multiplied, the largest possible product is \n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 20 \\qquad \\text{(C)}\\ 30 \\qquad \\text{(D)}\\ 40 \\qquad \\text{(E)}\\ 60$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_232", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When three different numbers from the set $\\{ -3, -2, -1, 4, 5 \\}$ are multiplied, the largest possible product is \n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 20 \\qquad \\text{(C)}\\ 30 \\qquad \\text{(D)}\\ 40 \\qquad \\text{(E)}\\ 60$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three times Dick's age plus Tom's age equals twice Harry's age. \nDouble the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. \nTheir respective ages are relatively prime to each other. The sum of the squares of their ages is\n$\\textbf{(A) }42\\qquad \\textbf{(B) }46\\qquad \\textbf{(C) }122\\qquad \\textbf{(D) }290\\qquad \\textbf{(E) }326$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_233", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three times Dick's age plus Tom's age equals twice Harry's age. \nDouble the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. \nTheir respective ages are relatively prime to each other. The sum of the squares of their ages is\n$\\textbf{(A) }42\\qquad \\textbf{(B) }46\\qquad \\textbf{(C) }122\\qquad \\textbf{(D) }290\\qquad \\textbf{(E) }326$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many unordered pairs of edges of a given cube determine a plane?\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 28 \\qquad \\textbf{(C) } 36\\qquad \\textbf{(D) } 42 \\qquad \\textbf{(E) } 66$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_234", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many unordered pairs of edges of a given cube determine a plane?\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 28 \\qquad \\textbf{(C) } 36\\qquad \\textbf{(D) } 42 \\qquad \\textbf{(E) } 66$" + } + }, + { + "question": "Return your final response within \\boxed{}. Real numbers $x$, $y$, and $z$ satisfy the inequalities\n$0 \\frac {1}{4} \\left( 1 + \\frac {1}{2} + \\cdots + \\frac {1}{n} \\right).\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\sum_{j=1}^n a_j^2 > \\frac{1}{4} \\left( 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n} \\right)", + "index": "Sky-T1_10k_262", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\, a_1, a_2, a_3, \\ldots \\,$ be a sequence of positive real numbers satisfying $\\, \\sum_{j = 1}^n a_j \\geq \\sqrt {n} \\,$ for all $\\, n \\geq 1$. Prove that, for all $\\, n \\geq 1, \\,$\n\\[\\sum_{j = 1}^n a_j^2 > \\frac {1}{4} \\left( 1 + \\frac {1}{2} + \\cdots + \\frac {1}{n} \\right).\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?\n$\\textbf{(A) }\\frac{1}{21}\\qquad\\textbf{(B) }\\frac{1}{14}\\qquad\\textbf{(C) }\\frac{5}{63}\\qquad\\textbf{(D) }\\frac{2}{21}\\qquad\\textbf{(E) }\\frac{1}{7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_263", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?\n$\\textbf{(A) }\\frac{1}{21}\\qquad\\textbf{(B) }\\frac{1}{14}\\qquad\\textbf{(C) }\\frac{5}{63}\\qquad\\textbf{(D) }\\frac{2}{21}\\qquad\\textbf{(E) }\\frac{1}{7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Karl bought five folders from Pay-A-Lot at a cost of $\\textdollar 2.50$ each.\nPay-A-Lot had a 20%-off sale the following day. How much could\nKarl have saved on the purchase by waiting a day?\n$\\textbf{(A)}\\ \\textdollar 1.00 \\qquad\\textbf{(B)}\\ \\textdollar 2.00 \\qquad\\textbf{(C)}\\ \\textdollar 2.50\\qquad\\textbf{(D)}\\ \\textdollar 2.75 \\qquad\\textbf{(E)}\\ \\textdollar 5.00$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_264", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Karl bought five folders from Pay-A-Lot at a cost of $\\textdollar 2.50$ each.\nPay-A-Lot had a 20%-off sale the following day. How much could\nKarl have saved on the purchase by waiting a day?\n$\\textbf{(A)}\\ \\textdollar 1.00 \\qquad\\textbf{(B)}\\ \\textdollar 2.00 \\qquad\\textbf{(C)}\\ \\textdollar 2.50\\qquad\\textbf{(D)}\\ \\textdollar 2.75 \\qquad\\textbf{(E)}\\ \\textdollar 5.00$" + } + }, + { + "question": "Return your final response within \\boxed{}. Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\\sin(2\\angle BAD)$?\n\n$\\textbf{(A) } \\dfrac{1}{3} \\qquad\\textbf{(B) } \\dfrac{\\sqrt{2}}{2} \\qquad\\textbf{(C) } \\dfrac{3}{4} \\qquad\\textbf{(D) } \\dfrac{7}{9} \\qquad\\textbf{(E) } \\dfrac{\\sqrt{3}}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_265", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\\sin(2\\angle BAD)$?\n\n$\\textbf{(A) } \\dfrac{1}{3} \\qquad\\textbf{(B) } \\dfrac{\\sqrt{2}}{2} \\qquad\\textbf{(C) } \\dfrac{3}{4} \\qquad\\textbf{(D) } \\dfrac{7}{9} \\qquad\\textbf{(E) } \\dfrac{\\sqrt{3}}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many positive integers $n$ satisfy the following condition:\n$(130n)^{50} > n^{100} > 2^{200}\\ ?$\n$\\textbf{(A) } 0\\qquad \\textbf{(B) } 7\\qquad \\textbf{(C) } 12\\qquad \\textbf{(D) } 65\\qquad \\textbf{(E) } 125$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_266", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many positive integers $n$ satisfy the following condition:\n$(130n)^{50} > n^{100} > 2^{200}\\ ?$\n$\\textbf{(A) } 0\\qquad \\textbf{(B) } 7\\qquad \\textbf{(C) } 12\\qquad \\textbf{(D) } 65\\qquad \\textbf{(E) } 125$" + } + }, + { + "question": "Return your final response within \\boxed{}. Real numbers $x$, $y$, and $z$ satisfy the inequalities\n$00$, and let $S=(1,x,x^2,\\ldots ,x^{100})$. If $A^{100}(S)=(1/2^{50})$, then what is $x$?\n$\\mathrm{(A) \\ } 1-\\frac{\\sqrt{2}}{2}\\qquad \\mathrm{(B) \\ } \\sqrt{2}-1\\qquad \\mathrm{(C) \\ } \\frac{1}{2}\\qquad \\mathrm{(D) \\ } 2-\\sqrt{2}\\qquad \\mathrm{(E) \\ } \\frac{\\sqrt{2}}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_272", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given a finite sequence $S=(a_1,a_2,\\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence \n$\\left(\\frac{a_1+a_2}{2},\\frac{a_2+a_3}{2},\\ldots ,\\frac{a_{n-1}+a_n}{2}\\right)$\nof $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\\le m\\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(1,x,x^2,\\ldots ,x^{100})$. If $A^{100}(S)=(1/2^{50})$, then what is $x$?\n$\\mathrm{(A) \\ } 1-\\frac{\\sqrt{2}}{2}\\qquad \\mathrm{(B) \\ } \\sqrt{2}-1\\qquad \\mathrm{(C) \\ } \\frac{1}{2}\\qquad \\mathrm{(D) \\ } 2-\\sqrt{2}\\qquad \\mathrm{(E) \\ } \\frac{\\sqrt{2}}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$?\n$\\textbf{(A)}\\ \\frac{3}{2} \\qquad\\textbf{(B)}\\ \\frac{90-40\\sqrt{3}}{11} \\qquad\\textbf{(C)}\\ 2 \\qquad\\textbf{(D)}\\ \\frac{144-25\\sqrt{3}}{44} \\qquad\\textbf{(E)}\\ \\frac{5}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_273", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$?\n$\\textbf{(A)}\\ \\frac{3}{2} \\qquad\\textbf{(B)}\\ \\frac{90-40\\sqrt{3}}{11} \\qquad\\textbf{(C)}\\ 2 \\qquad\\textbf{(D)}\\ \\frac{144-25\\sqrt{3}}{44} \\qquad\\textbf{(E)}\\ \\frac{5}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A circle is inscribed in a triangle with side lengths $8, 13$, and $17$. Let the segments of the side of length $8$, \nmade by a point of tangency, be $r$ and $s$, with $r0,M \\ne 1, N \\ne 1$, then $MN$ equals:\n$\\text{(A) } \\frac{1}{2} \\quad \\text{(B) } 1 \\quad \\text{(C) } 2 \\quad \\text{(D) } 10 \\\\ \\text{(E) a number greater than 2 and less than 10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_289", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $Log_M{N}=Log_N{M},M \\ne N,MN>0,M \\ne 1, N \\ne 1$, then $MN$ equals:\n$\\text{(A) } \\frac{1}{2} \\quad \\text{(B) } 1 \\quad \\text{(C) } 2 \\quad \\text{(D) } 10 \\\\ \\text{(E) a number greater than 2 and less than 10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a, a', b,$ and $b'$ be real numbers with $a$ and $a'$ nonzero. The solution to $ax+b=0$ is less than the solution to $a'x+b'=0$ if and only if \n$\\mathrm{(A)\\ } a'b0,t \\ne 1$, a relation between $x$ and $y$ is: \n$\\text{(A) } y^x=x^{1/y}\\quad \\text{(B) } y^{1/x}=x^{y}\\quad \\text{(C) } y^x=x^y\\quad \\text{(D) } x^x=y^y\\quad \\text{(E) none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_317", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x=t^{1/(t-1)}$ and $y=t^{t/(t-1)},t>0,t \\ne 1$, a relation between $x$ and $y$ is: \n$\\text{(A) } y^x=x^{1/y}\\quad \\text{(B) } y^{1/x}=x^{y}\\quad \\text{(C) } y^x=x^y\\quad \\text{(D) } x^x=y^y\\quad \\text{(E) none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Moe uses a mower to cut his rectangular $90$-foot by $150$-foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn?\n$\\textbf{(A) } 0.75 \\qquad\\textbf{(B) } 0.8 \\qquad\\textbf{(C) } 1.35 \\qquad\\textbf{(D) } 1.5 \\qquad\\textbf{(E) } 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_318", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Moe uses a mower to cut his rectangular $90$-foot by $150$-foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn?\n$\\textbf{(A) } 0.75 \\qquad\\textbf{(B) } 0.8 \\qquad\\textbf{(C) } 1.35 \\qquad\\textbf{(D) } 1.5 \\qquad\\textbf{(E) } 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 22\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 25\\qquad\\textbf{(E)}\\ 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_319", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 22\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 25\\qquad\\textbf{(E)}\\ 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. What is the value of $\\frac{A}{B}$?\n$\\textbf{(A)}\\ \\frac{64}{225} \\qquad \\textbf{(B)}\\ \\frac{8}{15} \\qquad \\textbf{(C)}\\ 1 \\qquad \\textbf{(D)}\\ \\frac{15}{8} \\qquad \\textbf{(E)}\\ \\frac{225}{64}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_320", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. What is the value of $\\frac{A}{B}$?\n$\\textbf{(A)}\\ \\frac{64}{225} \\qquad \\textbf{(B)}\\ \\frac{8}{15} \\qquad \\textbf{(C)}\\ 1 \\qquad \\textbf{(D)}\\ \\frac{15}{8} \\qquad \\textbf{(E)}\\ \\frac{225}{64}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Liliane has $50\\%$ more soda than Jacqueline, and Alice has $25\\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?\n$\\textbf{(A) }$ Liliane has $20\\%$ more soda than Alice.\n$\\textbf{(B) }$ Liliane has $25\\%$ more soda than Alice.\n$\\textbf{(C) }$ Liliane has $45\\%$ more soda than Alice.\n$\\textbf{(D) }$ Liliane has $75\\%$ more soda than Alice.\n$\\textbf{(E) }$ Liliane has $100\\%$ more soda than Alice.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_321", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Liliane has $50\\%$ more soda than Jacqueline, and Alice has $25\\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?\n$\\textbf{(A) }$ Liliane has $20\\%$ more soda than Alice.\n$\\textbf{(B) }$ Liliane has $25\\%$ more soda than Alice.\n$\\textbf{(C) }$ Liliane has $45\\%$ more soda than Alice.\n$\\textbf{(D) }$ Liliane has $75\\%$ more soda than Alice.\n$\\textbf{(E) }$ Liliane has $100\\%$ more soda than Alice." + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\left (a+\\frac{1}{a} \\right )^2=3$, then $a^3+\\frac{1}{a^3}$ equals: \n$\\textbf{(A)}\\ \\frac{10\\sqrt{3}}{3}\\qquad\\textbf{(B)}\\ 3\\sqrt{3}\\qquad\\textbf{(C)}\\ 0\\qquad\\textbf{(D)}\\ 7\\sqrt{7}\\qquad\\textbf{(E)}\\ 6\\sqrt{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_322", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\left (a+\\frac{1}{a} \\right )^2=3$, then $a^3+\\frac{1}{a^3}$ equals: \n$\\textbf{(A)}\\ \\frac{10\\sqrt{3}}{3}\\qquad\\textbf{(B)}\\ 3\\sqrt{3}\\qquad\\textbf{(C)}\\ 0\\qquad\\textbf{(D)}\\ 7\\sqrt{7}\\qquad\\textbf{(E)}\\ 6\\sqrt{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the $100\\text{th}$ number in the [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence): $1,5,9,13,17,21,25,...$?\n$\\text{(A)}\\ 397 \\qquad \\text{(B)}\\ 399 \\qquad \\text{(C)}\\ 401 \\qquad \\text{(D)}\\ 403 \\qquad \\text{(E)}\\ 405$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_323", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the $100\\text{th}$ number in the [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence): $1,5,9,13,17,21,25,...$?\n$\\text{(A)}\\ 397 \\qquad \\text{(B)}\\ 399 \\qquad \\text{(C)}\\ 401 \\qquad \\text{(D)}\\ 403 \\qquad \\text{(E)}\\ 405$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many positive integers $m$ is \n\\[\\frac{2002}{m^2 -2}\\]\na positive integer?\n$\\text{(A) one} \\qquad \\text{(B) two} \\qquad \\text{(C) three} \\qquad \\text{(D) four} \\qquad \\text{(E) more than four}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_324", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many positive integers $m$ is \n\\[\\frac{2002}{m^2 -2}\\]\na positive integer?\n$\\text{(A) one} \\qquad \\text{(B) two} \\qquad \\text{(C) three} \\qquad \\text{(D) four} \\qquad \\text{(E) more than four}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$\n\\[\\begin{tabular}{|c|c|c|}\\hline 1 & &\\\\ \\hline & 2 & A\\\\ \\hline & & B\\\\ \\hline\\end{tabular}\\]\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_325", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$\n\\[\\begin{tabular}{|c|c|c|}\\hline 1 & &\\\\ \\hline & 2 & A\\\\ \\hline & & B\\\\ \\hline\\end{tabular}\\]\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\\leq m\\leq 2012$ and \\[5^n<2^m<2^{m+2}<5^{n+1}?\\]\n$\\textbf{(A) }278\\qquad \\textbf{(B) }279\\qquad \\textbf{(C) }280\\qquad \\textbf{(D) }281\\qquad \\textbf{(E) }282\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_326", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\\leq m\\leq 2012$ and \\[5^n<2^m<2^{m+2}<5^{n+1}?\\]\n$\\textbf{(A) }278\\qquad \\textbf{(B) }279\\qquad \\textbf{(C) }280\\qquad \\textbf{(D) }281\\qquad \\textbf{(E) }282\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Evaluate $(x^x)^{(x^x)}$ at $x = 2$. \n$\\text{(A)} \\ 16 \\qquad \\text{(B)} \\ 64 \\qquad \\text{(C)} \\ 256 \\qquad \\text{(D)} \\ 1024 \\qquad \\text{(E)} \\ 65,536$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_327", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Evaluate $(x^x)^{(x^x)}$ at $x = 2$. \n$\\text{(A)} \\ 16 \\qquad \\text{(B)} \\ 64 \\qquad \\text{(C)} \\ 256 \\qquad \\text{(D)} \\ 1024 \\qquad \\text{(E)} \\ 65,536$" + } + }, + { + "question": "Return your final response within \\boxed{}. Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number?\n$\\text{(A)}\\ \\dfrac{1}{3} \\qquad \\text{(B)}\\ \\dfrac{5}{12} \\qquad \\text{(C)}\\ \\dfrac{4}{9} \\qquad \\text{(D)}\\ \\dfrac{17}{36} \\qquad \\text{(E)}\\ \\dfrac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_328", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number?\n$\\text{(A)}\\ \\dfrac{1}{3} \\qquad \\text{(B)}\\ \\dfrac{5}{12} \\qquad \\text{(C)}\\ \\dfrac{4}{9} \\qquad \\text{(D)}\\ \\dfrac{17}{36} \\qquad \\text{(E)}\\ \\dfrac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the correct ordering of the three numbers $\\frac{5}{19}$, $\\frac{7}{21}$, and $\\frac{9}{23}$, in increasing order?\n$\\textbf{(A)}\\hspace{.05in}\\frac{9}{23}<\\frac{7}{21}<\\frac{5}{19}\\quad\\textbf{(B)}\\hspace{.05in}\\frac{5}{19}<\\frac{7}{21}<\\frac{9}{23}\\quad\\textbf{(C)}\\hspace{.05in}\\frac{9}{23}<\\frac{5}{19}<\\frac{7}{21}$\n$\\textbf{(D)}\\hspace{.05in}\\frac{5}{19}<\\frac{9}{23}<\\frac{7}{21}\\quad\\textbf{(E)}\\hspace{.05in}\\frac{7}{21}<\\frac{5}{19}<\\frac{9}{23}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_329", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the correct ordering of the three numbers $\\frac{5}{19}$, $\\frac{7}{21}$, and $\\frac{9}{23}$, in increasing order?\n$\\textbf{(A)}\\hspace{.05in}\\frac{9}{23}<\\frac{7}{21}<\\frac{5}{19}\\quad\\textbf{(B)}\\hspace{.05in}\\frac{5}{19}<\\frac{7}{21}<\\frac{9}{23}\\quad\\textbf{(C)}\\hspace{.05in}\\frac{9}{23}<\\frac{5}{19}<\\frac{7}{21}$\n$\\textbf{(D)}\\hspace{.05in}\\frac{5}{19}<\\frac{9}{23}<\\frac{7}{21}\\quad\\textbf{(E)}\\hspace{.05in}\\frac{7}{21}<\\frac{5}{19}<\\frac{9}{23}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The set of solutions of the equation $\\log_{10}\\left( a^2-15a\\right)=2$ consists of\n$\\textbf{(A)}\\ \\text{two integers } \\qquad\\textbf{(B)}\\ \\text{one integer and one fraction}\\qquad \\textbf{(C)}\\ \\text{two irrational numbers }\\qquad\\textbf{(D)}\\ \\text{two non-real numbers} \\qquad\\textbf{(E)}\\ \\text{no numbers, that is, the empty set}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_330", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The set of solutions of the equation $\\log_{10}\\left( a^2-15a\\right)=2$ consists of\n$\\textbf{(A)}\\ \\text{two integers } \\qquad\\textbf{(B)}\\ \\text{one integer and one fraction}\\qquad \\textbf{(C)}\\ \\text{two irrational numbers }\\qquad\\textbf{(D)}\\ \\text{two non-real numbers} \\qquad\\textbf{(E)}\\ \\text{no numbers, that is, the empty set}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is\n\n$\\text{(A)}\\ 75 \\qquad \\text{(B)}\\ 76 \\qquad \\text{(C)}\\ 78 \\qquad \\text{(D)}\\ 80 \\qquad \\text{(E)}\\ 81$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_331", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is\n\n$\\text{(A)}\\ 75 \\qquad \\text{(B)}\\ 76 \\qquad \\text{(C)}\\ 78 \\qquad \\text{(D)}\\ 80 \\qquad \\text{(E)}\\ 81$" + } + }, + { + "question": "Return your final response within \\boxed{}. Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?\n$\\mathrm{(A)}\\ 10\\qquad \\mathrm{(B)}\\ 12\\qquad \\mathrm{(C)}\\ 15\\qquad \\mathrm{(D)}\\ 18\\qquad \\mathrm{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_332", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?\n$\\mathrm{(A)}\\ 10\\qquad \\mathrm{(B)}\\ 12\\qquad \\mathrm{(C)}\\ 15\\qquad \\mathrm{(D)}\\ 18\\qquad \\mathrm{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?\n$\\textbf{(A)}\\ 25\\qquad\\textbf{(B)}\\ 40\\qquad\\textbf{(C)}\\ 64\\qquad\\textbf{(D)}\\ 100\\qquad\\textbf{(E)}\\ 160$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_333", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?\n$\\textbf{(A)}\\ 25\\qquad\\textbf{(B)}\\ 40\\qquad\\textbf{(C)}\\ 64\\qquad\\textbf{(D)}\\ 100\\qquad\\textbf{(E)}\\ 160$" + } + }, + { + "question": "Return your final response within \\boxed{}. The vertices of $\\triangle ABC$ are $A = (0,0)\\,$, $B = (0,420)\\,$, and $C = (560,0)\\,$. The six faces of a die are labeled with two $A\\,$'s, two $B\\,$'s, and two $C\\,$'s. Point $P_1 = (k,m)\\,$ is chosen in the interior of $\\triangle ABC$, and points $P_2\\,$, $P_3\\,$, $P_4, \\dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L\\,$, where $L \\in \\{A, B, C\\}$, and $P_n\\,$ is the most recently obtained point, then $P_{n + 1}^{}$ is the midpoint of $\\overline{P_n L}$. Given that $P_7 = (14,92)\\,$, what is $k + m\\,$?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "344", + "index": "Sky-T1_10k_334", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The vertices of $\\triangle ABC$ are $A = (0,0)\\,$, $B = (0,420)\\,$, and $C = (560,0)\\,$. The six faces of a die are labeled with two $A\\,$'s, two $B\\,$'s, and two $C\\,$'s. Point $P_1 = (k,m)\\,$ is chosen in the interior of $\\triangle ABC$, and points $P_2\\,$, $P_3\\,$, $P_4, \\dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L\\,$, where $L \\in \\{A, B, C\\}$, and $P_n\\,$ is the most recently obtained point, then $P_{n + 1}^{}$ is the midpoint of $\\overline{P_n L}$. Given that $P_7 = (14,92)\\,$, what is $k + m\\,$?" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\\log_x w = 24$, $\\log_y w = 40$ and $\\log_{xyz} w = 12$. Find $\\log_z w$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "60", + "index": "Sky-T1_10k_335", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\\log_x w = 24$, $\\log_y w = 40$ and $\\log_{xyz} w = 12$. Find $\\log_z w$." + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{1000^2}{252^2-248^2}$ equals\n$\\mathrm{(A) \\ }62,500 \\qquad \\mathrm{(B) \\ }1,000 \\qquad \\mathrm{(C) \\ } 500\\qquad \\mathrm{(D) \\ }250 \\qquad \\mathrm{(E) \\ } \\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_336", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{1000^2}{252^2-248^2}$ equals\n$\\mathrm{(A) \\ }62,500 \\qquad \\mathrm{(B) \\ }1,000 \\qquad \\mathrm{(C) \\ } 500\\qquad \\mathrm{(D) \\ }250 \\qquad \\mathrm{(E) \\ } \\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 13 \\qquad \\textbf{(C)}\\ 14 \\qquad \\textbf{(D)}\\ 15 \\qquad \\textbf{(E)} 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_337", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 13 \\qquad \\textbf{(C)}\\ 14 \\qquad \\textbf{(D)}\\ 15 \\qquad \\textbf{(E)} 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit?\n\n$\\textbf{(A)}\\ 52 \\qquad\\textbf{(B)}\\ 60 \\qquad\\textbf{(C)}\\ 66 \\qquad\\textbf{(D)}\\ 68 \\qquad\\textbf{(E)}\\ 70$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_338", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit?\n\n$\\textbf{(A)}\\ 52 \\qquad\\textbf{(B)}\\ 60 \\qquad\\textbf{(C)}\\ 66 \\qquad\\textbf{(D)}\\ 68 \\qquad\\textbf{(E)}\\ 70$" + } + }, + { + "question": "Return your final response within \\boxed{}. A fair $6$ sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?\n$\\textbf{(A) }\\dfrac16\\qquad\\textbf{(B) }\\dfrac5{12}\\qquad\\textbf{(C) }\\dfrac12\\qquad\\textbf{(D) }\\dfrac7{12}\\qquad\\textbf{(E) }\\dfrac56$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_339", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A fair $6$ sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?\n$\\textbf{(A) }\\dfrac16\\qquad\\textbf{(B) }\\dfrac5{12}\\qquad\\textbf{(C) }\\dfrac12\\qquad\\textbf{(D) }\\dfrac7{12}\\qquad\\textbf{(E) }\\dfrac56$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\\frac{bc}{ad}?$\n$\\textbf{(A) } 6 \\qquad\\textbf{(B) } 19 \\qquad\\textbf{(C) } 24 \\qquad\\textbf{(D) } 26 \\qquad\\textbf{(E) } 38$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_340", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\\frac{bc}{ad}?$\n$\\textbf{(A) } 6 \\qquad\\textbf{(B) } 19 \\qquad\\textbf{(C) } 24 \\qquad\\textbf{(D) } 26 \\qquad\\textbf{(E) } 38$" + } + }, + { + "question": "Return your final response within \\boxed{}. Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at a point $X$ inside the square. How far is $X$ from the side of $CD$?\n$\\text{(A) } \\tfrac{1}{2} s(\\sqrt{3}+4)\\quad \\text{(B) } \\tfrac{1}{2} s\\sqrt{3}\\quad \\text{(C) } \\tfrac{1}{2} s(1+\\sqrt{3})\\quad \\text{(D) } \\tfrac{1}{2} s(\\sqrt{3}-1)\\quad \\text{(E) } \\tfrac{1}{2} s(2-\\sqrt{3})$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_341", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at a point $X$ inside the square. How far is $X$ from the side of $CD$?\n$\\text{(A) } \\tfrac{1}{2} s(\\sqrt{3}+4)\\quad \\text{(B) } \\tfrac{1}{2} s\\sqrt{3}\\quad \\text{(C) } \\tfrac{1}{2} s(1+\\sqrt{3})\\quad \\text{(D) } \\tfrac{1}{2} s(\\sqrt{3}-1)\\quad \\text{(E) } \\tfrac{1}{2} s(2-\\sqrt{3})$" + } + }, + { + "question": "Return your final response within \\boxed{}. Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\\sqrt 2$, and $\\angle BAC = 45^\\circ$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers?\n$\\textbf{(A)}\\ 30\\ \\text{and}\\ 31 \\qquad\\textbf{(B)}\\ 31\\ \\text{and}\\ 32 \\qquad\\textbf{(C)}\\ 32\\ \\text{and}\\ 33 \\qquad\\textbf{(D)}\\ 33\\ \\text{and}\\ 34 \\qquad\\textbf{(E)}\\ 34\\ \\text{and}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_342", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\\sqrt 2$, and $\\angle BAC = 45^\\circ$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers?\n$\\textbf{(A)}\\ 30\\ \\text{and}\\ 31 \\qquad\\textbf{(B)}\\ 31\\ \\text{and}\\ 32 \\qquad\\textbf{(C)}\\ 32\\ \\text{and}\\ 33 \\qquad\\textbf{(D)}\\ 33\\ \\text{and}\\ 34 \\qquad\\textbf{(E)}\\ 34\\ \\text{and}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $1$; $4$; $\\ldots$ and $9$; $16$; $\\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2004$ terms of each sequence. How many distinct numbers are in $S$?\n$\\mathrm{(A) \\ } 3722 \\qquad \\mathrm{(B) \\ } 3732 \\qquad \\mathrm{(C) \\ } 3914 \\qquad \\mathrm{(D) \\ } 3924 \\qquad \\mathrm{(E) \\ } 4007$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_343", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $1$; $4$; $\\ldots$ and $9$; $16$; $\\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2004$ terms of each sequence. How many distinct numbers are in $S$?\n$\\mathrm{(A) \\ } 3722 \\qquad \\mathrm{(B) \\ } 3732 \\qquad \\mathrm{(C) \\ } 3914 \\qquad \\mathrm{(D) \\ } 3924 \\qquad \\mathrm{(E) \\ } 4007$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is \n$\\mathrm{(A) \\ 10 } \\qquad \\mathrm{(B) \\ 12 } \\qquad \\mathrm{(C) \\ 14 } \\qquad \\mathrm{(D) \\ 16 } \\qquad \\mathrm{(E) \\ 18 }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_344", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is \n$\\mathrm{(A) \\ 10 } \\qquad \\mathrm{(B) \\ 12 } \\qquad \\mathrm{(C) \\ 14 } \\qquad \\mathrm{(D) \\ 16 } \\qquad \\mathrm{(E) \\ 18 }$" + } + }, + { + "question": "Return your final response within \\boxed{}. The first three terms of a geometric progression are $\\sqrt{2}, \\sqrt[3]{2}, \\sqrt[6]{2}$. Find the fourth term.\n$\\textbf{(A)}\\ 1\\qquad \\textbf{(B)}\\ \\sqrt[7]{2}\\qquad \\textbf{(C)}\\ \\sqrt[8]{2}\\qquad \\textbf{(D)}\\ \\sqrt[9]{2}\\qquad \\textbf{(E)}\\ \\sqrt[10]{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_345", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The first three terms of a geometric progression are $\\sqrt{2}, \\sqrt[3]{2}, \\sqrt[6]{2}$. Find the fourth term.\n$\\textbf{(A)}\\ 1\\qquad \\textbf{(B)}\\ \\sqrt[7]{2}\\qquad \\textbf{(C)}\\ \\sqrt[8]{2}\\qquad \\textbf{(D)}\\ \\sqrt[9]{2}\\qquad \\textbf{(E)}\\ \\sqrt[10]{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?\n$\\textbf{(A)}\\ \\frac{bde}{ac}\\qquad\\textbf{(B)}\\ \\frac{ac}{bde}\\qquad\\textbf{(C)}\\ \\frac{abde}{c}\\qquad\\textbf{(D)}\\ \\frac{bcde}{a}\\qquad\\textbf{(E)}\\ \\frac{abc}{de}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_346", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?\n$\\textbf{(A)}\\ \\frac{bde}{ac}\\qquad\\textbf{(B)}\\ \\frac{ac}{bde}\\qquad\\textbf{(C)}\\ \\frac{abde}{c}\\qquad\\textbf{(D)}\\ \\frac{bcde}{a}\\qquad\\textbf{(E)}\\ \\frac{abc}{de}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ratio of the areas of two concentric circles is $1: 3$. If the radius of the smaller is $r$, then the difference between the \nradii is best approximated by: \n$\\textbf{(A)}\\ 0.41r \\qquad \\textbf{(B)}\\ 0.73 \\qquad \\textbf{(C)}\\ 0.75 \\qquad \\textbf{(D)}\\ 0.73r \\qquad \\textbf{(E)}\\ 0.75r$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_347", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ratio of the areas of two concentric circles is $1: 3$. If the radius of the smaller is $r$, then the difference between the \nradii is best approximated by: \n$\\textbf{(A)}\\ 0.41r \\qquad \\textbf{(B)}\\ 0.73 \\qquad \\textbf{(C)}\\ 0.75 \\qquad \\textbf{(D)}\\ 0.73r \\qquad \\textbf{(E)}\\ 0.75r$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter\n$AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto\nlines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle\nformed by lines $PQ$ and $RS$ is half the size of $\\angle XOZ$, where\n$O$ is the midpoint of segment $AB$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The acute angle between PQ and RS is half the measure of \\angle XOZ.", + "index": "Sky-T1_10k_348", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter\n$AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto\nlines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle\nformed by lines $PQ$ and $RS$ is half the size of $\\angle XOZ$, where\n$O$ is the midpoint of segment $AB$." + } + }, + { + "question": "Return your final response within \\boxed{}. The solution of the equations \n\\begin{align*}2x-3y &=7 \\\\ 4x-6y &=20\\end{align*}\nis: \n$\\textbf{(A)}\\ x=18, y=12 \\qquad \\textbf{(B)}\\ x=0, y=0 \\qquad \\textbf{(C)}\\ \\text{There is no solution} \\\\ \\textbf{(D)}\\ \\text{There are an unlimited number of solutions}\\qquad \\textbf{(E)}\\ x=8, y=5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_349", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The solution of the equations \n\\begin{align*}2x-3y &=7 \\\\ 4x-6y &=20\\end{align*}\nis: \n$\\textbf{(A)}\\ x=18, y=12 \\qquad \\textbf{(B)}\\ x=0, y=0 \\qquad \\textbf{(C)}\\ \\text{There is no solution} \\\\ \\textbf{(D)}\\ \\text{There are an unlimited number of solutions}\\qquad \\textbf{(E)}\\ x=8, y=5$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many ways are there to paint each of the integers $2, 3, \\cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors?\n$\\textbf{(A)}~144\\qquad\\textbf{(B)}~216\\qquad\\textbf{(C)}~256\\qquad\\textbf{(D)}~384\\qquad\\textbf{(E)}~432$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_350", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many ways are there to paint each of the integers $2, 3, \\cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors?\n$\\textbf{(A)}~144\\qquad\\textbf{(B)}~216\\qquad\\textbf{(C)}~256\\qquad\\textbf{(D)}~384\\qquad\\textbf{(E)}~432$" + } + }, + { + "question": "Return your final response within \\boxed{}. One of the factors of $x^4+4$ is: \n$\\textbf{(A)}\\ x^2+2 \\qquad \\textbf{(B)}\\ x+1 \\qquad \\textbf{(C)}\\ x^2-2x+2 \\qquad \\textbf{(D)}\\ x^2-4\\\\ \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_351", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One of the factors of $x^4+4$ is: \n$\\textbf{(A)}\\ x^2+2 \\qquad \\textbf{(B)}\\ x+1 \\qquad \\textbf{(C)}\\ x^2-2x+2 \\qquad \\textbf{(D)}\\ x^2-4\\\\ \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of positive integers less than $1000$ divisible by neither $5$ nor $7$ is:\n$\\text{(A) } 688 \\quad \\text{(B) } 686 \\quad \\text{(C) } 684 \\quad \\text{(D) } 658 \\quad \\text{(E) } 630$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_352", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of positive integers less than $1000$ divisible by neither $5$ nor $7$ is:\n$\\text{(A) } 688 \\quad \\text{(B) } 686 \\quad \\text{(C) } 684 \\quad \\text{(D) } 658 \\quad \\text{(E) } 630$" + } + }, + { + "question": "Return your final response within \\boxed{}. An $n$-term sequence $(x_1, x_2, \\ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length n containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.\n(proposed by Kiran Kedlaya)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "b_{n+1} = 2a_n", + "index": "Sky-T1_10k_353", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An $n$-term sequence $(x_1, x_2, \\ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length n containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.\n(proposed by Kiran Kedlaya)" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a$ and $b$ are real numbers, the equation $3x-5+a=bx+1$ has a unique solution $x$ [The symbol $a \\neq 0$ means that $a$ is different from zero]:\n$\\text{(A) for all a and b} \\qquad \\text{(B) if a }\\neq\\text{2b}\\qquad \\text{(C) if a }\\neq 6\\qquad \\\\ \\text{(D) if b }\\neq 0\\qquad \\text{(E) if b }\\neq 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_354", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a$ and $b$ are real numbers, the equation $3x-5+a=bx+1$ has a unique solution $x$ [The symbol $a \\neq 0$ means that $a$ is different from zero]:\n$\\text{(A) for all a and b} \\qquad \\text{(B) if a }\\neq\\text{2b}\\qquad \\text{(C) if a }\\neq 6\\qquad \\\\ \\text{(D) if b }\\neq 0\\qquad \\text{(E) if b }\\neq 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. If one uses only the tabular information $10^3=1000$, $10^4=10,000$, $2^{10}=1024$, $2^{11}=2048$, $2^{12}=4096$, $2^{13}=8192$, then the strongest statement one can make for $\\log_{10}{2}$ is that it lies between:\n$\\textbf{(A)}\\ \\frac{3}{10} \\; \\text{and} \\; \\frac{4}{11}\\qquad \\textbf{(B)}\\ \\frac{3}{10} \\; \\text{and} \\; \\frac{4}{12}\\qquad \\textbf{(C)}\\ \\frac{3}{10} \\; \\text{and} \\; \\frac{4}{13}\\qquad \\textbf{(D)}\\ \\frac{3}{10} \\; \\text{and} \\; \\frac{40}{132}\\qquad \\textbf{(E)}\\ \\frac{3}{11} \\; \\text{and} \\; \\frac{40}{132}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_355", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If one uses only the tabular information $10^3=1000$, $10^4=10,000$, $2^{10}=1024$, $2^{11}=2048$, $2^{12}=4096$, $2^{13}=8192$, then the strongest statement one can make for $\\log_{10}{2}$ is that it lies between:\n$\\textbf{(A)}\\ \\frac{3}{10} \\; \\text{and} \\; \\frac{4}{11}\\qquad \\textbf{(B)}\\ \\frac{3}{10} \\; \\text{and} \\; \\frac{4}{12}\\qquad \\textbf{(C)}\\ \\frac{3}{10} \\; \\text{and} \\; \\frac{4}{13}\\qquad \\textbf{(D)}\\ \\frac{3}{10} \\; \\text{and} \\; \\frac{40}{132}\\qquad \\textbf{(E)}\\ \\frac{3}{11} \\; \\text{and} \\; \\frac{40}{132}$" + } + }, + { + "question": "Return your final response within \\boxed{}. First $a$ is chosen at random from the set $\\{1,2,3,\\cdots,99,100\\}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is\n$\\text{(A) } \\frac{1}{16}\\quad \\text{(B) } \\frac{1}{8}\\quad \\text{(C) } \\frac{3}{16}\\quad \\text{(D) } \\frac{1}{5}\\quad \\text{(E) } \\frac{1}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_356", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. First $a$ is chosen at random from the set $\\{1,2,3,\\cdots,99,100\\}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is\n$\\text{(A) } \\frac{1}{16}\\quad \\text{(B) } \\frac{1}{8}\\quad \\text{(C) } \\frac{3}{16}\\quad \\text{(D) } \\frac{1}{5}\\quad \\text{(E) } \\frac{1}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Zoran Sunik) A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $n$. Show that if $k\\ge 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^i k$ is greater than the minimum number of jumps needed to reach $2^i$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The minimum number of jumps to reach 2^i k is greater than to reach 2^i.", + "index": "Sky-T1_10k_357", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Zoran Sunik) A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $n$. Show that if $k\\ge 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^i k$ is greater than the minimum number of jumps needed to reach $2^i$." + } + }, + { + "question": "Return your final response within \\boxed{}. Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is:\n$\\textbf{(A)}\\ \\frac{16}{9}\\qquad \\textbf{(B)}\\ \\frac{16}{3}\\qquad \\textbf{(C)}\\ \\frac{4}{9}\\qquad \\textbf{(D)}\\ \\frac{4}{3}\\qquad \\textbf{(E)}\\ -\\frac{4}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_358", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is:\n$\\textbf{(A)}\\ \\frac{16}{9}\\qquad \\textbf{(B)}\\ \\frac{16}{3}\\qquad \\textbf{(C)}\\ \\frac{4}{9}\\qquad \\textbf{(D)}\\ \\frac{4}{3}\\qquad \\textbf{(E)}\\ -\\frac{4}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ pages. On the third day she read $1/3$ of the remaining pages plus $18$ pages. She then realized that there were only $62$ pages left to read, which she read the next day. How many pages are in this book? \n$\\textbf{(A)}\\ 120 \\qquad\\textbf{(B)}\\ 180\\qquad\\textbf{(C)}\\ 240\\qquad\\textbf{(D)}\\ 300\\qquad\\textbf{(E)}\\ 360$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_359", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ pages. On the third day she read $1/3$ of the remaining pages plus $18$ pages. She then realized that there were only $62$ pages left to read, which she read the next day. How many pages are in this book? \n$\\textbf{(A)}\\ 120 \\qquad\\textbf{(B)}\\ 180\\qquad\\textbf{(C)}\\ 240\\qquad\\textbf{(D)}\\ 300\\qquad\\textbf{(E)}\\ 360$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\\[n = f_1\\cdot f_2\\cdots f_k,\\]where $k\\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\\cdot 3$, and $3\\cdot2$, so $D(6) = 3$. What is $D(96)$?\n$\\textbf{(A) } 112 \\qquad\\textbf{(B) } 128 \\qquad\\textbf{(C) } 144 \\qquad\\textbf{(D) } 172 \\qquad\\textbf{(E) } 184$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_360", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\\[n = f_1\\cdot f_2\\cdots f_k,\\]where $k\\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\\cdot 3$, and $3\\cdot2$, so $D(6) = 3$. What is $D(96)$?\n$\\textbf{(A) } 112 \\qquad\\textbf{(B) } 128 \\qquad\\textbf{(C) } 144 \\qquad\\textbf{(D) } 172 \\qquad\\textbf{(E) } 184$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?\n$\\textbf{(A)}\\ 2520\\qquad\\textbf{(B)}\\ 2880\\qquad\\textbf{(C)}\\ 3120\\qquad\\textbf{(D)}\\ 3250\\qquad\\textbf{(E)}\\ 3750$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_361", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?\n$\\textbf{(A)}\\ 2520\\qquad\\textbf{(B)}\\ 2880\\qquad\\textbf{(C)}\\ 3120\\qquad\\textbf{(D)}\\ 3250\\qquad\\textbf{(E)}\\ 3750$" + } + }, + { + "question": "Return your final response within \\boxed{}. John scores $93$ on this year's AHSME. Had the old scoring system still been in effect, he would score only $84$ for the same answers. \nHow many questions does he leave unanswered? (In the new scoring system that year, one receives $5$ points for each correct answer,\n$0$ points for each wrong answer, and $2$ points for each problem left unanswered. In the previous scoring system, one started with $30$ points, received $4$ more for each correct answer, lost $1$ point for each wrong answer, and neither gained nor lost points for unanswered questions.)\n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 9\\qquad \\textbf{(C)}\\ 11\\qquad \\textbf{(D)}\\ 14\\qquad \\textbf{(E)}\\ \\text{Not uniquely determined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_362", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. John scores $93$ on this year's AHSME. Had the old scoring system still been in effect, he would score only $84$ for the same answers. \nHow many questions does he leave unanswered? (In the new scoring system that year, one receives $5$ points for each correct answer,\n$0$ points for each wrong answer, and $2$ points for each problem left unanswered. In the previous scoring system, one started with $30$ points, received $4$ more for each correct answer, lost $1$ point for each wrong answer, and neither gained nor lost points for unanswered questions.)\n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 9\\qquad \\textbf{(C)}\\ 11\\qquad \\textbf{(D)}\\ 14\\qquad \\textbf{(E)}\\ \\text{Not uniquely determined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Melanie Wood)\nAlice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from on to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "Bob", + "index": "Sky-T1_10k_363", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Melanie Wood)\nAlice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from on to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players." + } + }, + { + "question": "Return your final response within \\boxed{}. For $t = 1, 2, 3, 4$, define $S_t = \\sum_{i = 1}^{350}a_i^t$, where $a_i \\in \\{1,2,3,4\\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "905", + "index": "Sky-T1_10k_364", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For $t = 1, 2, 3, 4$, define $S_t = \\sum_{i = 1}^{350}a_i^t$, where $a_i \\in \\{1,2,3,4\\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$." + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\\{1, 2, ..., 30\\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "295", + "index": "Sky-T1_10k_365", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\\{1, 2, ..., 30\\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$." + } + }, + { + "question": "Return your final response within \\boxed{}. A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture?\n$\\text{(A)}\\ 1\\frac{1}{2} \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 9\\frac{1}{2} \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 22$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_366", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture?\n$\\text{(A)}\\ 1\\frac{1}{2} \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 9\\frac{1}{2} \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 22$" + } + }, + { + "question": "Return your final response within \\boxed{}. A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $$48$, and a group of 10th graders buys tickets costing a total of $$64$. How many values for $x$ are possible?\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_367", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $$48$, and a group of 10th graders buys tickets costing a total of $$64$. How many values for $x$ are possible?\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to\n[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy]\n$\\textbf{(A) }50 \\qquad \\textbf{(B) }52 \\qquad \\textbf{(C) }54 \\qquad \\textbf{(D) }56 \\qquad \\textbf{(E) }58$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_368", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to\n[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy]\n$\\textbf{(A) }50 \\qquad \\textbf{(B) }52 \\qquad \\textbf{(C) }54 \\qquad \\textbf{(D) }56 \\qquad \\textbf{(E) }58$" + } + }, + { + "question": "Return your final response within \\boxed{}. The base three representation of $x$ is \n\\[12112211122211112222\\]\nThe first digit (on the left) of the base nine representation of $x$ is\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_369", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The base three representation of $x$ is \n\\[12112211122211112222\\]\nThe first digit (on the left) of the base nine representation of $x$ is\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The average cost of a long-distance call in the USA in $1985$ was\n$41$ cents per minute, and the average cost of a long-distance\ncall in the USA in $2005$ was $7$ cents per minute. Find the\napproximate percent decrease in the cost per minute of a long-\ndistance call.\n$\\mathrm{(A)}\\ 7 \\qquad\\mathrm{(B)}\\ 17 \\qquad\\mathrm{(C)}\\ 34 \\qquad\\mathrm{(D)}\\ 41 \\qquad\\mathrm{(E)}\\ 80$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_370", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The average cost of a long-distance call in the USA in $1985$ was\n$41$ cents per minute, and the average cost of a long-distance\ncall in the USA in $2005$ was $7$ cents per minute. Find the\napproximate percent decrease in the cost per minute of a long-\ndistance call.\n$\\mathrm{(A)}\\ 7 \\qquad\\mathrm{(B)}\\ 17 \\qquad\\mathrm{(C)}\\ 34 \\qquad\\mathrm{(D)}\\ 41 \\qquad\\mathrm{(E)}\\ 80$" + } + }, + { + "question": "Return your final response within \\boxed{}. At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?\n$\\textbf{(A)}\\ 324 \\qquad \\textbf{(B)}\\ 441 \\qquad \\textbf{(C)}\\ 630 \\qquad \\textbf{(D)}\\ 648 \\qquad \\textbf{(E)}\\ 882$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_371", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?\n$\\textbf{(A)}\\ 324 \\qquad \\textbf{(B)}\\ 441 \\qquad \\textbf{(C)}\\ 630 \\qquad \\textbf{(D)}\\ 648 \\qquad \\textbf{(E)}\\ 882$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a rectangle and let $\\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\\overline{DM}$ has integer length, and the lengths of $\\overline{MA},\\overline{MC},$ and $\\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$\n$\\textbf{(A) }24\\sqrt5 \\qquad \\textbf{(B) }60 \\qquad \\textbf{(C) }28\\sqrt5\\qquad \\textbf{(D) }66 \\qquad \\textbf{(E) }8\\sqrt{70}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_372", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a rectangle and let $\\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\\overline{DM}$ has integer length, and the lengths of $\\overline{MA},\\overline{MC},$ and $\\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$\n$\\textbf{(A) }24\\sqrt5 \\qquad \\textbf{(B) }60 \\qquad \\textbf{(C) }28\\sqrt5\\qquad \\textbf{(D) }66 \\qquad \\textbf{(E) }8\\sqrt{70}$" + } + }, + { + "question": "Return your final response within \\boxed{}. [Pdfresizer.com-pdf-convert.png](https://artofproblemsolving.com/wiki/index.php/File:Pdfresizer.com-pdf-convert.png)\nIn the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, \nand sides $BC$ and $ED$. Each side has length $1$. Also, $\\angle FAB = \\angle BCD = 60^\\circ$. \nThe area of the figure is \n$\\textbf{(A)} \\ \\frac{\\sqrt 3}{2} \\qquad \\textbf{(B)} \\ 1 \\qquad \\textbf{(C)} \\ \\frac{3}{2} \\qquad \\textbf{(D)}\\ \\sqrt{3}\\qquad \\textbf{(E)}\\ 2$\nSolution", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_373", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [Pdfresizer.com-pdf-convert.png](https://artofproblemsolving.com/wiki/index.php/File:Pdfresizer.com-pdf-convert.png)\nIn the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, \nand sides $BC$ and $ED$. Each side has length $1$. Also, $\\angle FAB = \\angle BCD = 60^\\circ$. \nThe area of the figure is \n$\\textbf{(A)} \\ \\frac{\\sqrt 3}{2} \\qquad \\textbf{(B)} \\ 1 \\qquad \\textbf{(C)} \\ \\frac{3}{2} \\qquad \\textbf{(D)}\\ \\sqrt{3}\\qquad \\textbf{(E)}\\ 2$\nSolution" + } + }, + { + "question": "Return your final response within \\boxed{}. $N$ is the north pole. $A$ and $B$ are points on a great circle through $N$ equidistant from $N$. $C$ is a point on the equator. Show that the great circle through $C$ and $N$ bisects the angle $ACB$ in the spherical triangle $ABC$ (a spherical triangle has great circle arcs as sides).", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The great circle through C and N bisects the angle \\angle ACB.", + "index": "Sky-T1_10k_374", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $N$ is the north pole. $A$ and $B$ are points on a great circle through $N$ equidistant from $N$. $C$ is a point on the equator. Show that the great circle through $C$ and $N$ bisects the angle $ACB$ in the spherical triangle $ABC$ (a spherical triangle has great circle arcs as sides)." + } + }, + { + "question": "Return your final response within \\boxed{}. Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?\n$\\textbf{(A)}\\ \\dfrac16\\qquad\\textbf{(B)}\\ \\dfrac{13}{72}\\qquad\\textbf{(C)}\\ \\dfrac7{36}\\qquad\\textbf{(D)}\\ \\dfrac5{24}\\qquad\\textbf{(E)}\\ \\dfrac29$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_375", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?\n$\\textbf{(A)}\\ \\dfrac16\\qquad\\textbf{(B)}\\ \\dfrac{13}{72}\\qquad\\textbf{(C)}\\ \\dfrac7{36}\\qquad\\textbf{(D)}\\ \\dfrac5{24}\\qquad\\textbf{(E)}\\ \\dfrac29$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following sets of whole numbers has the largest average?\n$\\text{(A)}\\ \\text{multiples of 2 between 1 and 101} \\qquad \\text{(B)}\\ \\text{multiples of 3 between 1 and 101}$\n$\\text{(C)}\\ \\text{multiples of 4 between 1 and 101} \\qquad \\text{(D)}\\ \\text{multiples of 5 between 1 and 101}$\n$\\text{(E)}\\ \\text{multiples of 6 between 1 and 101}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_376", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following sets of whole numbers has the largest average?\n$\\text{(A)}\\ \\text{multiples of 2 between 1 and 101} \\qquad \\text{(B)}\\ \\text{multiples of 3 between 1 and 101}$\n$\\text{(C)}\\ \\text{multiples of 4 between 1 and 101} \\qquad \\text{(D)}\\ \\text{multiples of 5 between 1 and 101}$\n$\\text{(E)}\\ \\text{multiples of 6 between 1 and 101}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the base ten number system the number $526$ means $5 \\times 10^2+2 \\times 10 + 6$. \nIn the Land of Mathesis, however, numbers are written in the base $r$. \nJones purchases an automobile there for $440$ monetary units (abbreviated m.u). \nHe gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The base $r$ is:\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 5\\qquad \\textbf{(C)}\\ 7\\qquad \\textbf{(D)}\\ 8\\qquad \\textbf{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_377", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the base ten number system the number $526$ means $5 \\times 10^2+2 \\times 10 + 6$. \nIn the Land of Mathesis, however, numbers are written in the base $r$. \nJones purchases an automobile there for $440$ monetary units (abbreviated m.u). \nHe gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The base $r$ is:\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 5\\qquad \\textbf{(C)}\\ 7\\qquad \\textbf{(D)}\\ 8\\qquad \\textbf{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. The \"Middle School Eight\" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the \"Middle School Eight\" teams?\n$\\textbf{(A) }60\\qquad\\textbf{(B) }88\\qquad\\textbf{(C) }96\\qquad\\textbf{(D) }144\\qquad \\textbf{(E) }160$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_378", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The \"Middle School Eight\" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the \"Middle School Eight\" teams?\n$\\textbf{(A) }60\\qquad\\textbf{(B) }88\\qquad\\textbf{(C) }96\\qquad\\textbf{(D) }144\\qquad \\textbf{(E) }160$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $|x-\\log y|=x+\\log y$ where $x$ and $\\log y$ are real, then\n$\\textbf{(A) }x=0\\qquad \\textbf{(B) }y=1\\qquad \\textbf{(C) }x=0\\text{ and }y=1\\qquad\\\\ \\textbf{(D) }x(y-1)=0\\qquad \\textbf{(E) }\\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_379", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $|x-\\log y|=x+\\log y$ where $x$ and $\\log y$ are real, then\n$\\textbf{(A) }x=0\\qquad \\textbf{(B) }y=1\\qquad \\textbf{(C) }x=0\\text{ and }y=1\\qquad\\\\ \\textbf{(D) }x(y-1)=0\\qquad \\textbf{(E) }\\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x_{k+1} = x_k + \\frac12$ for $k=1, 2, \\dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \\dots + x_n$. \n$\\textbf{(A)}\\ \\frac{n+1}{2}\\qquad\\textbf{(B)}\\ \\frac{n+3}{2}\\qquad\\textbf{(C)}\\ \\frac{n^2-1}{2}\\qquad\\textbf{(D)}\\ \\frac{n^2+n}{4}\\qquad\\textbf{(E)}\\ \\frac{n^2+3n}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_380", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x_{k+1} = x_k + \\frac12$ for $k=1, 2, \\dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \\dots + x_n$. \n$\\textbf{(A)}\\ \\frac{n+1}{2}\\qquad\\textbf{(B)}\\ \\frac{n+3}{2}\\qquad\\textbf{(C)}\\ \\frac{n^2-1}{2}\\qquad\\textbf{(D)}\\ \\frac{n^2+n}{4}\\qquad\\textbf{(E)}\\ \\frac{n^2+3n}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define $x\\otimes y=x^3-y$. What is $h\\otimes (h\\otimes h)$?\n$\\textbf{(A)}\\ -h\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ h\\qquad\\textbf{(D)}\\ 2h\\qquad\\textbf{(E)}\\ h^3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_381", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define $x\\otimes y=x^3-y$. What is $h\\otimes (h\\otimes h)$?\n$\\textbf{(A)}\\ -h\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ h\\qquad\\textbf{(D)}\\ 2h\\qquad\\textbf{(E)}\\ h^3$" + } + }, + { + "question": "Return your final response within \\boxed{}. Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_382", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Indicate in which one of the following equations $y$ is neither directly nor inversely proportional to $x$:\n$\\textbf{(A)}\\ x + y = 0 \\qquad\\textbf{(B)}\\ 3xy = 10 \\qquad\\textbf{(C)}\\ x = 5y \\qquad\\textbf{(D)}\\ 3x + y = 10$\n$\\textbf{(E)}\\ \\frac {x}{y} = \\sqrt {3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_383", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Indicate in which one of the following equations $y$ is neither directly nor inversely proportional to $x$:\n$\\textbf{(A)}\\ x + y = 0 \\qquad\\textbf{(B)}\\ 3xy = 10 \\qquad\\textbf{(C)}\\ x = 5y \\qquad\\textbf{(D)}\\ 3x + y = 10$\n$\\textbf{(E)}\\ \\frac {x}{y} = \\sqrt {3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{2}{10}+\\frac{4}{100}+\\frac{6}{1000}=$\n$\\text{(A)}\\ .012 \\qquad \\text{(B)}\\ .0246 \\qquad \\text{(C)}\\ .12 \\qquad \\text{(D)}\\ .246 \\qquad \\text{(E)}\\ 246$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_384", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{2}{10}+\\frac{4}{100}+\\frac{6}{1000}=$\n$\\text{(A)}\\ .012 \\qquad \\text{(B)}\\ .0246 \\qquad \\text{(C)}\\ .12 \\qquad \\text{(D)}\\ .246 \\qquad \\text{(E)}\\ 246$" + } + }, + { + "question": "Return your final response within \\boxed{}. [asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP(\"$A$\",A)); D(MP(\"$B$\",B)); D(MP(\"$C$\",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP(\"$B'$\",BB[0],NW);MP(\"$C'$\",CC[1],NE); //Credit to TheMaskedMagician for the diagram[/asy]\nCircles with centers $A ,~ B$, and $C$ each have radius $r$, where $1 < r < 2$. \nThe distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ \nwhich is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, \nthen length $B'C'$ equals\n$\\textbf{(A) }3r-2\\qquad \\textbf{(B) }r^2\\qquad \\textbf{(C) }r+\\sqrt{3(r-1)}\\qquad\\\\ \\textbf{(D) }1+\\sqrt{3(r^2-1)}\\qquad \\textbf{(E) }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_385", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP(\"$A$\",A)); D(MP(\"$B$\",B)); D(MP(\"$C$\",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP(\"$B'$\",BB[0],NW);MP(\"$C'$\",CC[1],NE); //Credit to TheMaskedMagician for the diagram[/asy]\nCircles with centers $A ,~ B$, and $C$ each have radius $r$, where $1 < r < 2$. \nThe distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ \nwhich is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, \nthen length $B'C'$ equals\n$\\textbf{(A) }3r-2\\qquad \\textbf{(B) }r^2\\qquad \\textbf{(C) }r+\\sqrt{3(r-1)}\\qquad\\\\ \\textbf{(D) }1+\\sqrt{3(r^2-1)}\\qquad \\textbf{(E) }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\\frac{7}{51}$ is $0.\\overline{23}_k = 0.232323..._k$. What is $k$?\n$\\textbf{(A) } 13 \\qquad\\textbf{(B) } 14 \\qquad\\textbf{(C) } 15 \\qquad\\textbf{(D) } 16 \\qquad\\textbf{(E) } 17$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_386", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\\frac{7}{51}$ is $0.\\overline{23}_k = 0.232323..._k$. What is $k$?\n$\\textbf{(A) } 13 \\qquad\\textbf{(B) } 14 \\qquad\\textbf{(C) } 15 \\qquad\\textbf{(D) } 16 \\qquad\\textbf{(E) } 17$" + } + }, + { + "question": "Return your final response within \\boxed{}. A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is :\n$\\text{(A) } 100\\quad \\text{(B) } 97\\quad \\text{(C) } 83\\quad \\text{(D) } 17\\quad \\text{(E) } 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_387", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is :\n$\\text{(A) } 100\\quad \\text{(B) } 97\\quad \\text{(C) } 83\\quad \\text{(D) } 17\\quad \\text{(E) } 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. The domain of the function $f(x)=\\log_{\\frac12}(\\log_4(\\log_{\\frac14}(\\log_{16}(\\log_{\\frac1{16}}x))))$ is an interval of length $\\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?\n$\\textbf{(A) }19\\qquad \\textbf{(B) }31\\qquad \\textbf{(C) }271\\qquad \\textbf{(D) }319\\qquad \\textbf{(E) }511\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_388", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The domain of the function $f(x)=\\log_{\\frac12}(\\log_4(\\log_{\\frac14}(\\log_{16}(\\log_{\\frac1{16}}x))))$ is an interval of length $\\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?\n$\\textbf{(A) }19\\qquad \\textbf{(B) }31\\qquad \\textbf{(C) }271\\qquad \\textbf{(D) }319\\qquad \\textbf{(E) }511\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. A rising number, such as $34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $\\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{\\text{th}}$ number in the list does not contain the digit\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_389", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rising number, such as $34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $\\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{\\text{th}}$ number in the list does not contain the digit\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?\n$\\textbf{(A)}\\ P^2Q \\qquad \\textbf{(B)}\\ P^nQ^m \\qquad \\textbf{(C)}\\ P^nQ^{2m} \\qquad \\textbf{(D)}\\ P^{2m}Q^n \\qquad \\textbf{(E)}\\ P^{2n}Q^m$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_390", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?\n$\\textbf{(A)}\\ P^2Q \\qquad \\textbf{(B)}\\ P^nQ^m \\qquad \\textbf{(C)}\\ P^nQ^{2m} \\qquad \\textbf{(D)}\\ P^{2m}Q^n \\qquad \\textbf{(E)}\\ P^{2n}Q^m$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the $1992^\\text{nd}$ letter in this sequence?\n\\[\\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\\cdots\\]\n$\\text{(A)}\\ \\text{A} \\qquad \\text{(B)}\\ \\text{B} \\qquad \\text{(C)}\\ \\text{C} \\qquad \\text{(D)}\\ \\text{D} \\qquad \\text{(E)}\\ \\text{E}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_391", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the $1992^\\text{nd}$ letter in this sequence?\n\\[\\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\\cdots\\]\n$\\text{(A)}\\ \\text{A} \\qquad \\text{(B)}\\ \\text{B} \\qquad \\text{(C)}\\ \\text{C} \\qquad \\text{(D)}\\ \\text{D} \\qquad \\text{(E)}\\ \\text{E}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2$?\n$\\textbf{(A) } 8 \\qquad\\textbf{(B) } 10 \\qquad\\textbf{(C) } 12 \\qquad\\textbf{(D) } 16 \\qquad\\textbf{(E) } 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_392", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2$?\n$\\textbf{(A) } 8 \\qquad\\textbf{(B) } 10 \\qquad\\textbf{(C) } 12 \\qquad\\textbf{(D) } 16 \\qquad\\textbf{(E) } 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. The area of a circle is doubled when its radius $r$ is increased by $n$. Then $r$ equals:\n$\\textbf{(A)}\\ n(\\sqrt{2} + 1)\\qquad \\textbf{(B)}\\ n(\\sqrt{2} - 1)\\qquad \\textbf{(C)}\\ n\\qquad \\textbf{(D)}\\ n(2 - \\sqrt{2})\\qquad \\textbf{(E)}\\ \\frac{n\\pi}{\\sqrt{2} + 1}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_393", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The area of a circle is doubled when its radius $r$ is increased by $n$. Then $r$ equals:\n$\\textbf{(A)}\\ n(\\sqrt{2} + 1)\\qquad \\textbf{(B)}\\ n(\\sqrt{2} - 1)\\qquad \\textbf{(C)}\\ n\\qquad \\textbf{(D)}\\ n(2 - \\sqrt{2})\\qquad \\textbf{(E)}\\ \\frac{n\\pi}{\\sqrt{2} + 1}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a=\\tfrac{1}{2}$ and $(a+1)(b+1)=2$ then the radian measure of $\\arctan a + \\arctan b$ equals\n$\\textbf{(A) }\\frac{\\pi}{2}\\qquad \\textbf{(B) }\\frac{\\pi}{3}\\qquad \\textbf{(C) }\\frac{\\pi}{4}\\qquad \\textbf{(D) }\\frac{\\pi}{5}\\qquad \\textbf{(E) }\\frac{\\pi}{6}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_394", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a=\\tfrac{1}{2}$ and $(a+1)(b+1)=2$ then the radian measure of $\\arctan a + \\arctan b$ equals\n$\\textbf{(A) }\\frac{\\pi}{2}\\qquad \\textbf{(B) }\\frac{\\pi}{3}\\qquad \\textbf{(C) }\\frac{\\pi}{4}\\qquad \\textbf{(D) }\\frac{\\pi}{5}\\qquad \\textbf{(E) }\\frac{\\pi}{6}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A painting $18$\" X $24$\" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is:\n$\\text{(A) } 1:3\\quad \\text{(B) } 1:2\\quad \\text{(C) } 2:3\\quad \\text{(D) } 3:4\\quad \\text{(E) } 1:1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_395", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A painting $18$\" X $24$\" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is:\n$\\text{(A) } 1:3\\quad \\text{(B) } 1:2\\quad \\text{(C) } 2:3\\quad \\text{(D) } 3:4\\quad \\text{(E) } 1:1$" + } + }, + { + "question": "Return your final response within \\boxed{}. In triangle $ABC, AB=AC$ and $\\measuredangle A=80^\\circ$. If points $D, E$, and $F$ lie on sides $BC, AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\\measuredangle EDF$ equals\n$\\textbf{(A) }30^\\circ\\qquad \\textbf{(B) }40^\\circ\\qquad \\textbf{(C) }50^\\circ\\qquad \\textbf{(D) }65^\\circ\\qquad \\textbf{(E) }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_396", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In triangle $ABC, AB=AC$ and $\\measuredangle A=80^\\circ$. If points $D, E$, and $F$ lie on sides $BC, AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\\measuredangle EDF$ equals\n$\\textbf{(A) }30^\\circ\\qquad \\textbf{(B) }40^\\circ\\qquad \\textbf{(C) }50^\\circ\\qquad \\textbf{(D) }65^\\circ\\qquad \\textbf{(E) }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. [Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The [inscribed circle](https://artofproblemsolving.com/wiki/index.php/Inscribed_circle) of [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $BEF$ is [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at [point](https://artofproblemsolving.com/wiki/index.php/Point) $Q.$ Find $PQ.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "259", + "index": "Sky-T1_10k_397", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The [inscribed circle](https://artofproblemsolving.com/wiki/index.php/Inscribed_circle) of [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $BEF$ is [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at [point](https://artofproblemsolving.com/wiki/index.php/Point) $Q.$ Find $PQ.$" + } + }, + { + "question": "Return your final response within \\boxed{}. An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding 2m to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "Exactly one vertex can be the winner for any initial configuration.", + "index": "Sky-T1_10k_398", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding 2m to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won." + } + }, + { + "question": "Return your final response within \\boxed{}. (Zuming Feng) Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \\parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct point $B_1$ in such a way that convex quadrilateral $APCB_1$ is cyclic, $QB_1 \\parallel BA$, and $B_1$ and $Q$ lie on opposite sides of line $AC$. Prove that points $B_1, C_1,P$, and $Q$ lie on a circle.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B_1, C_1, P, Q lie on a circle", + "index": "Sky-T1_10k_399", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Zuming Feng) Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \\parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct point $B_1$ in such a way that convex quadrilateral $APCB_1$ is cyclic, $QB_1 \\parallel BA$, and $B_1$ and $Q$ lie on opposite sides of line $AC$. Prove that points $B_1, C_1,P$, and $Q$ lie on a circle." + } + }, + { + "question": "Return your final response within \\boxed{}. How many terms are in the arithmetic sequence $13$, $16$, $19$, $\\dotsc$, $70$, $73$?\n$\\textbf{(A)}\\ 20 \\qquad\\textbf{(B)} \\ 21 \\qquad\\textbf{(C)} \\ 24 \\qquad\\textbf{(D)} \\ 60 \\qquad\\textbf{(E)} \\ 61$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_400", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many terms are in the arithmetic sequence $13$, $16$, $19$, $\\dotsc$, $70$, $73$?\n$\\textbf{(A)}\\ 20 \\qquad\\textbf{(B)} \\ 21 \\qquad\\textbf{(C)} \\ 24 \\qquad\\textbf{(D)} \\ 60 \\qquad\\textbf{(E)} \\ 61$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.\n\nSuppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$\n$\\textbf{(A) }81 \\qquad \\textbf{(B) }89 \\qquad \\textbf{(C) }97\\qquad \\textbf{(D) }105 \\qquad \\textbf{(E) }113$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_401", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.\n\nSuppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$\n$\\textbf{(A) }81 \\qquad \\textbf{(B) }89 \\qquad \\textbf{(C) }97\\qquad \\textbf{(D) }105 \\qquad \\textbf{(E) }113$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the numeration system with base $5$, counting is as follows: $1, 2, 3, 4, 10, 11, 12, 13, 14, 20,\\ldots$.\nThe number whose description in the decimal system is $69$, when described in the base $5$ system, is a number with:\n$\\textbf{(A)}\\ \\text{two consecutive digits} \\qquad\\textbf{(B)}\\ \\text{two non-consecutive digits} \\qquad \\\\ \\textbf{(C)}\\ \\text{three consecutive digits} \\qquad\\textbf{(D)}\\ \\text{three non-consecutive digits} \\qquad \\\\ \\textbf{(E)}\\ \\text{four digits}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_402", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the numeration system with base $5$, counting is as follows: $1, 2, 3, 4, 10, 11, 12, 13, 14, 20,\\ldots$.\nThe number whose description in the decimal system is $69$, when described in the base $5$ system, is a number with:\n$\\textbf{(A)}\\ \\text{two consecutive digits} \\qquad\\textbf{(B)}\\ \\text{two non-consecutive digits} \\qquad \\\\ \\textbf{(C)}\\ \\text{three consecutive digits} \\qquad\\textbf{(D)}\\ \\text{three non-consecutive digits} \\qquad \\\\ \\textbf{(E)}\\ \\text{four digits}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures\nFour friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.\n$\\circ$ Art's cookies are trapezoids: \n\n$\\circ$ Roger's cookies are rectangles: \n\n$\\circ$ Paul's cookies are parallelograms: \n\n$\\circ$ Trisha's cookies are triangles: \n\nHow many cookies will be in one batch of Trisha's cookies?\n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 18\\qquad\\textbf{(E)}\\ 24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_403", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures\nFour friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.\n$\\circ$ Art's cookies are trapezoids: \n\n$\\circ$ Roger's cookies are rectangles: \n\n$\\circ$ Paul's cookies are parallelograms: \n\n$\\circ$ Trisha's cookies are triangles: \n\nHow many cookies will be in one batch of Trisha's cookies?\n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 18\\qquad\\textbf{(E)}\\ 24$" + } + }, + { + "question": "Return your final response within \\boxed{}. A regular [octahedron](https://artofproblemsolving.com/wiki/index.php/Octahedron) has side length $1$. A [plane](https://artofproblemsolving.com/wiki/index.php/Plane) [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) to two of its opposite faces cuts the octahedron into the two congruent solids. The [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) formed by the intersection of the plane and the octahedron has area $\\frac {a\\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. What is $a + b + c$?\n$\\textbf{(A)}\\ 10\\qquad \\textbf{(B)}\\ 11\\qquad \\textbf{(C)}\\ 12\\qquad \\textbf{(D)}\\ 13\\qquad \\textbf{(E)}\\ 14$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_404", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A regular [octahedron](https://artofproblemsolving.com/wiki/index.php/Octahedron) has side length $1$. A [plane](https://artofproblemsolving.com/wiki/index.php/Plane) [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) to two of its opposite faces cuts the octahedron into the two congruent solids. The [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) formed by the intersection of the plane and the octahedron has area $\\frac {a\\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. What is $a + b + c$?\n$\\textbf{(A)}\\ 10\\qquad \\textbf{(B)}\\ 11\\qquad \\textbf{(C)}\\ 12\\qquad \\textbf{(D)}\\ 13\\qquad \\textbf{(E)}\\ 14$" + } + }, + { + "question": "Return your final response within \\boxed{}. If one side of a triangle is $12$ inches and the opposite angle is $30^{\\circ}$, then the diameter of the circumscribed circle is: \n$\\textbf{(A)}\\ 18\\text{ inches} \\qquad \\textbf{(B)}\\ 30\\text{ inches} \\qquad \\textbf{(C)}\\ 24\\text{ inches} \\qquad \\textbf{(D)}\\ 20\\text{ inches}\\\\ \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_405", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If one side of a triangle is $12$ inches and the opposite angle is $30^{\\circ}$, then the diameter of the circumscribed circle is: \n$\\textbf{(A)}\\ 18\\text{ inches} \\qquad \\textbf{(B)}\\ 30\\text{ inches} \\qquad \\textbf{(C)}\\ 24\\text{ inches} \\qquad \\textbf{(D)}\\ 20\\text{ inches}\\\\ \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, \nand $f$ jumps equals $g$ meters, then one meter equals how many skips?\n$\\textbf{(A)}\\ \\frac{bdg}{cef}\\qquad \\textbf{(B)}\\ \\frac{cdf}{beg}\\qquad \\textbf{(C)}\\ \\frac{cdg}{bef}\\qquad \\textbf{(D)}\\ \\frac{cef}{bdg}\\qquad \\textbf{(E)}\\ \\frac{ceg}{bdf}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_406", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, \nand $f$ jumps equals $g$ meters, then one meter equals how many skips?\n$\\textbf{(A)}\\ \\frac{bdg}{cef}\\qquad \\textbf{(B)}\\ \\frac{cdf}{beg}\\qquad \\textbf{(C)}\\ \\frac{cdg}{bef}\\qquad \\textbf{(D)}\\ \\frac{cef}{bdg}\\qquad \\textbf{(E)}\\ \\frac{ceg}{bdf}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The times between $7$ and $8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $84^{\\circ}$ are: \n$\\textbf{(A)}\\ \\text{7: 23 and 7: 53}\\qquad \\textbf{(B)}\\ \\text{7: 20 and 7: 50}\\qquad \\textbf{(C)}\\ \\text{7: 22 and 7: 53}\\\\ \\textbf{(D)}\\ \\text{7: 23 and 7: 52}\\qquad \\textbf{(E)}\\ \\text{7: 21 and 7: 49}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_407", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The times between $7$ and $8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $84^{\\circ}$ are: \n$\\textbf{(A)}\\ \\text{7: 23 and 7: 53}\\qquad \\textbf{(B)}\\ \\text{7: 20 and 7: 50}\\qquad \\textbf{(C)}\\ \\text{7: 22 and 7: 53}\\\\ \\textbf{(D)}\\ \\text{7: 23 and 7: 52}\\qquad \\textbf{(E)}\\ \\text{7: 21 and 7: 49}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \\tfrac{321}{400}$?\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 14 \\qquad \\textbf{(C) }16 \\qquad \\textbf{(D) } 18 \\qquad \\textbf{(E) } 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_408", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \\tfrac{321}{400}$?\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 14 \\qquad \\textbf{(C) }16 \\qquad \\textbf{(D) } 18 \\qquad \\textbf{(E) } 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define $a\\clubsuit b=a^2b-ab^2$. Which of the following describes the set of points $(x, y)$ for which $x\\clubsuit y=y\\clubsuit x$?\n$\\textbf{(A)}\\ \\text{a finite set of points}\\\\ \\qquad\\textbf{(B)}\\ \\text{one line}\\\\ \\qquad\\textbf{(C)}\\ \\text{two parallel lines}\\\\ \\qquad\\textbf{(D)}\\ \\text{two intersecting lines}\\\\ \\qquad\\textbf{(E)}\\ \\text{three lines}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_409", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define $a\\clubsuit b=a^2b-ab^2$. Which of the following describes the set of points $(x, y)$ for which $x\\clubsuit y=y\\clubsuit x$?\n$\\textbf{(A)}\\ \\text{a finite set of points}\\\\ \\qquad\\textbf{(B)}\\ \\text{one line}\\\\ \\qquad\\textbf{(C)}\\ \\text{two parallel lines}\\\\ \\qquad\\textbf{(D)}\\ \\text{two intersecting lines}\\\\ \\qquad\\textbf{(E)}\\ \\text{three lines}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $A$ can do a piece of work in $9$ days. $B$ is $50\\%$ more efficient than $A$. The number of days it takes $B$ to do the same piece of work is:\n$\\textbf{(A)}\\ 13\\frac {1}{2} \\qquad\\textbf{(B)}\\ 4\\frac {1}{2} \\qquad\\textbf{(C)}\\ 6 \\qquad\\textbf{(D)}\\ 3 \\qquad\\textbf{(E)}\\ \\text{none of these answers}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_410", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $A$ can do a piece of work in $9$ days. $B$ is $50\\%$ more efficient than $A$. The number of days it takes $B$ to do the same piece of work is:\n$\\textbf{(A)}\\ 13\\frac {1}{2} \\qquad\\textbf{(B)}\\ 4\\frac {1}{2} \\qquad\\textbf{(C)}\\ 6 \\qquad\\textbf{(D)}\\ 3 \\qquad\\textbf{(E)}\\ \\text{none of these answers}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph of the function $f(x) = 2x^3 - 7$ goes: \n$\\textbf{(A)}\\ \\text{up to the right and down to the left} \\\\ \\textbf{(B)}\\ \\text{down to the right and up to the left}\\\\ \\textbf{(C)}\\ \\text{up to the right and up to the left}\\\\ \\textbf{(D)}\\ \\text{down to the right and down to the left}\\\\ \\textbf{(E)}\\ \\text{none of these ways.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_411", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph of the function $f(x) = 2x^3 - 7$ goes: \n$\\textbf{(A)}\\ \\text{up to the right and down to the left} \\\\ \\textbf{(B)}\\ \\text{down to the right and up to the left}\\\\ \\textbf{(C)}\\ \\text{up to the right and up to the left}\\\\ \\textbf{(D)}\\ \\text{down to the right and down to the left}\\\\ \\textbf{(E)}\\ \\text{none of these ways.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The probability that event $A$ occurs is $\\frac{3}{4}$; the probability that event B occurs is $\\frac{2}{3}$. \nLet $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval\n$\\textbf{(A)}\\ \\Big[\\frac{1}{12},\\frac{1}{2}\\Big]\\qquad \\textbf{(B)}\\ \\Big[\\frac{5}{12},\\frac{1}{2}\\Big]\\qquad \\textbf{(C)}\\ \\Big[\\frac{1}{2},\\frac{2}{3}\\Big]\\qquad \\textbf{(D)}\\ \\Big[\\frac{5}{12},\\frac{2}{3}\\Big]\\qquad \\textbf{(E)}\\ \\Big[\\frac{1}{12},\\frac{2}{3}\\Big]$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_412", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The probability that event $A$ occurs is $\\frac{3}{4}$; the probability that event B occurs is $\\frac{2}{3}$. \nLet $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval\n$\\textbf{(A)}\\ \\Big[\\frac{1}{12},\\frac{1}{2}\\Big]\\qquad \\textbf{(B)}\\ \\Big[\\frac{5}{12},\\frac{1}{2}\\Big]\\qquad \\textbf{(C)}\\ \\Big[\\frac{1}{2},\\frac{2}{3}\\Big]\\qquad \\textbf{(D)}\\ \\Big[\\frac{5}{12},\\frac{2}{3}\\Big]\\qquad \\textbf{(E)}\\ \\Big[\\frac{1}{12},\\frac{2}{3}\\Big]$" + } + }, + { + "question": "Return your final response within \\boxed{}. If a number $N,N \\ne 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is\n$\\text{(A) } \\frac{1}{R}\\quad \\text{(B) } R\\quad \\text{(C) } 4\\quad \\text{(D) } \\frac{1}{4}\\quad \\text{(E) } -R$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_413", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If a number $N,N \\ne 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is\n$\\text{(A) } \\frac{1}{R}\\quad \\text{(B) } R\\quad \\text{(C) } 4\\quad \\text{(D) } \\frac{1}{4}\\quad \\text{(E) } -R$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?\n$\\text{(A)}\\ 200 \\qquad \\text{(B)}\\ 202 \\qquad \\text{(C)}\\ 220 \\qquad \\text{(D)}\\ 380 \\qquad \\text{(E)}\\ 398$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_414", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?\n$\\text{(A)}\\ 200 \\qquad \\text{(B)}\\ 202 \\qquad \\text{(C)}\\ 220 \\qquad \\text{(D)}\\ 380 \\qquad \\text{(E)}\\ 398$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many positive [integers](https://artofproblemsolving.com/wiki/index.php/Integer) $b$ have the property that $\\log_{b} 729$ is a positive integer?\n$\\mathrm{(A) \\ 0 } \\qquad \\mathrm{(B) \\ 1 } \\qquad \\mathrm{(C) \\ 2 } \\qquad \\mathrm{(D) \\ 3 } \\qquad \\mathrm{(E) \\ 4 }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_415", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many positive [integers](https://artofproblemsolving.com/wiki/index.php/Integer) $b$ have the property that $\\log_{b} 729$ is a positive integer?\n$\\mathrm{(A) \\ 0 } \\qquad \\mathrm{(B) \\ 1 } \\qquad \\mathrm{(C) \\ 2 } \\qquad \\mathrm{(D) \\ 3 } \\qquad \\mathrm{(E) \\ 4 }$" + } + }, + { + "question": "Return your final response within \\boxed{}. Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\\triangle{ABC}$?\n\n\n$\\textbf {(A) } 2\\sqrt{3} \\qquad \\textbf {(B) } 3\\sqrt{3} \\qquad \\textbf {(C) } 1+3\\sqrt{2} \\qquad \\textbf {(D) } 2+2\\sqrt{3} \\qquad \\textbf {(E) } 3+2\\sqrt{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_416", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\\triangle{ABC}$?\n\n\n$\\textbf {(A) } 2\\sqrt{3} \\qquad \\textbf {(B) } 3\\sqrt{3} \\qquad \\textbf {(C) } 1+3\\sqrt{2} \\qquad \\textbf {(D) } 2+2\\sqrt{3} \\qquad \\textbf {(E) } 3+2\\sqrt{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Arithmetic sequences $\\left(a_n\\right)$ and $\\left(b_n\\right)$ have integer terms with $a_1=b_1=1 \\sqrt{ab} \\qquad \\textbf{(B)}\\ \\frac{a+b}{2} < \\sqrt{ab} \\qquad \\textbf{(C)}\\ \\frac{a+b}{2}=\\sqrt{ab}\\\\ \\textbf{(D)}\\ \\frac{a+b}{2}\\leq\\sqrt{ab}\\qquad \\textbf{(E)}\\ \\frac{a+b}{2}\\geq\\sqrt{ab}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_478", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The lengths of two line segments are $a$ units and $b$ units respectively. Then the correct relation between them is: \n$\\textbf{(A)}\\ \\frac{a+b}{2} > \\sqrt{ab} \\qquad \\textbf{(B)}\\ \\frac{a+b}{2} < \\sqrt{ab} \\qquad \\textbf{(C)}\\ \\frac{a+b}{2}=\\sqrt{ab}\\\\ \\textbf{(D)}\\ \\frac{a+b}{2}\\leq\\sqrt{ab}\\qquad \\textbf{(E)}\\ \\frac{a+b}{2}\\geq\\sqrt{ab}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many integers $n$ between $1$ and $50$, inclusive, is \\[\\frac{(n^2-1)!}{(n!)^n}\\] an integer? (Recall that $0! = 1$.)\n$\\textbf{(A) } 31 \\qquad \\textbf{(B) } 32 \\qquad \\textbf{(C) } 33 \\qquad \\textbf{(D) } 34 \\qquad \\textbf{(E) } 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_479", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many integers $n$ between $1$ and $50$, inclusive, is \\[\\frac{(n^2-1)!}{(n!)^n}\\] an integer? (Recall that $0! = 1$.)\n$\\textbf{(A) } 31 \\qquad \\textbf{(B) } 32 \\qquad \\textbf{(C) } 33 \\qquad \\textbf{(D) } 34 \\qquad \\textbf{(E) } 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mr. Green receives a $10\\%$ raise every year. His salary after four such raises has gone up by what percent?\n$\\text{(A)}\\ \\text{less than }40\\% \\qquad \\text{(B)}\\ 40\\% \\qquad \\text{(C)}\\ 44\\% \\qquad \\text{(D)}\\ 45\\% \\qquad \\text{(E)}\\ \\text{more than }45\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_480", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mr. Green receives a $10\\%$ raise every year. His salary after four such raises has gone up by what percent?\n$\\text{(A)}\\ \\text{less than }40\\% \\qquad \\text{(B)}\\ 40\\% \\qquad \\text{(C)}\\ 44\\% \\qquad \\text{(D)}\\ 45\\% \\qquad \\text{(E)}\\ \\text{more than }45\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following values is largest?\n$\\textbf{(A) }2+0+1+7\\qquad\\textbf{(B) }2 \\times 0 +1+7\\qquad\\textbf{(C) }2+0 \\times 1 + 7\\qquad\\textbf{(D) }2+0+1 \\times 7\\qquad\\textbf{(E) }2 \\times 0 \\times 1 \\times 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_481", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following values is largest?\n$\\textbf{(A) }2+0+1+7\\qquad\\textbf{(B) }2 \\times 0 +1+7\\qquad\\textbf{(C) }2+0 \\times 1 + 7\\qquad\\textbf{(D) }2+0+1 \\times 7\\qquad\\textbf{(E) }2 \\times 0 \\times 1 \\times 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB=CD=EF$ and diagonals $AD,BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $\\frac{CP}{PE}=(\\frac{AC}{CE})^2$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\left( \\dfrac{AC}{CE} \\right)^2", + "index": "Sky-T1_10k_482", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB=CD=EF$ and diagonals $AD,BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $\\frac{CP}{PE}=(\\frac{AC}{CE})^2$." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0$?\n$\\textbf{(A) }\\frac{17}{32} \\qquad \\textbf{(B) }\\frac{11}{16} \\qquad \\textbf{(C) }\\frac79 \\qquad \\textbf{(D) }\\frac76\\qquad \\textbf{(E) }\\frac{25}{11}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_483", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0$?\n$\\textbf{(A) }\\frac{17}{32} \\qquad \\textbf{(B) }\\frac{11}{16} \\qquad \\textbf{(C) }\\frac79 \\qquad \\textbf{(D) }\\frac76\\qquad \\textbf{(E) }\\frac{25}{11}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{(3!)!}{3!}=$\n$\\text{(A)}\\ 1\\qquad\\text{(B)}\\ 2\\qquad\\text{(C)}\\ 6\\qquad\\text{(D)}\\ 40\\qquad\\text{(E)}\\ 120$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_484", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{(3!)!}{3!}=$\n$\\text{(A)}\\ 1\\qquad\\text{(B)}\\ 2\\qquad\\text{(C)}\\ 6\\qquad\\text{(D)}\\ 40\\qquad\\text{(E)}\\ 120$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x, y, z$ are reals such that $0\\le x, y, z \\le 1$, show that $\\frac{x}{y + z + 1} + \\frac{y}{z + x + 1} + \\frac{z}{x + y + 1} \\le 1 - (1 - x)(1 - y)(1 - z)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\frac{x}{y + z + 1} + \\frac{y}{z + x + 1} + \\frac{z}{x + y + 1} \\le 1 - (1 - x)(1 - y)(1 - z)", + "index": "Sky-T1_10k_485", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x, y, z$ are reals such that $0\\le x, y, z \\le 1$, show that $\\frac{x}{y + z + 1} + \\frac{y}{z + x + 1} + \\frac{z}{x + y + 1} \\le 1 - (1 - x)(1 - y)(1 - z)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of $2010$ boxes in a line contains a single red marble, and for $1 \\le k \\le 2010$, the box in the $k\\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \\frac{1}{2010}$?\n$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 63 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 201 \\qquad \\textbf{(E)}\\ 1005$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_486", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of $2010$ boxes in a line contains a single red marble, and for $1 \\le k \\le 2010$, the box in the $k\\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \\frac{1}{2010}$?\n$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 63 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 201 \\qquad \\textbf{(E)}\\ 1005$" + } + }, + { + "question": "Return your final response within \\boxed{}. The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings?\n\\[\\begin{array}{c|c|c}\\text{Child}&\\text{Eye Color}&\\text{Hair Color}\\\\ \\hline\\text{Benjamin}&\\text{Blue}&\\text{Black}\\\\ \\hline\\text{Jim}&\\text{Brown}&\\text{Blonde}\\\\ \\hline\\text{Nadeen}&\\text{Brown}&\\text{Black}\\\\ \\hline\\text{Austin}&\\text{Blue}&\\text{Blonde}\\\\ \\hline\\text{Tevyn}&\\text{Blue}&\\text{Black}\\\\ \\hline\\text{Sue}&\\text{Blue}&\\text{Blonde}\\\\ \\hline\\end{array}\\]\n$\\textbf{(A)}\\ \\text{Nadeen and Austin}\\qquad\\textbf{(B)}\\ \\text{Benjamin and Sue}\\qquad\\textbf{(C)}\\ \\text{Benjamin and Austin}\\qquad\\textbf{(D)}\\ \\text{Nadeen and Tevyn}$\n$\\textbf{(E)}\\ \\text{Austin and Sue}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_487", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings?\n\\[\\begin{array}{c|c|c}\\text{Child}&\\text{Eye Color}&\\text{Hair Color}\\\\ \\hline\\text{Benjamin}&\\text{Blue}&\\text{Black}\\\\ \\hline\\text{Jim}&\\text{Brown}&\\text{Blonde}\\\\ \\hline\\text{Nadeen}&\\text{Brown}&\\text{Black}\\\\ \\hline\\text{Austin}&\\text{Blue}&\\text{Blonde}\\\\ \\hline\\text{Tevyn}&\\text{Blue}&\\text{Black}\\\\ \\hline\\text{Sue}&\\text{Blue}&\\text{Blonde}\\\\ \\hline\\end{array}\\]\n$\\textbf{(A)}\\ \\text{Nadeen and Austin}\\qquad\\textbf{(B)}\\ \\text{Benjamin and Sue}\\qquad\\textbf{(C)}\\ \\text{Benjamin and Austin}\\qquad\\textbf{(D)}\\ \\text{Nadeen and Tevyn}$\n$\\textbf{(E)}\\ \\text{Austin and Sue}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)\n$\\textbf{(A)}\\ 210 \\qquad\\textbf{(B)}\\ 420 \\qquad\\textbf{(C)}\\ 630 \\qquad\\textbf{(D)}\\ 840 \\qquad\\textbf{(E)}\\ 1050$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_488", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)\n$\\textbf{(A)}\\ 210 \\qquad\\textbf{(B)}\\ 420 \\qquad\\textbf{(C)}\\ 630 \\qquad\\textbf{(D)}\\ 840 \\qquad\\textbf{(E)}\\ 1050$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\\overline{AB}$, and for $n \\geq 2$, let $C_n$ be the foot of the altitude to $\\overline{C_{n-2}B}$ in $\\triangle C_{n-2}C_{n-1}B$. The sum $\\sum_{n=2}^\\infty C_{n-2}C_{n-1} = 6p$. Find $p$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "182", + "index": "Sky-T1_10k_489", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\\overline{AB}$, and for $n \\geq 2$, let $C_n$ be the foot of the altitude to $\\overline{C_{n-2}B}$ in $\\triangle C_{n-2}C_{n-1}B$. The sum $\\sum_{n=2}^\\infty C_{n-2}C_{n-1} = 6p$. Find $p$." + } + }, + { + "question": "Return your final response within \\boxed{}. Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?\n$\\text{(A) } \\dfrac{1+\\sqrt2}4 \\quad \\text{(B) } \\dfrac{\\sqrt5-1}2 \\quad \\text{(C) } \\dfrac{\\sqrt3+1}4 \\quad \\text{(D) } \\dfrac{2\\sqrt3}5 \\quad \\text{(E) } \\dfrac{\\sqrt5}3$\n[asy] scale(200); draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle)); path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180); draw(p); p=rotate(90)*p; draw(p); p=rotate(90)*p; draw(p); p=rotate(90)*p; draw(p); draw(scale((sqrt(5)-1)/4)*unitcircle); [/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_490", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?\n$\\text{(A) } \\dfrac{1+\\sqrt2}4 \\quad \\text{(B) } \\dfrac{\\sqrt5-1}2 \\quad \\text{(C) } \\dfrac{\\sqrt3+1}4 \\quad \\text{(D) } \\dfrac{2\\sqrt3}5 \\quad \\text{(E) } \\dfrac{\\sqrt5}3$\n[asy] scale(200); draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle)); path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180); draw(p); p=rotate(90)*p; draw(p); p=rotate(90)*p; draw(p); p=rotate(90)*p; draw(p); draw(scale((sqrt(5)-1)/4)*unitcircle); [/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. If $3^{2x}+9=10\\left(3^{x}\\right)$, then the value of $(x^2+1)$ is \n$\\textbf{(A) }1\\text{ only}\\qquad \\textbf{(B) }5\\text{ only}\\qquad \\textbf{(C) }1\\text{ or }5\\qquad \\textbf{(D) }2\\qquad \\textbf{(E) }10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_491", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $3^{2x}+9=10\\left(3^{x}\\right)$, then the value of $(x^2+1)$ is \n$\\textbf{(A) }1\\text{ only}\\qquad \\textbf{(B) }5\\text{ only}\\qquad \\textbf{(C) }1\\text{ or }5\\qquad \\textbf{(D) }2\\qquad \\textbf{(E) }10$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{1}{10}+\\frac{2}{20}+\\frac{3}{30} =$\n$\\text{(A)}\\ .1 \\qquad \\text{(B)}\\ .123 \\qquad \\text{(C)}\\ .2 \\qquad \\text{(D)}\\ .3 \\qquad \\text{(E)}\\ .6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_492", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{1}{10}+\\frac{2}{20}+\\frac{3}{30} =$\n$\\text{(A)}\\ .1 \\qquad \\text{(B)}\\ .123 \\qquad \\text{(C)}\\ .2 \\qquad \\text{(D)}\\ .3 \\qquad \\text{(E)}\\ .6$" + } + }, + { + "question": "Return your final response within \\boxed{}. A [line](https://artofproblemsolving.com/wiki/index.php/Line) passes through $A\\ (1,1)$ and $B\\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?\n$(\\mathrm {A}) \\ 0 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 3 \\qquad (\\mathrm {D}) \\ 8 \\qquad (\\mathrm {E})\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_493", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A [line](https://artofproblemsolving.com/wiki/index.php/Line) passes through $A\\ (1,1)$ and $B\\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?\n$(\\mathrm {A}) \\ 0 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 3 \\qquad (\\mathrm {D}) \\ 8 \\qquad (\\mathrm {E})\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $$14$ each, $4$-pound rocks worth $$11$ each, and $1$-pound rocks worth $$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?\n$\\textbf{(A) } 48 \\qquad \\textbf{(B) } 49 \\qquad \\textbf{(C) } 50 \\qquad \\textbf{(D) } 51 \\qquad \\textbf{(E) } 52$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_494", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $$14$ each, $4$-pound rocks worth $$11$ each, and $1$-pound rocks worth $$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?\n$\\textbf{(A) } 48 \\qquad \\textbf{(B) } 49 \\qquad \\textbf{(C) } 50 \\qquad \\textbf{(D) } 51 \\qquad \\textbf{(E) } 52$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$? (Note: Both months have $31$ days.)\n$\\textrm{(A)}\\ \\text{Monday} \\qquad \\textrm{(B)}\\ \\text{Tuesday} \\qquad \\textrm{(C)}\\ \\text{Wednesday} \\qquad \\textrm{(D)}\\ \\text{Thursday} \\qquad \\textrm{(E)}\\ \\text{Friday}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_495", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$? (Note: Both months have $31$ days.)\n$\\textrm{(A)}\\ \\text{Monday} \\qquad \\textrm{(B)}\\ \\text{Tuesday} \\qquad \\textrm{(C)}\\ \\text{Wednesday} \\qquad \\textrm{(D)}\\ \\text{Thursday} \\qquad \\textrm{(E)}\\ \\text{Friday}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 56\\qquad\\textbf{(C)}\\ 64\\qquad\\textbf{(D)}\\ 72\\qquad\\textbf{(E)}\\ 80$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_496", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 56\\qquad\\textbf{(C)}\\ 64\\qquad\\textbf{(D)}\\ 72\\qquad\\textbf{(E)}\\ 80$" + } + }, + { + "question": "Return your final response within \\boxed{}. $1000\\times 1993 \\times 0.1993 \\times 10 =$\n$\\text{(A)}\\ 1.993\\times 10^3 \\qquad \\text{(B)}\\ 1993.1993 \\qquad \\text{(C)}\\ (199.3)^2 \\qquad \\text{(D)}\\ 1,993,001.993 \\qquad \\text{(E)}\\ (1993)^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_497", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $1000\\times 1993 \\times 0.1993 \\times 10 =$\n$\\text{(A)}\\ 1.993\\times 10^3 \\qquad \\text{(B)}\\ 1993.1993 \\qquad \\text{(C)}\\ (199.3)^2 \\qquad \\text{(D)}\\ 1,993,001.993 \\qquad \\text{(E)}\\ (1993)^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?\n$\\textbf{(A)}\\ 2\\ \\text{by}\\ 4\\qquad\\textbf{(B)}\\ \\ 2\\ \\text{by}\\ 6\\qquad\\textbf{(C)}\\ \\ 2\\ \\text{by}\\ 8\\qquad\\textbf{(D)}\\ 4\\ \\text{by}\\ 4\\qquad\\textbf{(E)}\\ 4\\ \\text{by}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_498", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?\n$\\textbf{(A)}\\ 2\\ \\text{by}\\ 4\\qquad\\textbf{(B)}\\ \\ 2\\ \\text{by}\\ 6\\qquad\\textbf{(C)}\\ \\ 2\\ \\text{by}\\ 8\\qquad\\textbf{(D)}\\ 4\\ \\text{by}\\ 4\\qquad\\textbf{(E)}\\ 4\\ \\text{by}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?\n$\\textbf{(A) } 1\\qquad \\textbf{(B) } 2\\qquad \\textbf{(C) } 3\\qquad \\textbf{(D) } 4\\qquad \\textbf{(E) } 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_499", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?\n$\\textbf{(A) } 1\\qquad \\textbf{(B) } 2\\qquad \\textbf{(C) } 3\\qquad \\textbf{(D) } 4\\qquad \\textbf{(E) } 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. $a_1, a_2, \\ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at \nrandom. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\\frac14$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\frac{1}{4}", + "index": "Sky-T1_10k_500", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $a_1, a_2, \\ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at \nrandom. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\\frac14$." + } + }, + { + "question": "Return your final response within \\boxed{}. A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?\n$\\textbf{(A)}\\ 11 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 13 \\qquad \\textbf{(D)}\\ 14 \\qquad \\textbf{(E)}\\ 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_501", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?\n$\\textbf{(A)}\\ 11 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 13 \\qquad \\textbf{(D)}\\ 14 \\qquad \\textbf{(E)}\\ 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$, $b$, and $c$ denote three distinct integers, and let $P$ denote a polynomial having all integral coefficients. Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "It is impossible for such a polynomial P to exist.", + "index": "Sky-T1_10k_502", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$, $b$, and $c$ denote three distinct integers, and let $P$ denote a polynomial having all integral coefficients. Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$." + } + }, + { + "question": "Return your final response within \\boxed{}. Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A?\n$\\textbf{(A)}\\ 2 \\frac{3}{4} \\qquad\\textbf{(B)}\\ 3 \\frac{3}{4} \\qquad\\textbf{(C)}\\ 4 \\frac{1}{2} \\qquad\\textbf{(D)}\\ 5 \\frac{1}{2} \\qquad\\textbf{(E)}\\ 6 \\frac{3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_503", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A?\n$\\textbf{(A)}\\ 2 \\frac{3}{4} \\qquad\\textbf{(B)}\\ 3 \\frac{3}{4} \\qquad\\textbf{(C)}\\ 4 \\frac{1}{2} \\qquad\\textbf{(D)}\\ 5 \\frac{1}{2} \\qquad\\textbf{(E)}\\ 6 \\frac{3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A student attempted to compute the average $A$ of $x, y$ and $z$ by computing the average of $x$ and $y$, \nand then computing the average of the result and $z$. Whenever $x < y < z$, the student's final result is\n$\\textbf{(A)}\\ \\text{correct}\\quad \\textbf{(B)}\\ \\text{always less than A}\\quad \\textbf{(C)}\\ \\text{always greater than A}\\quad\\\\ \\textbf{(D)}\\ \\text{sometimes less than A and sometimes equal to A}\\quad\\\\ \\textbf{(E)}\\ \\text{sometimes greater than A and sometimes equal to A} \\quad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_504", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A student attempted to compute the average $A$ of $x, y$ and $z$ by computing the average of $x$ and $y$, \nand then computing the average of the result and $z$. Whenever $x < y < z$, the student's final result is\n$\\textbf{(A)}\\ \\text{correct}\\quad \\textbf{(B)}\\ \\text{always less than A}\\quad \\textbf{(C)}\\ \\text{always greater than A}\\quad\\\\ \\textbf{(D)}\\ \\text{sometimes less than A and sometimes equal to A}\\quad\\\\ \\textbf{(E)}\\ \\text{sometimes greater than A and sometimes equal to A} \\quad$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $p$ is a prime and both roots of $x^2+px-444p=0$ are integers, then\n$\\textbf{(A)}\\ 1b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$.\nHe picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\\to 3,4,1,2\\to 2,3,4,1\\to 2,4,3,1\\to 2,3,4,1$.)\nShow that no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The final number of swapped pairs equals the initial number.", + "index": "Sky-T1_10k_508", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Karl starts with $n$ cards labeled $1,2,3,\\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$.\nHe picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\\to 3,4,1,2\\to 2,3,4,1\\to 2,4,3,1\\to 2,3,4,1$.)\nShow that no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup." + } + }, + { + "question": "Return your final response within \\boxed{}. The acute angles of a right triangle are $a^{\\circ}$ and $b^{\\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 5 \\qquad\\textbf{(D)}\\ 7 \\qquad\\textbf{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_509", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The acute angles of a right triangle are $a^{\\circ}$ and $b^{\\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 5 \\qquad\\textbf{(D)}\\ 7 \\qquad\\textbf{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the true statement: If a quadrilateral is a square, then it is a rectangle. \nIt follows that, of the converse and the inverse of this true statement is:\n$\\textbf{(A)}\\ \\text{only the converse is true} \\qquad \\\\ \\textbf{(B)}\\ \\text{only the inverse is true }\\qquad \\\\ \\textbf{(C)}\\ \\text{both are true} \\qquad \\\\ \\textbf{(D)}\\ \\text{neither is true} \\qquad \\\\ \\textbf{(E)}\\ \\text{the inverse is true, but the converse is sometimes true}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_510", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the true statement: If a quadrilateral is a square, then it is a rectangle. \nIt follows that, of the converse and the inverse of this true statement is:\n$\\textbf{(A)}\\ \\text{only the converse is true} \\qquad \\\\ \\textbf{(B)}\\ \\text{only the inverse is true }\\qquad \\\\ \\textbf{(C)}\\ \\text{both are true} \\qquad \\\\ \\textbf{(D)}\\ \\text{neither is true} \\qquad \\\\ \\textbf{(E)}\\ \\text{the inverse is true, but the converse is sometimes true}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $b>1$, $x>0$, and $(2x)^{\\log_b 2}-(3x)^{\\log_b 3}=0$, then $x$ is\n$\\textbf{(A)}\\ \\dfrac{1}{216}\\qquad\\textbf{(B)}\\ \\dfrac{1}{6}\\qquad\\textbf{(C)}\\ 1\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ \\text{not uniquely determined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_511", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $b>1$, $x>0$, and $(2x)^{\\log_b 2}-(3x)^{\\log_b 3}=0$, then $x$ is\n$\\textbf{(A)}\\ \\dfrac{1}{216}\\qquad\\textbf{(B)}\\ \\dfrac{1}{6}\\qquad\\textbf{(C)}\\ 1\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ \\text{not uniquely determined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, $BD=8$, and $CE=12$. The area of triangle $ABC$ is \n\n$\\textbf{(A)}\\ 24\\qquad\\textbf{(B)}\\ 32\\qquad\\textbf{(C)}\\ 48\\qquad\\textbf{(D)}\\ 64\\qquad\\textbf{(E)}\\ 96$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_512", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, $BD=8$, and $CE=12$. The area of triangle $ABC$ is \n\n$\\textbf{(A)}\\ 24\\qquad\\textbf{(B)}\\ 32\\qquad\\textbf{(C)}\\ 48\\qquad\\textbf{(D)}\\ 64\\qquad\\textbf{(E)}\\ 96$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$\n$\\textbf{(A) }98\\qquad\\textbf{(B) }100\\qquad\\textbf{(C) }117\\qquad\\textbf{(D) }119\\qquad\\textbf{(E) }121$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_513", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$\n$\\textbf{(A) }98\\qquad\\textbf{(B) }100\\qquad\\textbf{(C) }117\\qquad\\textbf{(D) }119\\qquad\\textbf{(E) }121$" + } + }, + { + "question": "Return your final response within \\boxed{}. You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?\n$\\mathrm{(A) \\ } 250\\qquad \\mathrm{(B) \\ } 300 \\qquad \\mathrm{(C) \\ } 1500 \\qquad \\mathrm{(D) \\ } 7500 \\qquad \\mathrm{(E) \\ } 9000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_514", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?\n$\\mathrm{(A) \\ } 250\\qquad \\mathrm{(B) \\ } 300 \\qquad \\mathrm{(C) \\ } 1500 \\qquad \\mathrm{(D) \\ } 7500 \\qquad \\mathrm{(E) \\ } 9000$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is $\\left(20-\\left(2010-201\\right)\\right)+\\left(2010-\\left(201-20\\right)\\right)$?\n$\\textbf{(A)}\\ -4020 \\qquad \\textbf{(B)}\\ 0 \\qquad \\textbf{(C)}\\ 40 \\qquad \\textbf{(D)}\\ 401 \\qquad \\textbf{(E)}\\ 4020$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_515", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is $\\left(20-\\left(2010-201\\right)\\right)+\\left(2010-\\left(201-20\\right)\\right)$?\n$\\textbf{(A)}\\ -4020 \\qquad \\textbf{(B)}\\ 0 \\qquad \\textbf{(C)}\\ 40 \\qquad \\textbf{(D)}\\ 401 \\qquad \\textbf{(E)}\\ 4020$" + } + }, + { + "question": "Return your final response within \\boxed{}. The system of equations\n\\begin{eqnarray*}\\log_{10}(2000xy) - (\\log_{10}x)(\\log_{10}y) & = & 4 \\\\ \\log_{10}(2yz) - (\\log_{10}y)(\\log_{10}z) & = & 1 \\\\ \\log_{10}(zx) - (\\log_{10}z)(\\log_{10}x) & = & 0 \\\\ \\end{eqnarray*}\nhas two solutions $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$. Find $y_{1} + y_{2}$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "25", + "index": "Sky-T1_10k_516", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The system of equations\n\\begin{eqnarray*}\\log_{10}(2000xy) - (\\log_{10}x)(\\log_{10}y) & = & 4 \\\\ \\log_{10}(2yz) - (\\log_{10}y)(\\log_{10}z) & = & 1 \\\\ \\log_{10}(zx) - (\\log_{10}z)(\\log_{10}x) & = & 0 \\\\ \\end{eqnarray*}\nhas two solutions $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$. Find $y_{1} + y_{2}$." + } + }, + { + "question": "Return your final response within \\boxed{}. A non-zero [digit](https://artofproblemsolving.com/wiki/index.php/Digit) is chosen in such a way that the probability of choosing digit $d$ is $\\log_{10}{(d+1)}-\\log_{10}{d}$. The probability that the digit $2$ is chosen is exactly $\\frac{1}{2}$ the probability that the digit chosen is in the set\n$\\mathrm{(A)\\ } \\{2, 3\\} \\qquad \\mathrm{(B) \\ }\\{3, 4\\} \\qquad \\mathrm{(C) \\ } \\{4, 5, 6, 7, 8\\} \\qquad \\mathrm{(D) \\ } \\{5, 6, 7, 8, 9\\} \\qquad \\mathrm{(E) \\ }\\{4, 5, 6, 7, 8, 9\\}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_517", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A non-zero [digit](https://artofproblemsolving.com/wiki/index.php/Digit) is chosen in such a way that the probability of choosing digit $d$ is $\\log_{10}{(d+1)}-\\log_{10}{d}$. The probability that the digit $2$ is chosen is exactly $\\frac{1}{2}$ the probability that the digit chosen is in the set\n$\\mathrm{(A)\\ } \\{2, 3\\} \\qquad \\mathrm{(B) \\ }\\{3, 4\\} \\qquad \\mathrm{(C) \\ } \\{4, 5, 6, 7, 8\\} \\qquad \\mathrm{(D) \\ } \\{5, 6, 7, 8, 9\\} \\qquad \\mathrm{(E) \\ }\\{4, 5, 6, 7, 8, 9\\}$" + } + }, + { + "question": "Return your final response within \\boxed{}. On an algebra quiz, $10\\%$ of the students scored $70$ points, $35\\%$ scored $80$ points, $30\\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_518", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On an algebra quiz, $10\\%$ of the students scored $70$ points, $35\\%$ scored $80$ points, $30\\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define a sequence recursively by $x_0=5$ and \\[x_{n+1}=\\frac{x_n^2+5x_n+4}{x_n+6}\\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that\n\\[x_m\\leq 4+\\frac{1}{2^{20}}.\\]\nIn which of the following intervals does $m$ lie?\n$\\textbf{(A) } [9,26] \\qquad\\textbf{(B) } [27,80] \\qquad\\textbf{(C) } [81,242]\\qquad\\textbf{(D) } [243,728] \\qquad\\textbf{(E) } [729,\\infty)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_519", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define a sequence recursively by $x_0=5$ and \\[x_{n+1}=\\frac{x_n^2+5x_n+4}{x_n+6}\\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that\n\\[x_m\\leq 4+\\frac{1}{2^{20}}.\\]\nIn which of the following intervals does $m$ lie?\n$\\textbf{(A) } [9,26] \\qquad\\textbf{(B) } [27,80] \\qquad\\textbf{(C) } [81,242]\\qquad\\textbf{(D) } [243,728] \\qquad\\textbf{(E) } [729,\\infty)$" + } + }, + { + "question": "Return your final response within \\boxed{}. A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?\n$\\text{(A)}\\frac{1}{4} \\qquad \\text{(B)}\\frac{1}{3} \\qquad \\text{(C)}\\frac{1}{2} \\qquad \\text{(D)}\\frac{2}{3} \\qquad \\text{(E)}\\frac{3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_520", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?\n$\\text{(A)}\\frac{1}{4} \\qquad \\text{(B)}\\frac{1}{3} \\qquad \\text{(C)}\\frac{1}{2} \\qquad \\text{(D)}\\frac{2}{3} \\qquad \\text{(E)}\\frac{3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points \n$(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$. If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$, \nwhich of the following is necessarily the slope of the line which best fits the data? \n\"Best fits\" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.\n$\\textbf{(A)}\\ \\frac{y_{3}-y_{1}}{x_{3}-x_{1}}\\qquad \\textbf{(B)}\\ \\frac{(y_{2}-y_{1})-(y_{3}-y_{2})}{x_{3}-x_{1}}\\qquad\\\\ \\textbf{(C)}\\ \\frac{2y_{3}-y_{1}-y_{2}}{2x_{3}-x_{1}-x_{2}}\\qquad \\textbf{(D)}\\ \\frac{y_{2}-y_{1}}{x_{2}-x_{1}}+\\frac{y_{3}-y_{2}}{x_{3}-x_{2}}\\qquad\\\\ \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_521", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points \n$(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$. If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$, \nwhich of the following is necessarily the slope of the line which best fits the data? \n\"Best fits\" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.\n$\\textbf{(A)}\\ \\frac{y_{3}-y_{1}}{x_{3}-x_{1}}\\qquad \\textbf{(B)}\\ \\frac{(y_{2}-y_{1})-(y_{3}-y_{2})}{x_{3}-x_{1}}\\qquad\\\\ \\textbf{(C)}\\ \\frac{2y_{3}-y_{1}-y_{2}}{2x_{3}-x_{1}-x_{2}}\\qquad \\textbf{(D)}\\ \\frac{y_{2}-y_{1}}{x_{2}-x_{1}}+\\frac{y_{3}-y_{2}}{x_{3}-x_{2}}\\qquad\\\\ \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?\n$\\textbf{(A) }6\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }11\\qquad\\textbf{(E) }15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_522", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?\n$\\textbf{(A) }6\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }11\\qquad\\textbf{(E) }15$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Titu Andreescu) Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that\n\\[\\frac {1}{3}|AB^3 - AD^3| \\le |BC^3 - CD^3| \\le 3|AB^3 - AD^3|.\\]\nWhen does equality hold?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\frac{1}{3}|AB^3 - AD^3| \\le |BC^3 - CD^3| \\le 3|AB^3 - AD^3|", + "index": "Sky-T1_10k_523", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Titu Andreescu) Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that\n\\[\\frac {1}{3}|AB^3 - AD^3| \\le |BC^3 - CD^3| \\le 3|AB^3 - AD^3|.\\]\nWhen does equality hold?" + } + }, + { + "question": "Return your final response within \\boxed{}. Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle School?\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 32 \\qquad \\text{(C)}\\ 40 \\qquad \\text{(D)}\\ 48 \\qquad \\text{(E)}\\ 52$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_524", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle School?\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 32 \\qquad \\text{(C)}\\ 40 \\qquad \\text{(D)}\\ 48 \\qquad \\text{(E)}\\ 52$" + } + }, + { + "question": "Return your final response within \\boxed{}. Show that, for any fixed integer $\\,n \\geq 1,\\,$ the sequence\n\\[2, \\; 2^2, \\; 2^{2^2}, \\; 2^{2^{2^2}}, \\ldots \\pmod{n}\\]\nis eventually constant. \n[The tower of exponents is defined by $a_1 = 2, \\; a_{i+1} = 2^{a_i}$. Also $a_i \\pmod{n}$ means the remainder which results from dividing $\\,a_i\\,$ by $\\,n$.]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The sequence is eventually constant modulo \\( n \\).", + "index": "Sky-T1_10k_525", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Show that, for any fixed integer $\\,n \\geq 1,\\,$ the sequence\n\\[2, \\; 2^2, \\; 2^{2^2}, \\; 2^{2^{2^2}}, \\ldots \\pmod{n}\\]\nis eventually constant. \n[The tower of exponents is defined by $a_1 = 2, \\; a_{i+1} = 2^{a_i}$. Also $a_i \\pmod{n}$ means the remainder which results from dividing $\\,a_i\\,$ by $\\,n$.]" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the sum of digits of all the numbers in the sequence $1,2,3,4,\\cdots ,10000$.\n$\\text{(A) } 180001\\quad \\text{(B) } 154756\\quad \\text{(C) } 45001\\quad \\text{(D) } 154755\\quad \\text{(E) } 270001$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_526", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the sum of digits of all the numbers in the sequence $1,2,3,4,\\cdots ,10000$.\n$\\text{(A) } 180001\\quad \\text{(B) } 154756\\quad \\text{(C) } 45001\\quad \\text{(D) } 154755\\quad \\text{(E) } 270001$" + } + }, + { + "question": "Return your final response within \\boxed{}. On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd?\n\n[asy] draw(circle((0,0),6)); draw(circle((0,0),3)); draw((0,0)--(0,6)); draw((0,0)--(rotate(120)*(0,6))); draw((0,0)--(rotate(-120)*(0,6))); label('1',(rotate(60)*(0,3/2))); label('2',(rotate(-60)*(0,3/2))); label('2',(0,-3/2)); label('2',(rotate(60)*(0,9/2))); label('1',(rotate(-60)*(0,9/2))); label('1',(0,-9/2)); [/asy]\n$\\mathrm{(A)} \\frac{17}{36} \\qquad \\mathrm{(B)} \\frac{35}{72} \\qquad \\mathrm{(C)} \\frac{1}{2} \\qquad \\mathrm{(D)} \\frac{37}{72} \\qquad \\mathrm{(E)} \\frac{19}{36}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_527", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd?\n\n[asy] draw(circle((0,0),6)); draw(circle((0,0),3)); draw((0,0)--(0,6)); draw((0,0)--(rotate(120)*(0,6))); draw((0,0)--(rotate(-120)*(0,6))); label('1',(rotate(60)*(0,3/2))); label('2',(rotate(-60)*(0,3/2))); label('2',(0,-3/2)); label('2',(rotate(60)*(0,9/2))); label('1',(rotate(-60)*(0,9/2))); label('1',(0,-9/2)); [/asy]\n$\\mathrm{(A)} \\frac{17}{36} \\qquad \\mathrm{(B)} \\frac{35}{72} \\qquad \\mathrm{(C)} \\frac{1}{2} \\qquad \\mathrm{(D)} \\frac{37}{72} \\qquad \\mathrm{(E)} \\frac{19}{36}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is [obtuse](https://artofproblemsolving.com/wiki/index.php/Obtuse_triangle), and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible [perimeter](https://artofproblemsolving.com/wiki/index.php/Perimeter).", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "77", + "index": "Sky-T1_10k_528", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is [obtuse](https://artofproblemsolving.com/wiki/index.php/Obtuse_triangle), and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible [perimeter](https://artofproblemsolving.com/wiki/index.php/Perimeter)." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\\overline{AC}$ such that $\\angle ABY = \\angle CBY$ and $\\overline{BE}\\perp\\overline{AC}.$ Similarly, points $F$ and $Z$ are selected on $\\overline{AB}$ such that $\\angle ACZ = \\angle BCZ$ and $\\overline{CF}\\perp\\overline{AB}.$\nLines $I_B F$ and $I_C E$ meet at $P.$ Prove that $\\overline{PO}$ and $\\overline{YZ}$ are perpendicular.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\overline{PO} \\perp \\overline{YZ}", + "index": "Sky-T1_10k_529", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\\overline{AC}$ such that $\\angle ABY = \\angle CBY$ and $\\overline{BE}\\perp\\overline{AC}.$ Similarly, points $F$ and $Z$ are selected on $\\overline{AB}$ such that $\\angle ACZ = \\angle BCZ$ and $\\overline{CF}\\perp\\overline{AB}.$\nLines $I_B F$ and $I_C E$ meet at $P.$ Prove that $\\overline{PO}$ and $\\overline{YZ}$ are perpendicular." + } + }, + { + "question": "Return your final response within \\boxed{}. In a certain year the price of gasoline rose by $20\\%$ during January, fell by $20\\%$ during February, rose by $25\\%$ during March, and fell by $x\\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$\n$\\mathrm{(A)}\\ 12\\qquad \\mathrm{(B)}\\ 17\\qquad \\mathrm{(C)}\\ 20\\qquad \\mathrm{(D)}\\ 25\\qquad \\mathrm{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_530", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a certain year the price of gasoline rose by $20\\%$ during January, fell by $20\\%$ during February, rose by $25\\%$ during March, and fell by $x\\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$\n$\\mathrm{(A)}\\ 12\\qquad \\mathrm{(B)}\\ 17\\qquad \\mathrm{(C)}\\ 20\\qquad \\mathrm{(D)}\\ 25\\qquad \\mathrm{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\\tfrac{12}5\\sqrt2$. What is the volume of the tetrahedron?\n$\\textbf{(A) }3\\sqrt2\\qquad\\textbf{(B) }2\\sqrt5\\qquad\\textbf{(C) }\\dfrac{24}5\\qquad\\textbf{(D) }3\\sqrt3\\qquad\\textbf{(E) }\\dfrac{24}5\\sqrt2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_531", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\\tfrac{12}5\\sqrt2$. What is the volume of the tetrahedron?\n$\\textbf{(A) }3\\sqrt2\\qquad\\textbf{(B) }2\\sqrt5\\qquad\\textbf{(C) }\\dfrac{24}5\\qquad\\textbf{(D) }3\\sqrt3\\qquad\\textbf{(E) }\\dfrac{24}5\\sqrt2$" + } + }, + { + "question": "Return your final response within \\boxed{}. In racing over a distance $d$ at uniform speed, $A$ can beat $B$ by $20$ yards, $B$ can beat $C$ by $10$ yards, \nand $A$ can beat $C$ by $28$ yards. Then $d$, in yards, equals:\n$\\textbf{(A)}\\ \\text{Not determined by the given information} \\qquad \\textbf{(B)}\\ 58\\qquad \\textbf{(C)}\\ 100\\qquad \\textbf{(D)}\\ 116\\qquad \\textbf{(E)}\\ 120$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_532", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In racing over a distance $d$ at uniform speed, $A$ can beat $B$ by $20$ yards, $B$ can beat $C$ by $10$ yards, \nand $A$ can beat $C$ by $28$ yards. Then $d$, in yards, equals:\n$\\textbf{(A)}\\ \\text{Not determined by the given information} \\qquad \\textbf{(B)}\\ 58\\qquad \\textbf{(C)}\\ 100\\qquad \\textbf{(D)}\\ 116\\qquad \\textbf{(E)}\\ 120$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two distinct [numbers](https://artofproblemsolving.com/wiki/index.php/Number) a and b are chosen randomly from the [set](https://artofproblemsolving.com/wiki/index.php/Set) $\\{2, 2^2, 2^3, ..., 2^{25}\\}$. What is the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that $\\mathrm{log}_a b$ is an [integer](https://artofproblemsolving.com/wiki/index.php/Integer)?\n$\\mathrm{(A)}\\ \\frac{2}{25}\\qquad \\mathrm{(B)}\\ \\frac{31}{300}\\qquad \\mathrm{(C)}\\ \\frac{13}{100}\\qquad \\mathrm{(D)}\\ \\frac{7}{50}\\qquad \\mathrm{(E)}\\ \\frac 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_533", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two distinct [numbers](https://artofproblemsolving.com/wiki/index.php/Number) a and b are chosen randomly from the [set](https://artofproblemsolving.com/wiki/index.php/Set) $\\{2, 2^2, 2^3, ..., 2^{25}\\}$. What is the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that $\\mathrm{log}_a b$ is an [integer](https://artofproblemsolving.com/wiki/index.php/Integer)?\n$\\mathrm{(A)}\\ \\frac{2}{25}\\qquad \\mathrm{(B)}\\ \\frac{31}{300}\\qquad \\mathrm{(C)}\\ \\frac{13}{100}\\qquad \\mathrm{(D)}\\ \\frac{7}{50}\\qquad \\mathrm{(E)}\\ \\frac 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "132", + "index": "Sky-T1_10k_534", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes." + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\log_{k}{x}\\cdot \\log_{5}{k} = 3$, then $x$ equals:\n$\\textbf{(A)}\\ k^6\\qquad \\textbf{(B)}\\ 5k^3\\qquad \\textbf{(C)}\\ k^3\\qquad \\textbf{(D)}\\ 243\\qquad \\textbf{(E)}\\ 125$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_535", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\log_{k}{x}\\cdot \\log_{5}{k} = 3$, then $x$ equals:\n$\\textbf{(A)}\\ k^6\\qquad \\textbf{(B)}\\ 5k^3\\qquad \\textbf{(C)}\\ k^3\\qquad \\textbf{(D)}\\ 243\\qquad \\textbf{(E)}\\ 125$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose $n$ is a [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer) and $d$ is a single [digit](https://artofproblemsolving.com/wiki/index.php/Digit) in [base 10](https://artofproblemsolving.com/wiki/index.php/Base_10). Find $n$ if\n\n$\\frac{n}{810}=0.d25d25d25\\ldots$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "750", + "index": "Sky-T1_10k_536", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose $n$ is a [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer) and $d$ is a single [digit](https://artofproblemsolving.com/wiki/index.php/Digit) in [base 10](https://artofproblemsolving.com/wiki/index.php/Base_10). Find $n$ if\n\n$\\frac{n}{810}=0.d25d25d25\\ldots$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\bigtriangleup ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$. Moreover $\\overline{BX}$ and $\\overline{CX}$ have integer lengths. What is $BC$?\n\n$\\textbf{(A)} \\ 11 \\qquad \\textbf{(B)} \\ 28 \\qquad \\textbf{(C)} \\ 33 \\qquad \\textbf{(D)} \\ 61 \\qquad \\textbf{(E)} \\ 72$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_537", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\bigtriangleup ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$. Moreover $\\overline{BX}$ and $\\overline{CX}$ have integer lengths. What is $BC$?\n\n$\\textbf{(A)} \\ 11 \\qquad \\textbf{(B)} \\ 28 \\qquad \\textbf{(C)} \\ 33 \\qquad \\textbf{(D)} \\ 61 \\qquad \\textbf{(E)} \\ 72$" + } + }, + { + "question": "Return your final response within \\boxed{}. In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?\n$\\textbf{(A)}\\ 35\\qquad \\textbf{(B)}\\ 40\\qquad \\textbf{(C)}\\ 48\\qquad \\textbf{(D)}\\ 56\\qquad \\textbf{(E)}\\ 72$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_538", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?\n$\\textbf{(A)}\\ 35\\qquad \\textbf{(B)}\\ 40\\qquad \\textbf{(C)}\\ 48\\qquad \\textbf{(D)}\\ 56\\qquad \\textbf{(E)}\\ 72$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+\\cdots+(n+1)i^n$, where $i=\\sqrt{-1}$, equals:\n$\\textbf{(A) }1+i\\qquad\\textbf{(B) }\\frac{1}{2}(n+2)\\qquad\\textbf{(C) }\\frac{1}{2}(n+2-ni)\\qquad$\n$\\textbf{(D) }\\frac{1}{2}[(n+1)(1-i)+2]\\qquad \\textbf{(E) }\\frac{1}{8}(n^2+8-4ni)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_539", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+\\cdots+(n+1)i^n$, where $i=\\sqrt{-1}$, equals:\n$\\textbf{(A) }1+i\\qquad\\textbf{(B) }\\frac{1}{2}(n+2)\\qquad\\textbf{(C) }\\frac{1}{2}(n+2-ni)\\qquad$\n$\\textbf{(D) }\\frac{1}{2}[(n+1)(1-i)+2]\\qquad \\textbf{(E) }\\frac{1}{8}(n^2+8-4ni)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Set $A$ has $20$ elements, and set $B$ has $15$ elements. What is the smallest possible number of elements in $A \\cup B$?\n$\\textbf{(A)}5 \\qquad\\textbf{(B)}\\ 15 \\qquad\\textbf{(C)}\\ 20\\qquad\\textbf{(D)}\\ 35\\qquad\\textbf{(E)}\\ 300$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_540", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Set $A$ has $20$ elements, and set $B$ has $15$ elements. What is the smallest possible number of elements in $A \\cup B$?\n$\\textbf{(A)}5 \\qquad\\textbf{(B)}\\ 15 \\qquad\\textbf{(C)}\\ 20\\qquad\\textbf{(D)}\\ 35\\qquad\\textbf{(E)}\\ 300$" + } + }, + { + "question": "Return your final response within \\boxed{}. Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 60$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_541", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 60$" + } + }, + { + "question": "Return your final response within \\boxed{}. Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number of hours it will take her to shovel it clean is closest to\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_542", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number of hours it will take her to shovel it clean is closest to\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?\n$\\text{(A)}\\ 30\\qquad\\text{(B)}\\ 40\\qquad\\text{(C)}\\ 50\\qquad\\text{(D)}\\ 60\\qquad\\text{(E)}\\ 70$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_543", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?\n$\\text{(A)}\\ 30\\qquad\\text{(B)}\\ 40\\qquad\\text{(C)}\\ 50\\qquad\\text{(D)}\\ 60\\qquad\\text{(E)}\\ 70$" + } + }, + { + "question": "Return your final response within \\boxed{}. It takes $5$ seconds for a clock to strike $6$ o'clock beginning at $6:00$ o'clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike $12$ o'clock?\n$\\textbf{(A)}9\\frac{1}{5}\\qquad \\textbf{(B )}10\\qquad \\textbf{(C )}11\\qquad \\textbf{(D )}14\\frac{2}{5}\\qquad \\textbf{(E )}\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_544", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. It takes $5$ seconds for a clock to strike $6$ o'clock beginning at $6:00$ o'clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike $12$ o'clock?\n$\\textbf{(A)}9\\frac{1}{5}\\qquad \\textbf{(B )}10\\qquad \\textbf{(C )}11\\qquad \\textbf{(D )}14\\frac{2}{5}\\qquad \\textbf{(E )}\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of the letters $\\text{W}$, $\\text{X}$, $\\text{Y}$, and $\\text{Z}$ represents a different integer in the set $\\{ 1,2,3,4\\}$, but not necessarily in that order. If $\\dfrac{\\text{W}}{\\text{X}} - \\dfrac{\\text{Y}}{\\text{Z}}=1$, then the sum of $\\text{W}$ and $\\text{Y}$ is\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_545", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of the letters $\\text{W}$, $\\text{X}$, $\\text{Y}$, and $\\text{Z}$ represents a different integer in the set $\\{ 1,2,3,4\\}$, but not necessarily in that order. If $\\dfrac{\\text{W}}{\\text{X}} - \\dfrac{\\text{Y}}{\\text{Z}}=1$, then the sum of $\\text{W}$ and $\\text{Y}$ is\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?\n$\\textbf{(A)}\\ 12\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 28\\qquad\\textbf{(D)}\\ 32\\qquad\\textbf{(E)}\\ 40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_546", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?\n$\\textbf{(A)}\\ 12\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 28\\qquad\\textbf{(D)}\\ 32\\qquad\\textbf{(E)}\\ 40$" + } + }, + { + "question": "Return your final response within \\boxed{}. A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?\n$\\textbf{(A)}\\ 5\\sqrt{2} - 7 \\qquad \\textbf{(B)}\\ 7 - 4\\sqrt{3} \\qquad \\textbf{(C)}\\ \\frac{2\\sqrt{2}}{27} \\qquad \\textbf{(D)}\\ \\frac{\\sqrt{2}}{9} \\qquad \\textbf{(E)}\\ \\frac{\\sqrt{3}}{9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_547", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?\n$\\textbf{(A)}\\ 5\\sqrt{2} - 7 \\qquad \\textbf{(B)}\\ 7 - 4\\sqrt{3} \\qquad \\textbf{(C)}\\ \\frac{2\\sqrt{2}}{27} \\qquad \\textbf{(D)}\\ \\frac{\\sqrt{2}}{9} \\qquad \\textbf{(E)}\\ \\frac{\\sqrt{3}}{9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Chords $AA'$, $BB'$, and $CC'$ of a sphere meet at an interior point $P$ but are not contained in the same plane. The sphere through $A$, $B$, $C$, and $P$ is tangent to the sphere through $A'$, $B'$, $C'$, and $P$. Prove that $AA'=BB'=CC'$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "AA'=BB'=CC'", + "index": "Sky-T1_10k_548", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Chords $AA'$, $BB'$, and $CC'$ of a sphere meet at an interior point $P$ but are not contained in the same plane. The sphere through $A$, $B$, $C$, and $P$ is tangent to the sphere through $A'$, $B'$, $C'$, and $P$. Prove that $AA'=BB'=CC'$." + } + }, + { + "question": "Return your final response within \\boxed{}. The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is: \n$\\text{(A) greater than the area of the given rectangle} \\quad\\\\ \\text{(B) equal to the area of the given rectangle} \\quad\\\\ \\text{(C) equal to the area of a square with its side equal to the smaller side of the given rectangle} \\quad\\\\ \\text{(D) equal to the area of a square with its side equal to the greater side of the given rectangle} \\quad\\\\ \\text{(E) equal to the area of a rectangle whose dimensions are the diagonal and the shorter side of the given rectangle}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_549", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is: \n$\\text{(A) greater than the area of the given rectangle} \\quad\\\\ \\text{(B) equal to the area of the given rectangle} \\quad\\\\ \\text{(C) equal to the area of a square with its side equal to the smaller side of the given rectangle} \\quad\\\\ \\text{(D) equal to the area of a square with its side equal to the greater side of the given rectangle} \\quad\\\\ \\text{(E) equal to the area of a rectangle whose dimensions are the diagonal and the shorter side of the given rectangle}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If February is a month that contains Friday the $13^{\\text{th}}$, what day of the week is February 1?\n$\\textbf{(A)}\\ \\text{Sunday} \\qquad \\textbf{(B)}\\ \\text{Monday} \\qquad \\textbf{(C)}\\ \\text{Wednesday} \\qquad \\textbf{(D)}\\ \\text{Thursday}\\qquad \\textbf{(E)}\\ \\text{Saturday}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_550", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If February is a month that contains Friday the $13^{\\text{th}}$, what day of the week is February 1?\n$\\textbf{(A)}\\ \\text{Sunday} \\qquad \\textbf{(B)}\\ \\text{Monday} \\qquad \\textbf{(C)}\\ \\text{Wednesday} \\qquad \\textbf{(D)}\\ \\text{Thursday}\\qquad \\textbf{(E)}\\ \\text{Saturday}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ is positive and $\\log{x} \\ge \\log{2} + \\frac{1}{2}\\log{x}$, then:\n$\\textbf{(A)}\\ {x}\\text{ has no minimum or maximum value}\\qquad \\\\ \\textbf{(B)}\\ \\text{the maximum value of }{x}\\text{ is }{1}\\qquad \\\\ \\textbf{(C)}\\ \\text{the minimum value of }{x}\\text{ is }{1}\\qquad \\\\ \\textbf{(D)}\\ \\text{the maximum value of }{x}\\text{ is }{4}\\qquad \\\\ \\textbf{(E)}\\ \\text{the minimum value of }{x}\\text{ is }{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_551", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ is positive and $\\log{x} \\ge \\log{2} + \\frac{1}{2}\\log{x}$, then:\n$\\textbf{(A)}\\ {x}\\text{ has no minimum or maximum value}\\qquad \\\\ \\textbf{(B)}\\ \\text{the maximum value of }{x}\\text{ is }{1}\\qquad \\\\ \\textbf{(C)}\\ \\text{the minimum value of }{x}\\text{ is }{1}\\qquad \\\\ \\textbf{(D)}\\ \\text{the maximum value of }{x}\\text{ is }{4}\\qquad \\\\ \\textbf{(E)}\\ \\text{the minimum value of }{x}\\text{ is }{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A charity sells $140$ benefit tickets for a total of $$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?\n$\\textbf{(A) } \\textdollar 782 \\qquad \\textbf{(B) } \\textdollar 986 \\qquad \\textbf{(C) } \\textdollar 1158 \\qquad \\textbf{(D) } \\textdollar 1219 \\qquad \\textbf{(E) }\\ \\textdollar 1449$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_552", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A charity sells $140$ benefit tickets for a total of $$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?\n$\\textbf{(A) } \\textdollar 782 \\qquad \\textbf{(B) } \\textdollar 986 \\qquad \\textbf{(C) } \\textdollar 1158 \\qquad \\textbf{(D) } \\textdollar 1219 \\qquad \\textbf{(E) }\\ \\textdollar 1449$" + } + }, + { + "question": "Return your final response within \\boxed{}. A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\\frac{1}{2}$ foot from the top face. The second cut is $\\frac{1}{3}$ foot below the first cut, and the third cut is $\\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?\n\n\n$\\textbf{(A)}\\:6\\qquad\\textbf{(B)}\\:7\\qquad\\textbf{(C)}\\:\\frac{419}{51}\\qquad\\textbf{(D)}\\:\\frac{158}{17}\\qquad\\textbf{(E)}\\:11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_553", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\\frac{1}{2}$ foot from the top face. The second cut is $\\frac{1}{3}$ foot below the first cut, and the third cut is $\\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?\n\n\n$\\textbf{(A)}\\:6\\qquad\\textbf{(B)}\\:7\\qquad\\textbf{(C)}\\:\\frac{419}{51}\\qquad\\textbf{(D)}\\:\\frac{158}{17}\\qquad\\textbf{(E)}\\:11$" + } + }, + { + "question": "Return your final response within \\boxed{}. A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "945", + "index": "Sky-T1_10k_554", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting." + } + }, + { + "question": "Return your final response within \\boxed{}. A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $$17.71$. What was the cost of a pencil in cents?\n$\\textbf{(A)}\\ 7 \\qquad \\textbf{(B)}\\ 11 \\qquad \\textbf{(C)}\\ 17 \\qquad \\textbf{(D)}\\ 23 \\qquad \\textbf{(E)}\\ 77$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_555", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $$17.71$. What was the cost of a pencil in cents?\n$\\textbf{(A)}\\ 7 \\qquad \\textbf{(B)}\\ 11 \\qquad \\textbf{(C)}\\ 17 \\qquad \\textbf{(D)}\\ 23 \\qquad \\textbf{(E)}\\ 77$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following methods of proving a geometric figure a locus is not correct? \n$\\textbf{(A)}\\ \\text{Every point of the locus satisfies the conditions and every point not on the locus does}\\\\ \\text{not satisfy the conditions.}$\n$\\textbf{(B)}\\ \\text{Every point not satisfying the conditions is not on the locus and every point on the locus}\\\\ \\text{does satisfy the conditions.}$\n$\\textbf{(C)}\\ \\text{Every point satisfying the conditions is on the locus and every point on the locus satisfies}\\\\ \\text{the conditions.}$\n$\\textbf{(D)}\\ \\text{Every point not on the locus does not satisfy the conditions and every point not satisfying}\\\\ \\text{the conditions is not on the locus.}$\n$\\textbf{(E)}\\ \\text{Every point satisfying the conditions is on the locus and every point not satisfying the} \\\\ \\text{conditions is not on the locus.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_556", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following methods of proving a geometric figure a locus is not correct? \n$\\textbf{(A)}\\ \\text{Every point of the locus satisfies the conditions and every point not on the locus does}\\\\ \\text{not satisfy the conditions.}$\n$\\textbf{(B)}\\ \\text{Every point not satisfying the conditions is not on the locus and every point on the locus}\\\\ \\text{does satisfy the conditions.}$\n$\\textbf{(C)}\\ \\text{Every point satisfying the conditions is on the locus and every point on the locus satisfies}\\\\ \\text{the conditions.}$\n$\\textbf{(D)}\\ \\text{Every point not on the locus does not satisfy the conditions and every point not satisfying}\\\\ \\text{the conditions is not on the locus.}$\n$\\textbf{(E)}\\ \\text{Every point satisfying the conditions is on the locus and every point not satisfying the} \\\\ \\text{conditions is not on the locus.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\le 4$. Prove that\n\\[\\frac{ab + 1}{(a + b)^2} + \\frac{bc + 1}{(b + c)^2} + \\frac{ca + 1}{(c + a)^2} \\ge 3.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "3", + "index": "Sky-T1_10k_557", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\le 4$. Prove that\n\\[\\frac{ab + 1}{(a + b)^2} + \\frac{bc + 1}{(b + c)^2} + \\frac{ca + 1}{(c + a)^2} \\ge 3.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. If $m$ and $b$ are real numbers and $mb>0$, then the line whose equation is $y=mx+b$ cannot contain the point\n$\\textbf{(A)}\\ (0,1997)\\qquad\\textbf{(B)}\\ (0,-1997)\\qquad\\textbf{(C)}\\ (19,97)\\qquad\\textbf{(D)}\\ (19,-97)\\qquad\\textbf{(E)}\\ (1997,0)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_558", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $m$ and $b$ are real numbers and $mb>0$, then the line whose equation is $y=mx+b$ cannot contain the point\n$\\textbf{(A)}\\ (0,1997)\\qquad\\textbf{(B)}\\ (0,-1997)\\qquad\\textbf{(C)}\\ (19,97)\\qquad\\textbf{(D)}\\ (19,-97)\\qquad\\textbf{(E)}\\ (1997,0)$" + } + }, + { + "question": "Return your final response within \\boxed{}. A housewife saved $\\textdollar{2.50}$ in buying a dress on sale. If she spent $\\textdollar{25}$ for the dress, she saved about: \n$\\textbf{(A)}\\ 8 \\% \\qquad \\textbf{(B)}\\ 9 \\% \\qquad \\textbf{(C)}\\ 10 \\% \\qquad \\textbf{(D)}\\ 11 \\% \\qquad \\textbf{(E)}\\ 12\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_559", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A housewife saved $\\textdollar{2.50}$ in buying a dress on sale. If she spent $\\textdollar{25}$ for the dress, she saved about: \n$\\textbf{(A)}\\ 8 \\% \\qquad \\textbf{(B)}\\ 9 \\% \\qquad \\textbf{(C)}\\ 10 \\% \\qquad \\textbf{(D)}\\ 11 \\% \\qquad \\textbf{(E)}\\ 12\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "231", + "index": "Sky-T1_10k_560", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$." + } + }, + { + "question": "Return your final response within \\boxed{}. Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? \n$\\mathrm{(A)}\\ 678 \\qquad \\mathrm{(B)}\\ 768 \\qquad \\mathrm{(C)}\\ 786 \\qquad \\mathrm{(D)}\\ 867 \\qquad \\mathrm{(E)}\\ 876$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_561", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? \n$\\mathrm{(A)}\\ 678 \\qquad \\mathrm{(B)}\\ 768 \\qquad \\mathrm{(C)}\\ 786 \\qquad \\mathrm{(D)}\\ 867 \\qquad \\mathrm{(E)}\\ 876$" + } + }, + { + "question": "Return your final response within \\boxed{}. When one ounce of water is added to a mixture of acid and water, the new mixture is $20\\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\\frac13\\%$ acid. The percentage of acid in the original mixture is\n$\\textbf{(A)}\\ 22\\% \\qquad \\textbf{(B)}\\ 24\\% \\qquad \\textbf{(C)}\\ 25\\% \\qquad \\textbf{(D)}\\ 30\\% \\qquad \\textbf{(E)}\\ 33\\frac13 \\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_562", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When one ounce of water is added to a mixture of acid and water, the new mixture is $20\\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\\frac13\\%$ acid. The percentage of acid in the original mixture is\n$\\textbf{(A)}\\ 22\\% \\qquad \\textbf{(B)}\\ 24\\% \\qquad \\textbf{(C)}\\ 25\\% \\qquad \\textbf{(D)}\\ 30\\% \\qquad \\textbf{(E)}\\ 33\\frac13 \\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. A mixture of $30$ liters of paint is $25\\%$ red tint, $30\\%$ yellow\ntint and $45\\%$ water. Five liters of yellow tint are added to\nthe original mixture. What is the percent of yellow tint\nin the new mixture?\n$\\mathrm{(A)}\\ 25 \\qquad \\mathrm{(B)}\\ 35 \\qquad \\mathrm{(C)}\\ 40 \\qquad \\mathrm{(D)}\\ 45 \\qquad \\mathrm{(E)}\\ 50$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_563", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A mixture of $30$ liters of paint is $25\\%$ red tint, $30\\%$ yellow\ntint and $45\\%$ water. Five liters of yellow tint are added to\nthe original mixture. What is the percent of yellow tint\nin the new mixture?\n$\\mathrm{(A)}\\ 25 \\qquad \\mathrm{(B)}\\ 35 \\qquad \\mathrm{(C)}\\ 40 \\qquad \\mathrm{(D)}\\ 45 \\qquad \\mathrm{(E)}\\ 50$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 =25$ and $(x-4)^2 + 9y^2 = 81 ?$\n$\\textbf{(A)}\\ 24\\qquad\\textbf{(B)}\\ 27\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 37.5\\qquad\\textbf{(E)}\\ 42$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_564", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 =25$ and $(x-4)^2 + 9y^2 = 81 ?$\n$\\textbf{(A)}\\ 24\\qquad\\textbf{(B)}\\ 27\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 37.5\\qquad\\textbf{(E)}\\ 42$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $B$ and $C$ lie on $\\overline{AD}$. The length of $\\overline{AB}$ is $4$ times the length of $\\overline{BD}$, and the length of $\\overline{AC}$ is $9$ times the length of $\\overline{CD}$. The length of $\\overline{BC}$ is what fraction of the length of $\\overline{AD}$?\n$\\textbf{(A)}\\ \\frac{1}{36}\\qquad\\textbf{(B)}\\ \\frac{1}{13}\\qquad\\textbf{(C)}\\ \\frac{1}{10}\\qquad\\textbf{(D)}\\ \\frac{5}{36}\\qquad\\textbf{(E)}\\ \\frac{1}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_565", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $B$ and $C$ lie on $\\overline{AD}$. The length of $\\overline{AB}$ is $4$ times the length of $\\overline{BD}$, and the length of $\\overline{AC}$ is $9$ times the length of $\\overline{CD}$. The length of $\\overline{BC}$ is what fraction of the length of $\\overline{AD}$?\n$\\textbf{(A)}\\ \\frac{1}{36}\\qquad\\textbf{(B)}\\ \\frac{1}{13}\\qquad\\textbf{(C)}\\ \\frac{1}{10}\\qquad\\textbf{(D)}\\ \\frac{5}{36}\\qquad\\textbf{(E)}\\ \\frac{1}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?\n$\\textbf{(A) } 22 \\qquad\\textbf{(B) } 24 \\qquad\\textbf{(C) } 25 \\qquad\\textbf{(D) } 26 \\qquad\\textbf{(E) } 28$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_566", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?\n$\\textbf{(A) } 22 \\qquad\\textbf{(B) } 24 \\qquad\\textbf{(C) } 25 \\qquad\\textbf{(D) } 26 \\qquad\\textbf{(E) } 28$" + } + }, + { + "question": "Return your final response within \\boxed{}. The perimeter of an isosceles right triangle is $2p$. Its area is:\n$\\textbf{(A)}\\ (2+\\sqrt{2})p \\qquad \\textbf{(B)}\\ (2-\\sqrt{2})p \\qquad \\textbf{(C)}\\ (3-2\\sqrt{2})p^2\\\\ \\textbf{(D)}\\ (1-2\\sqrt{2})p^2\\qquad \\textbf{(E)}\\ (3+2\\sqrt{2})p^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_567", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The perimeter of an isosceles right triangle is $2p$. Its area is:\n$\\textbf{(A)}\\ (2+\\sqrt{2})p \\qquad \\textbf{(B)}\\ (2-\\sqrt{2})p \\qquad \\textbf{(C)}\\ (3-2\\sqrt{2})p^2\\\\ \\textbf{(D)}\\ (1-2\\sqrt{2})p^2\\qquad \\textbf{(E)}\\ (3+2\\sqrt{2})p^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. At his usual rate a man rows 15 miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is:\n$\\text{(A) } 2 \\quad \\text{(B) } \\frac{5}{2} \\quad \\text{(C) } 3 \\quad \\text{(D) } \\frac{7}{2} \\quad \\text{(E) } 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_568", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. At his usual rate a man rows 15 miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is:\n$\\text{(A) } 2 \\quad \\text{(B) } \\frac{5}{2} \\quad \\text{(C) } 3 \\quad \\text{(D) } \\frac{7}{2} \\quad \\text{(E) } 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. If two poles $20''$ and $80''$ high are $100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is: \n$\\textbf{(A)}\\ 50''\\qquad\\textbf{(B)}\\ 40''\\qquad\\textbf{(C)}\\ 16''\\qquad\\textbf{(D)}\\ 60''\\qquad\\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_569", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If two poles $20''$ and $80''$ high are $100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is: \n$\\textbf{(A)}\\ 50''\\qquad\\textbf{(B)}\\ 40''\\qquad\\textbf{(C)}\\ 16''\\qquad\\textbf{(D)}\\ 60''\\qquad\\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is the largest?\n$\\text{(A)}\\ \\dfrac{1}{3} \\qquad \\text{(B)}\\ \\dfrac{1}{4} \\qquad \\text{(C)}\\ \\dfrac{3}{8} \\qquad \\text{(D)}\\ \\dfrac{5}{12} \\qquad \\text{(E)}\\ \\dfrac{7}{24}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_570", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is the largest?\n$\\text{(A)}\\ \\dfrac{1}{3} \\qquad \\text{(B)}\\ \\dfrac{1}{4} \\qquad \\text{(C)}\\ \\dfrac{3}{8} \\qquad \\text{(D)}\\ \\dfrac{5}{12} \\qquad \\text{(E)}\\ \\dfrac{7}{24}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$?\n$\\textbf{(A)}\\ 60^\\circ \\qquad \\textbf{(B)}\\ 75^\\circ \\qquad \\textbf{(C)}\\ 90^\\circ \\qquad \\textbf{(D)}\\ 105^\\circ \\qquad \\textbf{(E)}\\ 120^\\circ$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_571", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$?\n$\\textbf{(A)}\\ 60^\\circ \\qquad \\textbf{(B)}\\ 75^\\circ \\qquad \\textbf{(C)}\\ 90^\\circ \\qquad \\textbf{(D)}\\ 105^\\circ \\qquad \\textbf{(E)}\\ 120^\\circ$" + } + }, + { + "question": "Return your final response within \\boxed{}. Penni Precisely buys $100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then\n$\\text{(A)}\\ A=B=C \\qquad \\text{(B)}\\ A=B0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\\Delta$, such that each unit equilateral triangle has sides parallel to $\\Delta$, but with opposite orientation. (An example with $n=2$ is drawn below.)\n[asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy]\nProve that\\[n \\leq \\frac{2}{3} L^{2}.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "n \\leq \\frac{2}{3} L^{2}", + "index": "Sky-T1_10k_590", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An equilateral triangle $\\Delta$ of side length $L>0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\\Delta$, such that each unit equilateral triangle has sides parallel to $\\Delta$, but with opposite orientation. (An example with $n=2$ is drawn below.)\n[asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy]\nProve that\\[n \\leq \\frac{2}{3} L^{2}.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 54\\qquad\\textbf{(C)}\\ 60\\qquad\\textbf{(D)}\\ 66\\qquad\\textbf{(E)}\\ 72$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_591", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 54\\qquad\\textbf{(C)}\\ 60\\qquad\\textbf{(D)}\\ 66\\qquad\\textbf{(E)}\\ 72$" + } + }, + { + "question": "Return your final response within \\boxed{}. Heather compares the price of a new computer at two different stores. Store $A$ offers $15\\%$ off the sticker price followed by a $$90$ rebate, and store $B$ offers $25\\%$ off the same sticker price with no rebate. Heather saves $$15$ by buying the computer at store $A$ instead of store $B$. What is the sticker price of the computer, in dollars?\n$\\mathrm{(A)}\\ 750\\qquad\\mathrm{(B)}\\ 900\\qquad\\mathrm{(C)}\\ 1000\\qquad\\mathrm{(D)}\\ 1050\\qquad\\mathrm{(E)}\\ 1500$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_592", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Heather compares the price of a new computer at two different stores. Store $A$ offers $15\\%$ off the sticker price followed by a $$90$ rebate, and store $B$ offers $25\\%$ off the same sticker price with no rebate. Heather saves $$15$ by buying the computer at store $A$ instead of store $B$. What is the sticker price of the computer, in dollars?\n$\\mathrm{(A)}\\ 750\\qquad\\mathrm{(B)}\\ 900\\qquad\\mathrm{(C)}\\ 1000\\qquad\\mathrm{(D)}\\ 1050\\qquad\\mathrm{(E)}\\ 1500$" + } + }, + { + "question": "Return your final response within \\boxed{}. [asy] draw(circle((0,0),10),black+linewidth(1)); draw(circle((-1.25,2.5),4.5),black+linewidth(1)); dot((0,0)); dot((-1.25,2.5)); draw((-sqrt(96),-2)--(-2,sqrt(96)),black+linewidth(.5)); draw((-2,sqrt(96))--(sqrt(96),-2),black+linewidth(.5)); draw((-sqrt(96),-2)--(sqrt(96)-2.5,7),black+linewidth(.5)); draw((-sqrt(96),-2)--(sqrt(96),-2),black+linewidth(.5)); MP(\"O'\", (0,0), W); MP(\"O\", (-2,2), W); MP(\"A\", (-10,-2), W); MP(\"B\", (10,-2), E); MP(\"C\", (-2,sqrt(96)), N); MP(\"D\", (sqrt(96)-2.5,7), NE); [/asy]\nTriangle $ABC$ is inscribed in a circle with center $O'$. A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$. Then we must have:\n$\\text{(A) } CD=BD=O'D \\quad \\text{(B) } AO=CO=OD \\quad \\text{(C) } CD=CO=BD \\\\ \\text{(D) } CD=OD=BD \\quad \\text{(E) } O'B=O'C=OD$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_593", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [asy] draw(circle((0,0),10),black+linewidth(1)); draw(circle((-1.25,2.5),4.5),black+linewidth(1)); dot((0,0)); dot((-1.25,2.5)); draw((-sqrt(96),-2)--(-2,sqrt(96)),black+linewidth(.5)); draw((-2,sqrt(96))--(sqrt(96),-2),black+linewidth(.5)); draw((-sqrt(96),-2)--(sqrt(96)-2.5,7),black+linewidth(.5)); draw((-sqrt(96),-2)--(sqrt(96),-2),black+linewidth(.5)); MP(\"O'\", (0,0), W); MP(\"O\", (-2,2), W); MP(\"A\", (-10,-2), W); MP(\"B\", (10,-2), E); MP(\"C\", (-2,sqrt(96)), N); MP(\"D\", (sqrt(96)-2.5,7), NE); [/asy]\nTriangle $ABC$ is inscribed in a circle with center $O'$. A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$. Then we must have:\n$\\text{(A) } CD=BD=O'D \\quad \\text{(B) } AO=CO=OD \\quad \\text{(C) } CD=CO=BD \\\\ \\text{(D) } CD=OD=BD \\quad \\text{(E) } O'B=O'C=OD$" + } + }, + { + "question": "Return your final response within \\boxed{}. Values for $A,B,C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$)\n$\\textbf{(A) }30 \\qquad \\textbf{(B) }60 \\qquad \\textbf{(C) }90 \\qquad \\textbf{(D) }180 \\qquad \\textbf{(E) }360$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_594", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Values for $A,B,C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$)\n$\\textbf{(A) }30 \\qquad \\textbf{(B) }60 \\qquad \\textbf{(C) }90 \\qquad \\textbf{(D) }180 \\qquad \\textbf{(E) }360$" + } + }, + { + "question": "Return your final response within \\boxed{}. Last year Mr. Jon Q. Public received an inheritance. He paid $20\\%$ in federal taxes on the inheritance, and paid $10\\%$ of what he had left in state taxes. He paid a total of $\\textdollar10500$ for both taxes. How many dollars was his inheritance?\n$(\\mathrm {A})\\ 30000 \\qquad (\\mathrm {B})\\ 32500 \\qquad(\\mathrm {C})\\ 35000 \\qquad(\\mathrm {D})\\ 37500 \\qquad(\\mathrm {E})\\ 40000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_595", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Last year Mr. Jon Q. Public received an inheritance. He paid $20\\%$ in federal taxes on the inheritance, and paid $10\\%$ of what he had left in state taxes. He paid a total of $\\textdollar10500$ for both taxes. How many dollars was his inheritance?\n$(\\mathrm {A})\\ 30000 \\qquad (\\mathrm {B})\\ 32500 \\qquad(\\mathrm {C})\\ 35000 \\qquad(\\mathrm {D})\\ 37500 \\qquad(\\mathrm {E})\\ 40000$" + } + }, + { + "question": "Return your final response within \\boxed{}. For nonnegative integers $a$ and $b$ with $a + b \\leq 6$, let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \\leq 6$. Find the remainder when $S$ is divided by $1000$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "564", + "index": "Sky-T1_10k_596", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For nonnegative integers $a$ and $b$ with $a + b \\leq 6$, let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \\leq 6$. Find the remainder when $S$ is divided by $1000$." + } + }, + { + "question": "Return your final response within \\boxed{}. For how many positive integers $x$ is $\\log_{10}(x-40) + \\log_{10}(60-x) < 2$ ?\n$\\textbf{(A) }10\\qquad \\textbf{(B) }18\\qquad \\textbf{(C) }19\\qquad \\textbf{(D) }20\\qquad \\textbf{(E) }\\text{infinitely many}\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_597", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many positive integers $x$ is $\\log_{10}(x-40) + \\log_{10}(60-x) < 2$ ?\n$\\textbf{(A) }10\\qquad \\textbf{(B) }18\\qquad \\textbf{(C) }19\\qquad \\textbf{(D) }20\\qquad \\textbf{(E) }\\text{infinitely many}\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $F=\\frac{6x^2+16x+3m}{6}$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between:\n$\\text{(A) } 3 \\text{ and } 4\\quad \\text{(B) } 4 \\text{ and } 5\\quad \\text{(C) } 5 \\text{ and } 6\\quad \\text{(D) } -4 \\text{ and } -3\\quad \\text{(E) } -6 \\text{ and } -5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_598", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $F=\\frac{6x^2+16x+3m}{6}$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between:\n$\\text{(A) } 3 \\text{ and } 4\\quad \\text{(B) } 4 \\text{ and } 5\\quad \\text{(C) } 5 \\text{ and } 6\\quad \\text{(D) } -4 \\text{ and } -3\\quad \\text{(E) } -6 \\text{ and } -5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given two positive numbers $a$, $b$ such that $a 1$, which of the following statements is incorrect?\n$\\textbf{(A)}\\ \\text{If }x=1,y=0 \\qquad\\\\ \\textbf{(B)}\\ \\text{If }x=a,y=1 \\qquad\\\\ \\textbf{(C)}\\ \\text{If }x=-1,y\\text{ is imaginary (complex)} \\qquad\\\\ \\textbf{(D)}\\ \\text{If }0 1$, which of the following statements is incorrect?\n$\\textbf{(A)}\\ \\text{If }x=1,y=0 \\qquad\\\\ \\textbf{(B)}\\ \\text{If }x=a,y=1 \\qquad\\\\ \\textbf{(C)}\\ \\text{If }x=-1,y\\text{ is imaginary (complex)} \\qquad\\\\ \\textbf{(D)}\\ \\text{If }01$ we have:\n$\\textbf{(A)}\\ (p-1)^{\\frac{1}{2}(p-1)}-1 \\; \\text{is divisible by} \\; p-2\\qquad \\textbf{(B)}\\ (p-1)^{\\frac{1}{2}(p-1)}+1 \\; \\text{is divisible by} \\; p\\\\ \\textbf{(C)}\\ (p-1)^{\\frac{1}{2}(p-1)} \\; \\text{is divisible by} \\; p\\qquad \\textbf{(D)}\\ (p-1)^{\\frac{1}{2}(p-1)}+1 \\; \\text{is divisible by} \\; p+1\\\\ \\textbf{(E)}\\ (p-1)^{\\frac{1}{2}(p-1)}-1 \\; \\text{is divisible by} \\; p-1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_614", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For every odd number $p>1$ we have:\n$\\textbf{(A)}\\ (p-1)^{\\frac{1}{2}(p-1)}-1 \\; \\text{is divisible by} \\; p-2\\qquad \\textbf{(B)}\\ (p-1)^{\\frac{1}{2}(p-1)}+1 \\; \\text{is divisible by} \\; p\\\\ \\textbf{(C)}\\ (p-1)^{\\frac{1}{2}(p-1)} \\; \\text{is divisible by} \\; p\\qquad \\textbf{(D)}\\ (p-1)^{\\frac{1}{2}(p-1)}+1 \\; \\text{is divisible by} \\; p+1\\\\ \\textbf{(E)}\\ (p-1)^{\\frac{1}{2}(p-1)}-1 \\; \\text{is divisible by} \\; p-1$" + } + }, + { + "question": "Return your final response within \\boxed{}. The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?\n$\\text{(A)}\\ \\frac{1}{2} \\qquad \\text{(B)}\\ \\frac{2}{3} \\qquad \\text{(C)}\\ \\frac{3}{4} \\qquad \\text{(D)}\\ \\frac{4}{5} \\qquad \\text{(E)}\\ \\frac{9}{10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_615", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?\n$\\text{(A)}\\ \\frac{1}{2} \\qquad \\text{(B)}\\ \\frac{2}{3} \\qquad \\text{(C)}\\ \\frac{3}{4} \\qquad \\text{(D)}\\ \\frac{4}{5} \\qquad \\text{(E)}\\ \\frac{9}{10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many subsets of two elements can be removed from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\\}$ so that the mean (average) of the remaining numbers is 6?\n$\\textbf{(A)}\\text{ 1}\\qquad\\textbf{(B)}\\text{ 2}\\qquad\\textbf{(C)}\\text{ 3}\\qquad\\textbf{(D)}\\text{ 5}\\qquad\\textbf{(E)}\\text{ 6}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_616", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many subsets of two elements can be removed from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\\}$ so that the mean (average) of the remaining numbers is 6?\n$\\textbf{(A)}\\text{ 1}\\qquad\\textbf{(B)}\\text{ 2}\\qquad\\textbf{(C)}\\text{ 3}\\qquad\\textbf{(D)}\\text{ 5}\\qquad\\textbf{(E)}\\text{ 6}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second? \n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ \\sqrt2 \\qquad \\textbf{(C)}\\ 1/2 \\qquad \\textbf{(D)}\\ 2\\sqrt2 \\qquad \\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_617", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second? \n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ \\sqrt2 \\qquad \\textbf{(C)}\\ 1/2 \\qquad \\textbf{(D)}\\ 2\\sqrt2 \\qquad \\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define $x\\otimes y=x^3-y$. What is $h\\otimes (h\\otimes h)$?\n$\\textbf{(A)}\\ -h\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ h\\qquad\\textbf{(D)}\\ 2h\\qquad\\textbf{(E)}\\ h^3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_618", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define $x\\otimes y=x^3-y$. What is $h\\otimes (h\\otimes h)$?\n$\\textbf{(A)}\\ -h\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ h\\qquad\\textbf{(D)}\\ 2h\\qquad\\textbf{(E)}\\ h^3$" + } + }, + { + "question": "Return your final response within \\boxed{}. The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "6", + "index": "Sky-T1_10k_619", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players." + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true?\n$\\text{(A) For all positive real numbers k, both roots are pure imaginary} \\quad\\\\ \\text{(B) For all negative real numbers k, both roots are pure imaginary} \\quad\\\\ \\text{(C) For all pure imaginary numbers k, both roots are real and rational} \\quad\\\\ \\text{(D) For all pure imaginary numbers k, both roots are real and irrational} \\quad\\\\ \\text{(E) For all complex numbers k, neither root is real}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_620", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true?\n$\\text{(A) For all positive real numbers k, both roots are pure imaginary} \\quad\\\\ \\text{(B) For all negative real numbers k, both roots are pure imaginary} \\quad\\\\ \\text{(C) For all pure imaginary numbers k, both roots are real and rational} \\quad\\\\ \\text{(D) For all pure imaginary numbers k, both roots are real and irrational} \\quad\\\\ \\text{(E) For all complex numbers k, neither root is real}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the area enclosed by the geoboard quadrilateral below?\n[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2; for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b) { dot((a,b)); }; draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy]\n$\\textbf{(A)}\\ 15\\qquad \\textbf{(B)}\\ 18\\frac12 \\qquad \\textbf{(C)}\\ 22\\frac12 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 41$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_621", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the area enclosed by the geoboard quadrilateral below?\n[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2; for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b) { dot((a,b)); }; draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy]\n$\\textbf{(A)}\\ 15\\qquad \\textbf{(B)}\\ 18\\frac12 \\qquad \\textbf{(C)}\\ 22\\frac12 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 41$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \\[\\min{\\left (\\frac{10a^2-5a+1}{b^2-5b+10},\\frac{10b^2-5b+1}{c^2-5c+10},\\frac{10c^2-5c+1}{a^2-5a+10}\\right )}\\leq abc\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\min{\\left (\\frac{10a^2-5a+1}{b^2-5b+10},\\frac{10b^2-5b+1}{c^2-5c+10},\\frac{10c^2-5c+1}{a^2-5a+10}\\right )}\\leq abc", + "index": "Sky-T1_10k_622", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \\[\\min{\\left (\\frac{10a^2-5a+1}{b^2-5b+10},\\frac{10b^2-5b+1}{c^2-5c+10},\\frac{10c^2-5c+1}{a^2-5a+10}\\right )}\\leq abc\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. The inequality $y-x<\\sqrt{x^2}$ is satisfied if and only if\n$\\textbf{(A) }y<0\\text{ or }y<2x\\text{ (or both inequalities hold)}\\qquad \\textbf{(B) }y>0\\text{ or }y<2x\\text{ (or both inequalities hold)}\\qquad \\textbf{(C) }y^2<2xy\\qquad \\textbf{(D) }y<0\\qquad \\textbf{(E) }x>0\\text{ and }y<2x$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_623", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The inequality $y-x<\\sqrt{x^2}$ is satisfied if and only if\n$\\textbf{(A) }y<0\\text{ or }y<2x\\text{ (or both inequalities hold)}\\qquad \\textbf{(B) }y>0\\text{ or }y<2x\\text{ (or both inequalities hold)}\\qquad \\textbf{(C) }y^2<2xy\\qquad \\textbf{(D) }y<0\\qquad \\textbf{(E) }x>0\\text{ and }y<2x$" + } + }, + { + "question": "Return your final response within \\boxed{}. The acute angles of a right triangle are $a^{\\circ}$ and $b^{\\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 5 \\qquad\\textbf{(D)}\\ 7 \\qquad\\textbf{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_624", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The acute angles of a right triangle are $a^{\\circ}$ and $b^{\\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 5 \\qquad\\textbf{(D)}\\ 7 \\qquad\\textbf{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. Connie multiplies a number by $2$ and gets $60$ as her answer. However, she should have divided the number by $2$ to get the correct answer. What is the correct answer?\n$\\textbf{(A)}\\ 7.5\\qquad\\textbf{(B)}\\ 15\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 120\\qquad\\textbf{(E)}\\ 240$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_625", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Connie multiplies a number by $2$ and gets $60$ as her answer. However, she should have divided the number by $2$ to get the correct answer. What is the correct answer?\n$\\textbf{(A)}\\ 7.5\\qquad\\textbf{(B)}\\ 15\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 120\\qquad\\textbf{(E)}\\ 240$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?\n$\\textbf{(A) } 1\\qquad\\textbf{(B) } 3\\qquad\\textbf{(C) }6 \\qquad\\textbf{(D) }12 \\qquad\\textbf{(E) }24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_626", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?\n$\\textbf{(A) } 1\\qquad\\textbf{(B) } 3\\qquad\\textbf{(C) }6 \\qquad\\textbf{(D) }12 \\qquad\\textbf{(E) }24$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "341", + "index": "Sky-T1_10k_627", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. Given the true statements: (1) If $a$ is greater than $b$, then $c$ is greater than $d$ (2) If $c$ is less than $d$, then $e$ is greater than $f$. A valid conclusion is:\n$\\textbf{(A)}\\ \\text{If }{a}\\text{ is less than }{b}\\text{, then }{e}\\text{ is greater than }{f}\\qquad \\\\ \\textbf{(B)}\\ \\text{If }{e}\\text{ is greater than }{f}\\text{, then }{a}\\text{ is less than }{b}\\qquad \\\\ \\textbf{(C)}\\ \\text{If }{e}\\text{ is less than }{f}\\text{, then }{a}\\text{ is greater than }{b}\\qquad \\\\ \\textbf{(D)}\\ \\text{If }{a}\\text{ is greater than }{b}\\text{, then }{e}\\text{ is less than }{f}\\qquad \\\\ \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_628", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the true statements: (1) If $a$ is greater than $b$, then $c$ is greater than $d$ (2) If $c$ is less than $d$, then $e$ is greater than $f$. A valid conclusion is:\n$\\textbf{(A)}\\ \\text{If }{a}\\text{ is less than }{b}\\text{, then }{e}\\text{ is greater than }{f}\\qquad \\\\ \\textbf{(B)}\\ \\text{If }{e}\\text{ is greater than }{f}\\text{, then }{a}\\text{ is less than }{b}\\qquad \\\\ \\textbf{(C)}\\ \\text{If }{e}\\text{ is less than }{f}\\text{, then }{a}\\text{ is greater than }{b}\\qquad \\\\ \\textbf{(D)}\\ \\text{If }{a}\\text{ is greater than }{b}\\text{, then }{e}\\text{ is less than }{f}\\qquad \\\\ \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?\n$\\textbf{(A)}\\ \\frac{31}{16}\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\frac{17}{8}\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ \\frac{65}{16}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_629", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?\n$\\textbf{(A)}\\ \\frac{31}{16}\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\frac{17}{8}\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ \\frac{65}{16}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\\triangle{A_1BC}$, prove that\n$I.Q. (A_1BC) > I.Q.(A_2BC)$,\nwhere the isoperimetric quotient of a figure $F$ is defined by\n$I.Q.(F) = \\frac{\\text{Area (F)}}{\\text{[Perimeter (F)]}^2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "I.Q. (A_1BC) > I.Q.(A_2BC)", + "index": "Sky-T1_10k_630", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\\triangle{A_1BC}$, prove that\n$I.Q. (A_1BC) > I.Q.(A_2BC)$,\nwhere the isoperimetric quotient of a figure $F$ is defined by\n$I.Q.(F) = \\frac{\\text{Area (F)}}{\\text{[Perimeter (F)]}^2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair is selected at random without replacement from the remaining $8$ coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all $4$ selected coins are genuine?\n$\\textbf{(A)}\\ \\frac{7}{11}\\qquad\\textbf{(B)}\\ \\frac{9}{13}\\qquad\\textbf{(C)}\\ \\frac{11}{15}\\qquad\\textbf{(D)}\\ \\frac{15}{19}\\qquad\\textbf{(E)}\\ \\frac{15}{16}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_631", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair is selected at random without replacement from the remaining $8$ coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all $4$ selected coins are genuine?\n$\\textbf{(A)}\\ \\frac{7}{11}\\qquad\\textbf{(B)}\\ \\frac{9}{13}\\qquad\\textbf{(C)}\\ \\frac{11}{15}\\qquad\\textbf{(D)}\\ \\frac{15}{19}\\qquad\\textbf{(E)}\\ \\frac{15}{16}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Determine the number of [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(a,b)$ of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) such that $\\log_a b + 6\\log_b a=5, 2 \\leq a \\leq 2005,$ and $2 \\leq b \\leq 2005.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "54", + "index": "Sky-T1_10k_632", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Determine the number of [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(a,b)$ of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) such that $\\log_a b + 6\\log_b a=5, 2 \\leq a \\leq 2005,$ and $2 \\leq b \\leq 2005.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\\tfrac{12}5\\sqrt2$. What is the volume of the tetrahedron?\n$\\textbf{(A) }3\\sqrt2\\qquad\\textbf{(B) }2\\sqrt5\\qquad\\textbf{(C) }\\dfrac{24}5\\qquad\\textbf{(D) }3\\sqrt3\\qquad\\textbf{(E) }\\dfrac{24}5\\sqrt2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_633", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\\tfrac{12}5\\sqrt2$. What is the volume of the tetrahedron?\n$\\textbf{(A) }3\\sqrt2\\qquad\\textbf{(B) }2\\sqrt5\\qquad\\textbf{(C) }\\dfrac{24}5\\qquad\\textbf{(D) }3\\sqrt3\\qquad\\textbf{(E) }\\dfrac{24}5\\sqrt2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?\n$\\textbf{(A)}\\; 8 \\qquad\\textbf{(B)}\\; 11 \\qquad\\textbf{(C)}\\; 14 \\qquad\\textbf{(D)}\\; 15 \\qquad\\textbf{(E)}\\; 18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_634", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?\n$\\textbf{(A)}\\; 8 \\qquad\\textbf{(B)}\\; 11 \\qquad\\textbf{(C)}\\; 14 \\qquad\\textbf{(D)}\\; 15 \\qquad\\textbf{(E)}\\; 18$" + } + }, + { + "question": "Return your final response within \\boxed{}. [Set](https://artofproblemsolving.com/wiki/index.php/Set) $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "201", + "index": "Sky-T1_10k_635", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [Set](https://artofproblemsolving.com/wiki/index.php/Set) $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "50", + "index": "Sky-T1_10k_636", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have?\n$\\textbf{(A)} ~600 \\qquad\\textbf{(B)} ~650 \\qquad\\textbf{(C)} ~1950 \\qquad\\textbf{(D)} ~2000\\qquad\\textbf{(E)} ~2050$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_637", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have?\n$\\textbf{(A)} ~600 \\qquad\\textbf{(B)} ~650 \\qquad\\textbf{(C)} ~1950 \\qquad\\textbf{(D)} ~2000\\qquad\\textbf{(E)} ~2050$" + } + }, + { + "question": "Return your final response within \\boxed{}. Monica is tiling the floor of her 12-foot by 16-foot living room. She plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use?\n\n$\\textbf{(A) }48\\qquad\\textbf{(B) }87\\qquad\\textbf{(C) }89\\qquad\\textbf{(D) }96\\qquad \\textbf{(E) }120$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_638", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Monica is tiling the floor of her 12-foot by 16-foot living room. She plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use?\n\n$\\textbf{(A) }48\\qquad\\textbf{(B) }87\\qquad\\textbf{(C) }89\\qquad\\textbf{(D) }96\\qquad \\textbf{(E) }120$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$ is: \n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B) }\\ 7 \\qquad \\textbf{(C) }\\ 14 \\qquad \\textbf{(D) }\\ 128 \\qquad \\textbf{(E) }\\ 128^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_639", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$ is: \n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B) }\\ 7 \\qquad \\textbf{(C) }\\ 14 \\qquad \\textbf{(D) }\\ 128 \\qquad \\textbf{(E) }\\ 128^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\\frac{p}{q}$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 4 \\qquad\\textbf{(C)}\\ 8 \\qquad\\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_640", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\\frac{p}{q}$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 4 \\qquad\\textbf{(C)}\\ 8 \\qquad\\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?\n[asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy]\n\n$\\textbf{(A) } \\frac{2}{7} \\qquad \\textbf{(B) } \\frac{5}{42} \\qquad \\textbf{(C) } \\frac{11}{14} \\qquad \\textbf{(D) } \\frac{5}{7} \\qquad \\textbf{(E) } \\frac{6}{7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_641", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?\n[asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy]\n\n$\\textbf{(A) } \\frac{2}{7} \\qquad \\textbf{(B) } \\frac{5}{42} \\qquad \\textbf{(C) } \\frac{11}{14} \\qquad \\textbf{(D) } \\frac{5}{7} \\qquad \\textbf{(E) } \\frac{6}{7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?\n$\\textbf{(A) } \\frac{7}{36} \\qquad\\textbf{(B) } \\frac{5}{24} \\qquad\\textbf{(C) } \\frac{2}{9} \\qquad\\textbf{(D) } \\frac{17}{72} \\qquad\\textbf{(E) } \\frac{1}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_642", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?\n$\\textbf{(A) } \\frac{7}{36} \\qquad\\textbf{(B) } \\frac{5}{24} \\qquad\\textbf{(C) } \\frac{2}{9} \\qquad\\textbf{(D) } \\frac{17}{72} \\qquad\\textbf{(E) } \\frac{1}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many ordered pairs of integers $(x, y)$ satisfy the equation \\[x^{2020}+y^2=2y?\\]\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } 2 \\qquad\\textbf{(C) } 3 \\qquad\\textbf{(D) } 4 \\qquad\\textbf{(E) } \\text{infinitely many}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_643", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many ordered pairs of integers $(x, y)$ satisfy the equation \\[x^{2020}+y^2=2y?\\]\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } 2 \\qquad\\textbf{(C) } 3 \\qquad\\textbf{(D) } 4 \\qquad\\textbf{(E) } \\text{infinitely many}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed?\n$\\textbf{(A)} ~10\\qquad\\textbf{(B)} ~13\\qquad\\textbf{(C)} ~15\\qquad\\textbf{(D)} ~17\\qquad\\textbf{(E)} ~20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_644", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed?\n$\\textbf{(A)} ~10\\qquad\\textbf{(B)} ~13\\qquad\\textbf{(C)} ~15\\qquad\\textbf{(D)} ~17\\qquad\\textbf{(E)} ~20$" + } + }, + { + "question": "Return your final response within \\boxed{}. On a certain math exam, $10\\%$ of the students got $70$ points, $25\\%$ got $80$ points, $20\\%$ got $85$ points, $15\\%$ got $90$ points, and the rest got $95$ points. What is the difference between the [mean](https://artofproblemsolving.com/wiki/index.php/Mean) and the [median](https://artofproblemsolving.com/wiki/index.php/Median) score on this exam?\n$\\textbf{(A) }\\ 0 \\qquad \\textbf{(B) }\\ 1 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 4 \\qquad \\textbf{(E) }\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_645", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On a certain math exam, $10\\%$ of the students got $70$ points, $25\\%$ got $80$ points, $20\\%$ got $85$ points, $15\\%$ got $90$ points, and the rest got $95$ points. What is the difference between the [mean](https://artofproblemsolving.com/wiki/index.php/Mean) and the [median](https://artofproblemsolving.com/wiki/index.php/Median) score on this exam?\n$\\textbf{(A) }\\ 0 \\qquad \\textbf{(B) }\\ 1 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 4 \\qquad \\textbf{(E) }\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM = CM$ and $\\angle BAD = \\angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\\angle ADO = \\angle HAN$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\angle ADO = \\angle HAN", + "index": "Sky-T1_10k_646", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM = CM$ and $\\angle BAD = \\angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\\angle ADO = \\angle HAN$." + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$ is: \n$\\textbf{(A)}\\ 32 \\qquad \\textbf{(B)}\\ 16 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 48 \\qquad \\textbf{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_647", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$ is: \n$\\textbf{(A)}\\ 32 \\qquad \\textbf{(B)}\\ 16 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 48 \\qquad \\textbf{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?\n$\\textbf{(A)}\\ 110 \\qquad \\textbf{(B)}\\ 165 \\qquad \\textbf{(C)}\\ 330 \\qquad \\textbf{(D)}\\ 625 \\qquad \\textbf{(E)}\\ 660$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_648", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?\n$\\textbf{(A)}\\ 110 \\qquad \\textbf{(B)}\\ 165 \\qquad \\textbf{(C)}\\ 330 \\qquad \\textbf{(D)}\\ 625 \\qquad \\textbf{(E)}\\ 660$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?\n$\\textbf{(A)}\\ 5724 \\qquad \\textbf{(B)}\\ 7245 \\qquad \\textbf{(C)}\\ 7254 \\qquad \\textbf{(D)}\\ 7425 \\qquad \\textbf{(E)}\\ 7542$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_649", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?\n$\\textbf{(A)}\\ 5724 \\qquad \\textbf{(B)}\\ 7245 \\qquad \\textbf{(C)}\\ 7254 \\qquad \\textbf{(D)}\\ 7425 \\qquad \\textbf{(E)}\\ 7542$" + } + }, + { + "question": "Return your final response within \\boxed{}. Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? \n$\\mathrm{(A)}\\ 678 \\qquad \\mathrm{(B)}\\ 768 \\qquad \\mathrm{(C)}\\ 786 \\qquad \\mathrm{(D)}\\ 867 \\qquad \\mathrm{(E)}\\ 876$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_650", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? \n$\\mathrm{(A)}\\ 678 \\qquad \\mathrm{(B)}\\ 768 \\qquad \\mathrm{(C)}\\ 786 \\qquad \\mathrm{(D)}\\ 867 \\qquad \\mathrm{(E)}\\ 876$" + } + }, + { + "question": "Return your final response within \\boxed{}. Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day? \n$\\text{(A)}\\mbox{ }525\\qquad\\text{(B)}\\mbox{ }560\\qquad\\text{(C)}\\mbox{ }595\\qquad\\text{(D)}\\mbox{ }665\\qquad\\text{(E)}\\mbox{ }735$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_651", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day? \n$\\text{(A)}\\mbox{ }525\\qquad\\text{(B)}\\mbox{ }560\\qquad\\text{(C)}\\mbox{ }595\\qquad\\text{(D)}\\mbox{ }665\\qquad\\text{(E)}\\mbox{ }735$" + } + }, + { + "question": "Return your final response within \\boxed{}. The arithmetic mean (average) of four numbers is $85$. If the largest of these numbers is $97$, then the mean of the remaining three numbers is \n$\\text{(A)}\\ 81.0 \\qquad \\text{(B)}\\ 82.7 \\qquad \\text{(C)}\\ 83.0 \\qquad \\text{(D)}\\ 84.0 \\qquad \\text{(E)}\\ 84.3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_652", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The arithmetic mean (average) of four numbers is $85$. If the largest of these numbers is $97$, then the mean of the remaining three numbers is \n$\\text{(A)}\\ 81.0 \\qquad \\text{(B)}\\ 82.7 \\qquad \\text{(C)}\\ 83.0 \\qquad \\text{(D)}\\ 84.0 \\qquad \\text{(E)}\\ 84.3$" + } + }, + { + "question": "Return your final response within \\boxed{}. A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?\n$\\textbf{(A) }\\frac{5}{54}\\qquad\\textbf{(B) }\\frac{1}{9}\\qquad\\textbf{(C) }\\frac{5}{27}\\qquad\\textbf{(D) }\\frac{2}{9}\\qquad\\textbf{(E) }\\frac{1}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_653", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?\n$\\textbf{(A) }\\frac{5}{54}\\qquad\\textbf{(B) }\\frac{1}{9}\\qquad\\textbf{(C) }\\frac{5}{27}\\qquad\\textbf{(D) }\\frac{2}{9}\\qquad\\textbf{(E) }\\frac{1}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 56\\qquad\\textbf{(C)}\\ 64\\qquad\\textbf{(D)}\\ 72\\qquad\\textbf{(E)}\\ 80$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_654", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 56\\qquad\\textbf{(C)}\\ 64\\qquad\\textbf{(D)}\\ 72\\qquad\\textbf{(E)}\\ 80$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }12\\qquad\\textbf{(E) }16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_655", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }12\\qquad\\textbf{(E) }16$" + } + }, + { + "question": "Return your final response within \\boxed{}. A charity sells $140$ benefit tickets for a total of $$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?\n$\\textbf{(A) } \\textdollar 782 \\qquad \\textbf{(B) } \\textdollar 986 \\qquad \\textbf{(C) } \\textdollar 1158 \\qquad \\textbf{(D) } \\textdollar 1219 \\qquad \\textbf{(E) }\\ \\textdollar 1449$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_656", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A charity sells $140$ benefit tickets for a total of $$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?\n$\\textbf{(A) } \\textdollar 782 \\qquad \\textbf{(B) } \\textdollar 986 \\qquad \\textbf{(C) } \\textdollar 1158 \\qquad \\textbf{(D) } \\textdollar 1219 \\qquad \\textbf{(E) }\\ \\textdollar 1449$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "112", + "index": "Sky-T1_10k_657", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$." + } + }, + { + "question": "Return your final response within \\boxed{}. If in the formula $C =\\frac{en}{R+nr}$, where $e$, $n$, $R$ and $r$ are all positive, $n$ is increased while $e$, $R$ and $r$ are kept constant, then $C$:\n$\\textbf{(A)}\\ \\text{Increases}\\qquad\\textbf{(B)}\\ \\text{Decreases}\\qquad\\textbf{(C)}\\ \\text{Remains constant}\\qquad\\textbf{(D)}\\ \\text{Increases and then decreases}\\qquad\\\\ \\textbf{(E)}\\ \\text{Decreases and then increases}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_658", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If in the formula $C =\\frac{en}{R+nr}$, where $e$, $n$, $R$ and $r$ are all positive, $n$ is increased while $e$, $R$ and $r$ are kept constant, then $C$:\n$\\textbf{(A)}\\ \\text{Increases}\\qquad\\textbf{(B)}\\ \\text{Decreases}\\qquad\\textbf{(C)}\\ \\text{Remains constant}\\qquad\\textbf{(D)}\\ \\text{Increases and then decreases}\\qquad\\\\ \\textbf{(E)}\\ \\text{Decreases and then increases}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of $5$ cups. What percent of the total capacity of the pitcher did each cup receive?\n$\\textbf{(A) }5 \\qquad \\textbf{(B) }10 \\qquad \\textbf{(C) }15 \\qquad \\textbf{(D) }20 \\qquad \\textbf{(E) }25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_659", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of $5$ cups. What percent of the total capacity of the pitcher did each cup receive?\n$\\textbf{(A) }5 \\qquad \\textbf{(B) }10 \\qquad \\textbf{(C) }15 \\qquad \\textbf{(D) }20 \\qquad \\textbf{(E) }25$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers? \n$\\mathrm{(A) \\ } 0\\qquad \\mathrm{(B) \\ } 1\\qquad \\mathrm{(C) \\ } 2\\qquad \\mathrm{(D) \\ } 2003\\qquad \\mathrm{(E) \\ } 4006$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_660", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers? \n$\\mathrm{(A) \\ } 0\\qquad \\mathrm{(B) \\ } 1\\qquad \\mathrm{(C) \\ } 2\\qquad \\mathrm{(D) \\ } 2003\\qquad \\mathrm{(E) \\ } 4006$" + } + }, + { + "question": "Return your final response within \\boxed{}. A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?\n$\\textbf{(A) }\\ 100 \\qquad \\textbf{(B) }\\ 200 \\qquad \\textbf{(C) }\\ 300 \\qquad \\textbf{(D) }\\ 400 \\qquad \\textbf{(E) }\\ 500$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_661", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?\n$\\textbf{(A) }\\ 100 \\qquad \\textbf{(B) }\\ 200 \\qquad \\textbf{(C) }\\ 300 \\qquad \\textbf{(D) }\\ 400 \\qquad \\textbf{(E) }\\ 500$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?\n\na square\na rectangle that is not a square\na rhombus that is not a square\na parallelogram that is not a rectangle or a rhombus\nan isosceles trapezoid that is not a parallelogram\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 2 \\qquad\\textbf{(C) } 3 \\qquad\\textbf{(D) } 4 \\qquad\\textbf{(E) } 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_662", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?\n\na square\na rectangle that is not a square\na rhombus that is not a square\na parallelogram that is not a rectangle or a rhombus\nan isosceles trapezoid that is not a parallelogram\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 2 \\qquad\\textbf{(C) } 3 \\qquad\\textbf{(D) } 4 \\qquad\\textbf{(E) } 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Assuming $a\\neq3$, $b\\neq4$, and $c\\neq5$, what is the value in simplest form of the following expression?\n\\[\\frac{a-3}{5-c} \\cdot \\frac{b-4}{3-a} \\cdot \\frac{c-5}{4-b}\\]\n$\\textbf{(A) } {-}1 \\qquad \\textbf{(B) } 1 \\qquad \\textbf{(C) } \\frac{abc}{60} \\qquad \\textbf{(D) } \\frac{1}{abc} - \\frac{1}{60} \\qquad \\textbf{(E) } \\frac{1}{60} - \\frac{1}{abc}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_663", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Assuming $a\\neq3$, $b\\neq4$, and $c\\neq5$, what is the value in simplest form of the following expression?\n\\[\\frac{a-3}{5-c} \\cdot \\frac{b-4}{3-a} \\cdot \\frac{c-5}{4-b}\\]\n$\\textbf{(A) } {-}1 \\qquad \\textbf{(B) } 1 \\qquad \\textbf{(C) } \\frac{abc}{60} \\qquad \\textbf{(D) } \\frac{1}{abc} - \\frac{1}{60} \\qquad \\textbf{(E) } \\frac{1}{60} - \\frac{1}{abc}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?\n$\\textbf{(A)}\\; \\text{3:10 PM} \\qquad\\textbf{(B)}\\; \\text{3:30 PM} \\qquad\\textbf{(C)}\\; \\text{4:00 PM} \\qquad\\textbf{(D)}\\; \\text{4:10 PM} \\qquad\\textbf{(E)}\\; \\text{4:30 PM}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_664", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?\n$\\textbf{(A)}\\; \\text{3:10 PM} \\qquad\\textbf{(B)}\\; \\text{3:30 PM} \\qquad\\textbf{(C)}\\; \\text{4:00 PM} \\qquad\\textbf{(D)}\\; \\text{4:10 PM} \\qquad\\textbf{(E)}\\; \\text{4:30 PM}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If you are given $\\log 8\\approx .9031$ and $\\log 9\\approx .9542$, then the only logarithm that cannot be found without the use of tables is:\n$\\textbf{(A)}\\ \\log 17\\qquad\\textbf{(B)}\\ \\log\\frac{5}{4}\\qquad\\textbf{(C)}\\ \\log 15\\qquad\\textbf{(D)}\\ \\log 600\\qquad\\textbf{(E)}\\ \\log .4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_665", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If you are given $\\log 8\\approx .9031$ and $\\log 9\\approx .9542$, then the only logarithm that cannot be found without the use of tables is:\n$\\textbf{(A)}\\ \\log 17\\qquad\\textbf{(B)}\\ \\log\\frac{5}{4}\\qquad\\textbf{(C)}\\ \\log 15\\qquad\\textbf{(D)}\\ \\log 600\\qquad\\textbf{(E)}\\ \\log .4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\\frac pq,$ where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3,17,$ and $10.$) What is $p+q?$\n$\\textbf{(A) }55 \\qquad \\textbf{(B) }56 \\qquad \\textbf{(C) }57\\qquad \\textbf{(D) }58 \\qquad \\textbf{(E) }59$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_666", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\\frac pq,$ where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3,17,$ and $10.$) What is $p+q?$\n$\\textbf{(A) }55 \\qquad \\textbf{(B) }56 \\qquad \\textbf{(C) }57\\qquad \\textbf{(D) }58 \\qquad \\textbf{(E) }59$" + } + }, + { + "question": "Return your final response within \\boxed{}. A store owner bought $1500$ pencils at $$ 0.10$ each. If he sells them for $$ 0.25$ each, how many of them must he sell to make a profit of exactly $$ 100.00$?\n$\\text{(A)}\\ 400 \\qquad \\text{(B)}\\ 667 \\qquad \\text{(C)}\\ 1000 \\qquad \\text{(D)}\\ 1500 \\qquad \\text{(E)}\\ 1900$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_667", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A store owner bought $1500$ pencils at $$ 0.10$ each. If he sells them for $$ 0.25$ each, how many of them must he sell to make a profit of exactly $$ 100.00$?\n$\\text{(A)}\\ 400 \\qquad \\text{(B)}\\ 667 \\qquad \\text{(C)}\\ 1000 \\qquad \\text{(D)}\\ 1500 \\qquad \\text{(E)}\\ 1900$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\frac{b}{a} = 2$ and $\\frac{c}{b} = 3$, what is the ratio of $a + b$ to $b + c$?\n$\\textbf{(A)}\\ \\frac{1}{3}\\qquad \\textbf{(B)}\\ \\frac{3}{8}\\qquad \\textbf{(C)}\\ \\frac{3}{5}\\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ \\frac{3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_668", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\frac{b}{a} = 2$ and $\\frac{c}{b} = 3$, what is the ratio of $a + b$ to $b + c$?\n$\\textbf{(A)}\\ \\frac{1}{3}\\qquad \\textbf{(B)}\\ \\frac{3}{8}\\qquad \\textbf{(C)}\\ \\frac{3}{5}\\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ \\frac{3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Rectangle $ABCD$ is inscribed in a semicircle with diameter $\\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$\n\n$\\textbf{(A) }240 \\qquad \\textbf{(B) }248 \\qquad \\textbf{(C) }256 \\qquad \\textbf{(D) }264 \\qquad \\textbf{(E) }272$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_669", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Rectangle $ABCD$ is inscribed in a semicircle with diameter $\\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$\n\n$\\textbf{(A) }240 \\qquad \\textbf{(B) }248 \\qquad \\textbf{(C) }256 \\qquad \\textbf{(D) }264 \\qquad \\textbf{(E) }272$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\\[t_n = \\frac{5t_{n-1}+1}{25t_{n-2}}\\]for all $n \\ge 3$. Then $t_{2020}$ can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "626", + "index": "Sky-T1_10k_670", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\\[t_n = \\frac{5t_{n-1}+1}{25t_{n-2}}\\]for all $n \\ge 3$. Then $t_{2020}$ can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$." + } + }, + { + "question": "Return your final response within \\boxed{}. The set of all real numbers for which \n\\[x+\\sqrt{x^2+1}-\\frac{1}{x+\\sqrt{x^2+1}}\\]\nis a rational number is the set of all\n(A) integers $x$ (B) rational $x$ (C) real $x$\n(D) $x$ for which $\\sqrt{x^2+1}$ is rational\n(E) $x$ for which $x+\\sqrt{x^2+1}$ is rational", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_671", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The set of all real numbers for which \n\\[x+\\sqrt{x^2+1}-\\frac{1}{x+\\sqrt{x^2+1}}\\]\nis a rational number is the set of all\n(A) integers $x$ (B) rational $x$ (C) real $x$\n(D) $x$ for which $\\sqrt{x^2+1}$ is rational\n(E) $x$ for which $x+\\sqrt{x^2+1}$ is rational" + } + }, + { + "question": "Return your final response within \\boxed{}. Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is\n\n$\\text{(A)}\\ 9 \\qquad \\text{(B)}\\ 18 \\qquad \\text{(C)}\\ 36 \\qquad \\text{(D)}\\ 72 \\qquad \\text{(D)}\\ 81$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_672", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is\n\n$\\text{(A)}\\ 9 \\qquad \\text{(B)}\\ 18 \\qquad \\text{(C)}\\ 36 \\qquad \\text{(D)}\\ 72 \\qquad \\text{(D)}\\ 81$" + } + }, + { + "question": "Return your final response within \\boxed{}. The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is:\n$\\text{(A) } -\\frac{2}{3}\\quad\\text{(B) } -\\frac{1}{3}\\quad\\text{(C) } 0\\quad\\text{(D) } \\frac{1}{4}\\quad\\text{(E) } \\frac{3}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_673", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is:\n$\\text{(A) } -\\frac{2}{3}\\quad\\text{(B) } -\\frac{1}{3}\\quad\\text{(C) } 0\\quad\\text{(D) } \\frac{1}{4}\\quad\\text{(E) } \\frac{3}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?\n$\\textbf{(A)}\\ \\frac14 \\qquad \\textbf{(B)}\\ \\frac13 \\qquad \\textbf{(C)}\\ \\frac38 \\qquad \\textbf{(D)}\\ \\frac12 \\qquad \\textbf{(E)}\\ \\frac23$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_674", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?\n$\\textbf{(A)}\\ \\frac14 \\qquad \\textbf{(B)}\\ \\frac13 \\qquad \\textbf{(C)}\\ \\frac38 \\qquad \\textbf{(D)}\\ \\frac12 \\qquad \\textbf{(E)}\\ \\frac23$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of distinct points in the $xy$-plane common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$ is\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } 3\\quad \\text{(E) } 4\\quad \\text{(F) } \\infty$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_675", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of distinct points in the $xy$-plane common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$ is\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } 3\\quad \\text{(E) } 4\\quad \\text{(F) } \\infty$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\\angle ABO=90^\\circ$ and $\\angle AOB=30^\\circ$. Suppose that $OA$ is rotated $90^\\circ$ counterclockwise about $O$. What are the coordinates of the image of $A$? \n$\\mathrm{(A)}\\ \\left( - \\frac {10}{3}\\sqrt {3},5\\right) \\qquad \\mathrm{(B)}\\ \\left( - \\frac {5}{3}\\sqrt {3},5\\right) \\qquad \\mathrm{(C)}\\ \\left(\\sqrt {3},5\\right) \\qquad \\mathrm{(D)}\\ \\left(\\frac {5}{3}\\sqrt {3},5\\right) \\qquad \\mathrm{(E)}\\ \\left(\\frac {10}{3}\\sqrt {3},5\\right)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_676", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\\angle ABO=90^\\circ$ and $\\angle AOB=30^\\circ$. Suppose that $OA$ is rotated $90^\\circ$ counterclockwise about $O$. What are the coordinates of the image of $A$? \n$\\mathrm{(A)}\\ \\left( - \\frac {10}{3}\\sqrt {3},5\\right) \\qquad \\mathrm{(B)}\\ \\left( - \\frac {5}{3}\\sqrt {3},5\\right) \\qquad \\mathrm{(C)}\\ \\left(\\sqrt {3},5\\right) \\qquad \\mathrm{(D)}\\ \\left(\\frac {5}{3}\\sqrt {3},5\\right) \\qquad \\mathrm{(E)}\\ \\left(\\frac {10}{3}\\sqrt {3},5\\right)$" + } + }, + { + "question": "Return your final response within \\boxed{}. A coin is altered so that the probability that it lands on heads is less than $\\frac{1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $\\frac{1}{6}$. What is the probability that the coin lands on heads?\n$\\textbf{(A)}\\ \\frac{\\sqrt{15}-3}{6} \\qquad \\textbf{(B)}\\ \\frac{6-\\sqrt{6\\sqrt{6}+2}}{12} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{2}-1}{2} \\qquad \\textbf{(D)}\\ \\frac{3-\\sqrt{3}}{6} \\qquad \\textbf{(E)}\\ \\frac{\\sqrt{3}-1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_677", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A coin is altered so that the probability that it lands on heads is less than $\\frac{1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $\\frac{1}{6}$. What is the probability that the coin lands on heads?\n$\\textbf{(A)}\\ \\frac{\\sqrt{15}-3}{6} \\qquad \\textbf{(B)}\\ \\frac{6-\\sqrt{6\\sqrt{6}+2}}{12} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{2}-1}{2} \\qquad \\textbf{(D)}\\ \\frac{3-\\sqrt{3}}{6} \\qquad \\textbf{(E)}\\ \\frac{\\sqrt{3}-1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $A$ and $B$ are on a circle of radius $5$ and $AB=6$. Point $C$ is the [midpoint](https://artofproblemsolving.com/wiki/index.php/Midpoint) of the minor arc $AB$. What is the length of the line segment $AC$?\n$\\mathrm{(A)}\\ \\sqrt{10}\\qquad\\mathrm{(B)}\\ \\frac{7}{2}\\qquad\\mathrm{(C)}\\ \\sqrt{14}\\qquad\\mathrm{(D)}\\ \\sqrt{15}\\qquad\\mathrm{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_678", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $A$ and $B$ are on a circle of radius $5$ and $AB=6$. Point $C$ is the [midpoint](https://artofproblemsolving.com/wiki/index.php/Midpoint) of the minor arc $AB$. What is the length of the line segment $AC$?\n$\\mathrm{(A)}\\ \\sqrt{10}\\qquad\\mathrm{(B)}\\ \\frac{7}{2}\\qquad\\mathrm{(C)}\\ \\sqrt{14}\\qquad\\mathrm{(D)}\\ \\sqrt{15}\\qquad\\mathrm{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $m,~n,~p$, and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q$, then the equation $f(g(x))=g(f(x))$ has a solution\n$\\textbf{(A) }\\text{for all choices of }m,~n,~p, \\text{ and } q\\qquad\\\\ \\textbf{(B) }\\text{if and only if }m=p\\text{ and }n=q\\qquad\\\\ \\textbf{(C) }\\text{if and only if }mq-np=0\\qquad\\\\ \\textbf{(D) }\\text{if and only if }n(1-p)-q(1-m)=0\\qquad\\\\ \\textbf{(E) }\\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_679", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $m,~n,~p$, and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q$, then the equation $f(g(x))=g(f(x))$ has a solution\n$\\textbf{(A) }\\text{for all choices of }m,~n,~p, \\text{ and } q\\qquad\\\\ \\textbf{(B) }\\text{if and only if }m=p\\text{ and }n=q\\qquad\\\\ \\textbf{(C) }\\text{if and only if }mq-np=0\\qquad\\\\ \\textbf{(D) }\\text{if and only if }n(1-p)-q(1-m)=0\\qquad\\\\ \\textbf{(E) }\\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC.$ if $OQ$ is 3 inches, then $OP$, in inches, is:\n$\\text{(A) } 3\\quad \\text{(B) } \\frac{9}{2}\\quad \\text{(C) } 6\\quad \\text{(D) } 9\\quad \\text{(E) } \\text{undetermined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_680", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC.$ if $OQ$ is 3 inches, then $OP$, in inches, is:\n$\\text{(A) } 3\\quad \\text{(B) } \\frac{9}{2}\\quad \\text{(C) } 6\\quad \\text{(D) } 9\\quad \\text{(E) } \\text{undetermined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?\n$\\textbf{(A) }\\text{6}\\qquad\\textbf{(B) }\\text{9}\\qquad\\textbf{(C) }\\text{12}\\qquad\\textbf{(D) }\\text{18}\\qquad\\textbf{(E) }\\text{24}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_681", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?\n$\\textbf{(A) }\\text{6}\\qquad\\textbf{(B) }\\text{9}\\qquad\\textbf{(C) }\\text{12}\\qquad\\textbf{(D) }\\text{18}\\qquad\\textbf{(E) }\\text{24}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is \n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 25 \\qquad \\textbf{(C)}\\ 44 \\qquad \\textbf{(D)}\\ \\text{infinity} \\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_682", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is \n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 25 \\qquad \\textbf{(C)}\\ 44 \\qquad \\textbf{(D)}\\ \\text{infinity} \\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?\n$\\text{(A)}\\hspace{.05in}3\\qquad\\text{(B)}\\hspace{.05in}4\\qquad\\text{(C)}\\hspace{.05in}5\\qquad\\text{(D)}\\hspace{.05in}6\\qquad\\text{(E)}\\hspace{.05in}7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_683", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?\n$\\text{(A)}\\hspace{.05in}3\\qquad\\text{(B)}\\hspace{.05in}4\\qquad\\text{(C)}\\hspace{.05in}5\\qquad\\text{(D)}\\hspace{.05in}6\\qquad\\text{(E)}\\hspace{.05in}7$" + } + }, + { + "question": "Return your final response within \\boxed{}. $AB$ is a fixed diameter of a circle whose center is $O$. From $C$, any point on the circle, a chord $CD$ is drawn perpendicular to $AB$. Then, as $C$ moves over a semicircle, the bisector of angle $OCD$ cuts the circle in a point that always: \n$\\textbf{(A)}\\ \\text{bisects the arc }AB\\qquad\\textbf{(B)}\\ \\text{trisects the arc }AB\\qquad\\textbf{(C)}\\ \\text{varies}$\n$\\textbf{(D)}\\ \\text{is as far from }AB\\text{ as from }D\\qquad\\textbf{(E)}\\ \\text{is equidistant from }B\\text{ and }C$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_684", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $AB$ is a fixed diameter of a circle whose center is $O$. From $C$, any point on the circle, a chord $CD$ is drawn perpendicular to $AB$. Then, as $C$ moves over a semicircle, the bisector of angle $OCD$ cuts the circle in a point that always: \n$\\textbf{(A)}\\ \\text{bisects the arc }AB\\qquad\\textbf{(B)}\\ \\text{trisects the arc }AB\\qquad\\textbf{(C)}\\ \\text{varies}$\n$\\textbf{(D)}\\ \\text{is as far from }AB\\text{ as from }D\\qquad\\textbf{(E)}\\ \\text{is equidistant from }B\\text{ and }C$" + } + }, + { + "question": "Return your final response within \\boxed{}. For one root of $ax^2 + bx + c = 0$ to be double the other, the coefficients $a,\\,b,\\,c$ must be related as follows:\n$\\textbf{(A)}\\ 4b^2 = 9c\\qquad \\textbf{(B)}\\ 2b^2 = 9ac\\qquad \\textbf{(C)}\\ 2b^2 = 9a\\qquad \\\\ \\textbf{(D)}\\ b^2 - 8ac = 0\\qquad \\textbf{(E)}\\ 9b^2 = 2ac$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_685", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For one root of $ax^2 + bx + c = 0$ to be double the other, the coefficients $a,\\,b,\\,c$ must be related as follows:\n$\\textbf{(A)}\\ 4b^2 = 9c\\qquad \\textbf{(B)}\\ 2b^2 = 9ac\\qquad \\textbf{(C)}\\ 2b^2 = 9a\\qquad \\\\ \\textbf{(D)}\\ b^2 - 8ac = 0\\qquad \\textbf{(E)}\\ 9b^2 = 2ac$" + } + }, + { + "question": "Return your final response within \\boxed{}. A bakery owner turns on his doughnut machine at $\\text{8:30}\\ {\\small\\text{AM}}$. At $\\text{11:10}\\ {\\small\\text{AM}}$ the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?\n$\\mathrm{(A)}\\ \\text{1:50}\\ {\\small\\text{PM}}\\qquad\\mathrm{(B)}\\ \\text{3:00}\\ {\\small\\text{PM}}\\qquad\\mathrm{(C)}\\ \\text{3:30}\\ {\\small\\text{PM}}\\qquad\\mathrm{(D)}\\ \\text{4:30}\\ {\\small\\text{PM}}\\qquad\\mathrm{(E)}\\ \\text{5:50}\\ {\\small\\text{PM}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_686", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A bakery owner turns on his doughnut machine at $\\text{8:30}\\ {\\small\\text{AM}}$. At $\\text{11:10}\\ {\\small\\text{AM}}$ the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?\n$\\mathrm{(A)}\\ \\text{1:50}\\ {\\small\\text{PM}}\\qquad\\mathrm{(B)}\\ \\text{3:00}\\ {\\small\\text{PM}}\\qquad\\mathrm{(C)}\\ \\text{3:30}\\ {\\small\\text{PM}}\\qquad\\mathrm{(D)}\\ \\text{4:30}\\ {\\small\\text{PM}}\\qquad\\mathrm{(E)}\\ \\text{5:50}\\ {\\small\\text{PM}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the sum of the prime factors of $2010$?\n$\\textbf{(A)}\\ 67\\qquad\\textbf{(B)}\\ 75\\qquad\\textbf{(C)}\\ 77\\qquad\\textbf{(D)}\\ 201\\qquad\\textbf{(E)}\\ 210$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_687", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the sum of the prime factors of $2010$?\n$\\textbf{(A)}\\ 67\\qquad\\textbf{(B)}\\ 75\\qquad\\textbf{(C)}\\ 77\\qquad\\textbf{(D)}\\ 201\\qquad\\textbf{(E)}\\ 210$" + } + }, + { + "question": "Return your final response within \\boxed{}. Assume that $x$ is a [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [real number](https://artofproblemsolving.com/wiki/index.php/Real_number). Which is equivalent to $\\sqrt[3]{x\\sqrt{x}}$?\n$\\mathrm{(A)}\\ x^{1/6}\\qquad\\mathrm{(B)}\\ x^{1/4}\\qquad\\mathrm{(C)}\\ x^{3/8}\\qquad\\mathrm{(D)}\\ x^{1/2}\\qquad\\mathrm{(E)}\\ x$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_688", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Assume that $x$ is a [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [real number](https://artofproblemsolving.com/wiki/index.php/Real_number). Which is equivalent to $\\sqrt[3]{x\\sqrt{x}}$?\n$\\mathrm{(A)}\\ x^{1/6}\\qquad\\mathrm{(B)}\\ x^{1/4}\\qquad\\mathrm{(C)}\\ x^{3/8}\\qquad\\mathrm{(D)}\\ x^{1/2}\\qquad\\mathrm{(E)}\\ x$" + } + }, + { + "question": "Return your final response within \\boxed{}. The formula $N=8 \\times 10^{8} \\times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars. \nThe lowest income, in dollars, of the wealthiest $800$ individuals is at least:\n$\\textbf{(A)}\\ 10^4\\qquad \\textbf{(B)}\\ 10^6\\qquad \\textbf{(C)}\\ 10^8\\qquad \\textbf{(D)}\\ 10^{12} \\qquad \\textbf{(E)}\\ 10^{16}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_689", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The formula $N=8 \\times 10^{8} \\times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars. \nThe lowest income, in dollars, of the wealthiest $800$ individuals is at least:\n$\\textbf{(A)}\\ 10^4\\qquad \\textbf{(B)}\\ 10^6\\qquad \\textbf{(C)}\\ 10^8\\qquad \\textbf{(D)}\\ 10^{12} \\qquad \\textbf{(E)}\\ 10^{16}$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\\gcd(a, b, c, d) = 77$ and $\\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$?\n$\\textbf{(A)}\\ 13,860\\qquad\\textbf{(B)}\\ 20,790\\qquad\\textbf{(C)}\\ 21,560 \\qquad\\textbf{(D)}\\ 27,720 \\qquad\\textbf{(E)}\\ 41,580$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_690", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\\gcd(a, b, c, d) = 77$ and $\\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$?\n$\\textbf{(A)}\\ 13,860\\qquad\\textbf{(B)}\\ 20,790\\qquad\\textbf{(C)}\\ 21,560 \\qquad\\textbf{(D)}\\ 27,720 \\qquad\\textbf{(E)}\\ 41,580$" + } + }, + { + "question": "Return your final response within \\boxed{}. Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?\n$\\textbf {(A) } \\frac{3}{2} \\qquad \\textbf {(B) } \\frac{5}{3} \\qquad \\textbf {(C) } \\frac{7}{4} \\qquad \\textbf {(D) } 2 \\qquad \\textbf {(E) } \\frac{13}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_691", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?\n$\\textbf {(A) } \\frac{3}{2} \\qquad \\textbf {(B) } \\frac{5}{3} \\qquad \\textbf {(C) } \\frac{7}{4} \\qquad \\textbf {(D) } 2 \\qquad \\textbf {(E) } \\frac{13}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In order to draw a graph of $ax^2+bx+c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $x$ were $3844, 3969, 4096, 4227, 4356, 4489, 4624$, and $4761$. The one which is incorrect is: \n$\\text{(A) } 4096 \\qquad \\text{(B) } 4356 \\qquad \\text{(C) } 4489 \\qquad \\text{(D) } 4761 \\qquad \\text{(E) } \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_692", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In order to draw a graph of $ax^2+bx+c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $x$ were $3844, 3969, 4096, 4227, 4356, 4489, 4624$, and $4761$. The one which is incorrect is: \n$\\text{(A) } 4096 \\qquad \\text{(B) } 4356 \\qquad \\text{(C) } 4489 \\qquad \\text{(D) } 4761 \\qquad \\text{(E) } \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $2-(-2)^{-2}$ ?\n$\\textbf{(A) } -2\\qquad\\textbf{(B) } \\dfrac{1}{16}\\qquad\\textbf{(C) } \\dfrac{7}{4}\\qquad\\textbf{(D) } \\dfrac{9}{4}\\qquad\\textbf{(E) } 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_693", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $2-(-2)^{-2}$ ?\n$\\textbf{(A) } -2\\qquad\\textbf{(B) } \\dfrac{1}{16}\\qquad\\textbf{(C) } \\dfrac{7}{4}\\qquad\\textbf{(D) } \\dfrac{9}{4}\\qquad\\textbf{(E) } 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. The [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) of boys to girls in Mr. Brown's math class is $2:3$. If there are $30$ students in the class, how many more girls than boys are in the class?\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_694", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) of boys to girls in Mr. Brown's math class is $2:3$. If there are $30$ students in the class, how many more girls than boys are in the class?\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Determine all non-negative integral solutions $(n_1,n_2,\\dots , n_{14})$ if any, apart from permutations, of the Diophantine Equation $n_1^4+n_2^4+\\cdots +n_{14}^4=1599$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "No solution exists", + "index": "Sky-T1_10k_695", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Determine all non-negative integral solutions $(n_1,n_2,\\dots , n_{14})$ if any, apart from permutations, of the Diophantine Equation $n_1^4+n_2^4+\\cdots +n_{14}^4=1599$." + } + }, + { + "question": "Return your final response within \\boxed{}. $1,000,000,000,000-777,777,777,777=$\n$\\text{(A)}\\ 222,222,222,222 \\qquad \\text{(B)}\\ 222,222,222,223 \\qquad \\text{(C)}\\ 233,333,333,333 \\qquad \\\\ \\text{(D)}\\ 322,222,222,223 \\qquad \\text{(E)}\\ 333,333,333,333$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_696", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $1,000,000,000,000-777,777,777,777=$\n$\\text{(A)}\\ 222,222,222,222 \\qquad \\text{(B)}\\ 222,222,222,223 \\qquad \\text{(C)}\\ 233,333,333,333 \\qquad \\\\ \\text{(D)}\\ 322,222,222,223 \\qquad \\text{(E)}\\ 333,333,333,333$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$ and $b$ be relatively prime positive integers with $a>b>0$ and $\\dfrac{a^3-b^3}{(a-b)^3} = \\dfrac{73}{3}$. What is $a-b$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad \\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_697", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$ and $b$ be relatively prime positive integers with $a>b>0$ and $\\dfrac{a^3-b^3}{(a-b)^3} = \\dfrac{73}{3}$. What is $a-b$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad \\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. For any positive integer $n$, define $\\boxed{n}$ to be the sum of the positive factors of $n$.\nFor example, $\\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\\boxed{\\boxed{11}}$ .\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 20 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 28 \\qquad \\textbf{(E)}\\ 30$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "n", + "index": "Sky-T1_10k_698", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For any positive integer $n$, define $\\boxed{n}$ to be the sum of the positive factors of $n$.\nFor example, $\\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\\boxed{\\boxed{11}}$ .\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 20 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 28 \\qquad \\textbf{(E)}\\ 30$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of integers $c$ such that the equation \\[\\left||20|x|-x^2|-c\\right|=21\\]has $12$ distinct real solutions.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "57", + "index": "Sky-T1_10k_699", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of integers $c$ such that the equation \\[\\left||20|x|-x^2|-c\\right|=21\\]has $12$ distinct real solutions." + } + }, + { + "question": "Return your final response within \\boxed{}. Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walters entered?\n$\\textbf{(A)} \\ 71 \\qquad \\textbf{(B)} \\ 76 \\qquad \\textbf{(C)} \\ 80 \\qquad \\textbf{(D)} \\ 82 \\qquad \\textbf{(E)} \\ 91$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_700", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walters entered?\n$\\textbf{(A)} \\ 71 \\qquad \\textbf{(B)} \\ 76 \\qquad \\textbf{(C)} \\ 80 \\qquad \\textbf{(D)} \\ 82 \\qquad \\textbf{(E)} \\ 91$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\\circ$. The enclosed area of the hexagon is $6\\sqrt{3}$. What is the perimeter of the hexagon?\n\n$\\textbf{(A)} \\: 4 \\qquad \\textbf{(B)} \\: 4\\sqrt3 \\qquad \\textbf{(C)} \\: 12 \\qquad \\textbf{(D)} \\: 18 \\qquad \\textbf{(E)} \\: 12\\sqrt3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_701", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\\circ$. The enclosed area of the hexagon is $6\\sqrt{3}$. What is the perimeter of the hexagon?\n\n$\\textbf{(A)} \\: 4 \\qquad \\textbf{(B)} \\: 4\\sqrt3 \\qquad \\textbf{(C)} \\: 12 \\qquad \\textbf{(D)} \\: 18 \\qquad \\textbf{(E)} \\: 12\\sqrt3$" + } + }, + { + "question": "Return your final response within \\boxed{}. Josh writes the numbers $1,2,3,\\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, skips $8$, marks out $10$, and so on to the end. Josh continues in this manner until only one number remains. What is that number?\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 32 \\qquad \\textbf{(C)}\\ 56 \\qquad \\textbf{(D)}\\ 64 \\qquad \\textbf{(E)}\\ 96$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_702", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Josh writes the numbers $1,2,3,\\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, skips $8$, marks out $10$, and so on to the end. Josh continues in this manner until only one number remains. What is that number?\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 32 \\qquad \\textbf{(C)}\\ 56 \\qquad \\textbf{(D)}\\ 64 \\qquad \\textbf{(E)}\\ 96$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$ is:\n$\\text{(A) } 0\\quad \\text{(B) } 3\\quad \\text{(C) } 4\\quad \\text{(D) } 12\\quad \\text{(E) } \\infty$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_703", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$ is:\n$\\text{(A) } 0\\quad \\text{(B) } 3\\quad \\text{(C) } 4\\quad \\text{(D) } 12\\quad \\text{(E) } \\infty$" + } + }, + { + "question": "Return your final response within \\boxed{}. Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates (2,2). What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?\n\n\nNOT TO SCALE\n$\\text{(A)}\\ (-6,0) \\qquad \\text{(B)}\\ (-4,0) \\qquad \\text{(C)}\\ (-2,0) \\qquad \\text{(D)}\\ (2,0) \\qquad \\text{(E)}\\ (4,0)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_704", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates (2,2). What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?\n\n\nNOT TO SCALE\n$\\text{(A)}\\ (-6,0) \\qquad \\text{(B)}\\ (-4,0) \\qquad \\text{(C)}\\ (-2,0) \\qquad \\text{(D)}\\ (2,0) \\qquad \\text{(E)}\\ (4,0)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "114", + "index": "Sky-T1_10k_705", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left." + } + }, + { + "question": "Return your final response within \\boxed{}. The value of $x$ that satisfies $\\log_{2^x} 3^{20} = \\log_{2^{x+3}} 3^{2020}$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "103", + "index": "Sky-T1_10k_706", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The value of $x$ that satisfies $\\log_{2^x} 3^{20} = \\log_{2^{x+3}} 3^{2020}$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?\n$\\mathrm{(A)}\\ 36\\qquad\\mathrm{(B)}\\ 40\\qquad\\textbf{(C)}\\ 44\\qquad\\mathrm{(D)}\\ 48\\qquad\\mathrm{(E)}\\ 52$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_707", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?\n$\\mathrm{(A)}\\ 36\\qquad\\mathrm{(B)}\\ 40\\qquad\\textbf{(C)}\\ 44\\qquad\\mathrm{(D)}\\ 48\\qquad\\mathrm{(E)}\\ 52$" + } + }, + { + "question": "Return your final response within \\boxed{}. The least value of the function $ax^2 + bx + c$ with $a>0$ is:\n$\\textbf{(A)}\\ -\\dfrac{b}{a} \\qquad \\textbf{(B)}\\ -\\dfrac{b}{2a} \\qquad \\textbf{(C)}\\ b^2-4ac \\qquad \\textbf{(D)}\\ \\dfrac{4ac-b^2}{4a}\\qquad \\textbf{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_708", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The least value of the function $ax^2 + bx + c$ with $a>0$ is:\n$\\textbf{(A)}\\ -\\dfrac{b}{a} \\qquad \\textbf{(B)}\\ -\\dfrac{b}{2a} \\qquad \\textbf{(C)}\\ b^2-4ac \\qquad \\textbf{(D)}\\ \\dfrac{4ac-b^2}{4a}\\qquad \\textbf{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\\pi$ radians is $180$ degrees.)\n\n[asy] size(85); fill((-30,0)..(-24,18)--(0,0)--(-24,-18)..cycle,gray(0.7)); fill((30,0)..(24,18)--(0,0)--(24,-18)..cycle,gray(0.7)); fill((-20,0)..(0,20)--(0,-20)..cycle,white); fill((20,0)..(0,20)--(0,-20)..cycle,white); fill((0,20)..(-16,12)--(0,0)--(16,12)..cycle,gray(0.7)); fill((0,-20)..(-16,-12)--(0,0)--(16,-12)..cycle,gray(0.7)); fill((0,10)..(-10,0)--(10,0)..cycle,white); fill((0,-10)..(-10,0)--(10,0)..cycle,white); fill((-10,0)..(-8,6)--(0,0)--(-8,-6)..cycle,gray(0.7)); fill((10,0)..(8,6)--(0,0)--(8,-6)..cycle,gray(0.7)); draw(Circle((0,0),10),linewidth(0.7)); draw(Circle((0,0),20),linewidth(0.7)); draw(Circle((0,0),30),linewidth(0.7)); draw((-28,-21)--(28,21),linewidth(0.7)); draw((-28,21)--(28,-21),linewidth(0.7));[/asy]\n$\\mathrm{(A) \\ } \\frac{\\pi}{8} \\qquad \\mathrm{(B) \\ } \\frac{\\pi}{7} \\qquad \\mathrm{(C) \\ } \\frac{\\pi}{6} \\qquad \\mathrm{(D) \\ } \\frac{\\pi}{5} \\qquad \\mathrm{(E) \\ } \\frac{\\pi}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_709", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\\pi$ radians is $180$ degrees.)\n\n[asy] size(85); fill((-30,0)..(-24,18)--(0,0)--(-24,-18)..cycle,gray(0.7)); fill((30,0)..(24,18)--(0,0)--(24,-18)..cycle,gray(0.7)); fill((-20,0)..(0,20)--(0,-20)..cycle,white); fill((20,0)..(0,20)--(0,-20)..cycle,white); fill((0,20)..(-16,12)--(0,0)--(16,12)..cycle,gray(0.7)); fill((0,-20)..(-16,-12)--(0,0)--(16,-12)..cycle,gray(0.7)); fill((0,10)..(-10,0)--(10,0)..cycle,white); fill((0,-10)..(-10,0)--(10,0)..cycle,white); fill((-10,0)..(-8,6)--(0,0)--(-8,-6)..cycle,gray(0.7)); fill((10,0)..(8,6)--(0,0)--(8,-6)..cycle,gray(0.7)); draw(Circle((0,0),10),linewidth(0.7)); draw(Circle((0,0),20),linewidth(0.7)); draw(Circle((0,0),30),linewidth(0.7)); draw((-28,-21)--(28,21),linewidth(0.7)); draw((-28,21)--(28,-21),linewidth(0.7));[/asy]\n$\\mathrm{(A) \\ } \\frac{\\pi}{8} \\qquad \\mathrm{(B) \\ } \\frac{\\pi}{7} \\qquad \\mathrm{(C) \\ } \\frac{\\pi}{6} \\qquad \\mathrm{(D) \\ } \\frac{\\pi}{5} \\qquad \\mathrm{(E) \\ } \\frac{\\pi}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A $4\\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?\n$\\begin{tabular}[t]{|c|c|c|c|} \\multicolumn{4}{c}{}\\\\\\hline 1&2&3&4\\\\\\hline 8&9&10&11\\\\\\hline 15&16&17&18\\\\\\hline 22&23&24&25\\\\\\hline \\end{tabular}$\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 8 \\qquad \\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_710", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A $4\\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?\n$\\begin{tabular}[t]{|c|c|c|c|} \\multicolumn{4}{c}{}\\\\\\hline 1&2&3&4\\\\\\hline 8&9&10&11\\\\\\hline 15&16&17&18\\\\\\hline 22&23&24&25\\\\\\hline \\end{tabular}$\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 8 \\qquad \\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "272", + "index": "Sky-T1_10k_711", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$." + } + }, + { + "question": "Return your final response within \\boxed{}. The discriminant of the equation $x^2+2x\\sqrt{3}+3=0$ is zero. Hence, its roots are: \n$\\textbf{(A)}\\ \\text{real and equal}\\qquad\\textbf{(B)}\\ \\text{rational and equal}\\qquad\\textbf{(C)}\\ \\text{rational and unequal}\\\\ \\textbf{(D)}\\ \\text{irrational and unequal}\\qquad\\textbf{(E)}\\ \\text{imaginary}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_712", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The discriminant of the equation $x^2+2x\\sqrt{3}+3=0$ is zero. Hence, its roots are: \n$\\textbf{(A)}\\ \\text{real and equal}\\qquad\\textbf{(B)}\\ \\text{rational and equal}\\qquad\\textbf{(C)}\\ \\text{rational and unequal}\\\\ \\textbf{(D)}\\ \\text{irrational and unequal}\\qquad\\textbf{(E)}\\ \\text{imaginary}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 54$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_713", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 54$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\\cdot 2 + 0\\cdot 3$ and $402\\cdot 2 + 404\\cdot 3$ are two such ways.)\n$\\textbf{(A)}\\ 236\\qquad\\textbf{(B)}\\ 336\\qquad\\textbf{(C)}\\ 337\\qquad\\textbf{(D)}\\ 403\\qquad\\textbf{(E)}\\ 672$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_714", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\\cdot 2 + 0\\cdot 3$ and $402\\cdot 2 + 404\\cdot 3$ are two such ways.)\n$\\textbf{(A)}\\ 236\\qquad\\textbf{(B)}\\ 336\\qquad\\textbf{(C)}\\ 337\\qquad\\textbf{(D)}\\ 403\\qquad\\textbf{(E)}\\ 672$" + } + }, + { + "question": "Return your final response within \\boxed{}. A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position?\n$\\textbf{(A)}\\ \\dfrac{1}{6} \\qquad \\textbf{(B)}\\ \\dfrac{1}{5} \\qquad \\textbf{(C)}\\ \\dfrac{1}{4} \\qquad \\textbf{(D)}\\ \\dfrac{1}{3} \\qquad \\textbf{(E)}\\ \\dfrac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_715", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position?\n$\\textbf{(A)}\\ \\dfrac{1}{6} \\qquad \\textbf{(B)}\\ \\dfrac{1}{5} \\qquad \\textbf{(C)}\\ \\dfrac{1}{4} \\qquad \\textbf{(D)}\\ \\dfrac{1}{3} \\qquad \\textbf{(E)}\\ \\dfrac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In right triangle $ABC$ the hypotenuse $\\overline{AB}=5$ and leg $\\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\\overline{PQ}=A_1B$ and leg $\\overline{PR}=A_1C$. If the bisector of angle $P$ meets the opposite side in $P_1$, the length of $PP_1$ is:\n$\\textbf{(A)}\\ \\frac{3\\sqrt{6}}{4}\\qquad \\textbf{(B)}\\ \\frac{3\\sqrt{5}}{4}\\qquad \\textbf{(C)}\\ \\frac{3\\sqrt{3}}{4}\\qquad \\textbf{(D)}\\ \\frac{3\\sqrt{2}}{2}\\qquad \\textbf{(E)}\\ \\frac{15\\sqrt{2}}{16}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_716", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In right triangle $ABC$ the hypotenuse $\\overline{AB}=5$ and leg $\\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\\overline{PQ}=A_1B$ and leg $\\overline{PR}=A_1C$. If the bisector of angle $P$ meets the opposite side in $P_1$, the length of $PP_1$ is:\n$\\textbf{(A)}\\ \\frac{3\\sqrt{6}}{4}\\qquad \\textbf{(B)}\\ \\frac{3\\sqrt{5}}{4}\\qquad \\textbf{(C)}\\ \\frac{3\\sqrt{3}}{4}\\qquad \\textbf{(D)}\\ \\frac{3\\sqrt{2}}{2}\\qquad \\textbf{(E)}\\ \\frac{15\\sqrt{2}}{16}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "89", + "index": "Sky-T1_10k_717", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$." + } + }, + { + "question": "Return your final response within \\boxed{}. In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0B>0$ and A is $x$% greater than $B$. What is $x$?\n$\\textbf {(A) } 100\\left(\\frac{A-B}{B}\\right) \\qquad \\textbf {(B) } 100\\left(\\frac{A+B}{B}\\right) \\qquad \\textbf {(C) } 100\\left(\\frac{A+B}{A}\\right)\\qquad \\textbf {(D) } 100\\left(\\frac{A-B}{A}\\right) \\qquad \\textbf {(E) } 100\\left(\\frac{A}{B}\\right)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_735", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose $A>B>0$ and A is $x$% greater than $B$. What is $x$?\n$\\textbf {(A) } 100\\left(\\frac{A-B}{B}\\right) \\qquad \\textbf {(B) } 100\\left(\\frac{A+B}{B}\\right) \\qquad \\textbf {(C) } 100\\left(\\frac{A+B}{A}\\right)\\qquad \\textbf {(D) } 100\\left(\\frac{A-B}{A}\\right) \\qquad \\textbf {(E) } 100\\left(\\frac{A}{B}\\right)$" + } + }, + { + "question": "Return your final response within \\boxed{}. The state income tax where Kristin lives is levied at the rate of $p\\%$ of the first\n$\\textdollar 28000$ of annual income plus $(p + 2)\\%$ of any amount above $\\textdollar 28000$. Kristin\nnoticed that the state income tax she paid amounted to $(p + 0.25)\\%$ of her\nannual income. What was her annual income? \n$\\textbf{(A)}\\,\\textdollar 28000 \\qquad \\textbf{(B)}\\,\\textdollar 32000 \\qquad \\textbf{(C)}\\,\\textdollar 35000 \\qquad \\textbf{(D)}\\,\\textdollar 42000 \\qquad \\textbf{(E)}\\,\\textdollar 56000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_736", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The state income tax where Kristin lives is levied at the rate of $p\\%$ of the first\n$\\textdollar 28000$ of annual income plus $(p + 2)\\%$ of any amount above $\\textdollar 28000$. Kristin\nnoticed that the state income tax she paid amounted to $(p + 0.25)\\%$ of her\nannual income. What was her annual income? \n$\\textbf{(A)}\\,\\textdollar 28000 \\qquad \\textbf{(B)}\\,\\textdollar 32000 \\qquad \\textbf{(C)}\\,\\textdollar 35000 \\qquad \\textbf{(D)}\\,\\textdollar 42000 \\qquad \\textbf{(E)}\\,\\textdollar 56000$" + } + }, + { + "question": "Return your final response within \\boxed{}. A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:\n$\\textbf{(A)}\\ 1080\\qquad \\textbf{(B)}\\ 900\\qquad \\textbf{(C)}\\ 720\\qquad \\textbf{(D)}\\ 540\\qquad \\textbf{(E)}\\ 360$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_737", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:\n$\\textbf{(A)}\\ 1080\\qquad \\textbf{(B)}\\ 900\\qquad \\textbf{(C)}\\ 720\\qquad \\textbf{(D)}\\ 540\\qquad \\textbf{(E)}\\ 360$" + } + }, + { + "question": "Return your final response within \\boxed{}. The solution of $\\sqrt{5x-1}+\\sqrt{x-1}=2$ is:\n$\\textbf{(A)}\\ x=2,x=1\\qquad\\textbf{(B)}\\ x=\\frac{2}{3}\\qquad\\textbf{(C)}\\ x=2\\qquad\\textbf{(D)}\\ x=1\\qquad\\textbf{(E)}\\ x=0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_738", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The solution of $\\sqrt{5x-1}+\\sqrt{x-1}=2$ is:\n$\\textbf{(A)}\\ x=2,x=1\\qquad\\textbf{(B)}\\ x=\\frac{2}{3}\\qquad\\textbf{(C)}\\ x=2\\qquad\\textbf{(D)}\\ x=1\\qquad\\textbf{(E)}\\ x=0$" + } + }, + { + "question": "Return your final response within \\boxed{}. A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?\n\n$\\textbf{(A) }36\\qquad\\textbf{(B) }60\\qquad\\textbf{(C) }72\\qquad\\textbf{(D) }84\\qquad\\textbf{(E) }108\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_739", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?\n\n$\\textbf{(A) }36\\qquad\\textbf{(B) }60\\qquad\\textbf{(C) }72\\qquad\\textbf{(D) }84\\qquad\\textbf{(E) }108\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Titu Andreescu)\nLet $a$, $b$, and $c$ be positive real numbers. Prove that\n\n\n$(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \\ge (a+b+c)^3$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \\ge (a + b + c)^3", + "index": "Sky-T1_10k_740", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Titu Andreescu)\nLet $a$, $b$, and $c$ be positive real numbers. Prove that\n\n\n$(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \\ge (a+b+c)^3$." + } + }, + { + "question": "Return your final response within \\boxed{}. Ralph went to the store and bought 12 pairs of socks for a total of $$24$. Some of the socks he bought cost $$1$ a pair, some of the socks he bought cost $$3$ a pair, and some of the socks he bought cost $$4$ a pair. If he bought at least one pair of each type, how many pairs of $$1$ socks did Ralph buy?\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 5 \\qquad \\textbf{(C) } 6 \\qquad \\textbf{(D) } 7 \\qquad \\textbf{(E) } 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_741", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ralph went to the store and bought 12 pairs of socks for a total of $$24$. Some of the socks he bought cost $$1$ a pair, some of the socks he bought cost $$3$ a pair, and some of the socks he bought cost $$4$ a pair. If he bought at least one pair of each type, how many pairs of $$1$ socks did Ralph buy?\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 5 \\qquad \\textbf{(C) } 6 \\qquad \\textbf{(D) } 7 \\qquad \\textbf{(E) } 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?\n$\\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \\cdots, a_n$ is defined to be\n\\[\\left[\\frac{a_1^2 + a_2^2 + \\cdots + a_n^2}n\\right]^{1/2}\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "337", + "index": "Sky-T1_10k_742", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?\n$\\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \\cdots, a_n$ is defined to be\n\\[\\left[\\frac{a_1^2 + a_2^2 + \\cdots + a_n^2}n\\right]^{1/2}\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\\triangle GEM$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "25", + "index": "Sky-T1_10k_743", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\\triangle GEM$." + } + }, + { + "question": "Return your final response within \\boxed{}. A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?\n[asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy]\n$\\textbf{(A)} \\:1 : 6 \\qquad\\textbf{ (B)}\\: 7 : 36 \\qquad\\textbf{(C)}\\: 1 : 5 \\qquad\\textbf{(D)}\\: 7 : 30\\qquad\\textbf{ (E)}\\: 6 : 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_744", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?\n[asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy]\n$\\textbf{(A)} \\:1 : 6 \\qquad\\textbf{ (B)}\\: 7 : 36 \\qquad\\textbf{(C)}\\: 1 : 5 \\qquad\\textbf{(D)}\\: 7 : 30\\qquad\\textbf{ (E)}\\: 6 : 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following equations have the same graph?\n$I.\\quad y=x-2 \\qquad II.\\quad y=\\frac{x^2-4}{x+2}\\qquad III.\\quad (x+2)y=x^2-4$\n$\\text{(A) I and II only} \\quad \\text{(B) I and III only} \\quad \\text{(C) II and III only} \\quad \\text{(D) I,II,and III} \\quad \\\\ \\text{(E) None. All of the equations have different graphs}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_745", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following equations have the same graph?\n$I.\\quad y=x-2 \\qquad II.\\quad y=\\frac{x^2-4}{x+2}\\qquad III.\\quad (x+2)y=x^2-4$\n$\\text{(A) I and II only} \\quad \\text{(B) I and III only} \\quad \\text{(C) II and III only} \\quad \\text{(D) I,II,and III} \\quad \\\\ \\text{(E) None. All of the equations have different graphs}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ratio of the length to the width of a rectangle is $4$ : $3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?\n$\\textbf{(A)}\\ \\frac{2}{7}\\qquad\\textbf{(B)}\\ \\frac{3}{7}\\qquad\\textbf{(C)}\\ \\frac{12}{25}\\qquad\\textbf{(D)}\\ \\frac{16}{25}\\qquad\\textbf{(E)}\\ \\frac{3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_746", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ratio of the length to the width of a rectangle is $4$ : $3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?\n$\\textbf{(A)}\\ \\frac{2}{7}\\qquad\\textbf{(B)}\\ \\frac{3}{7}\\qquad\\textbf{(C)}\\ \\frac{12}{25}\\qquad\\textbf{(D)}\\ \\frac{16}{25}\\qquad\\textbf{(E)}\\ \\frac{3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$?\n$\\textbf{(A) } 25 \\qquad\\textbf{(B) } 33 \\qquad\\textbf{(C) } 44\\qquad\\textbf{(D) } 66 \\qquad\\textbf{(E) } 78$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_747", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$?\n$\\textbf{(A) } 25 \\qquad\\textbf{(B) } 33 \\qquad\\textbf{(C) } 44\\qquad\\textbf{(D) } 66 \\qquad\\textbf{(E) } 78$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x\\not=0$ or $4$ and $y\\not=0$ or $6$, then $\\frac{2}{x}+\\frac{3}{y}=\\frac{1}{2}$ is equivalent to\n$\\mathrm{(A)\\ } 4x+3y=xy \\qquad \\mathrm{(B) \\ }y=\\frac{4x}{6-y} \\qquad \\mathrm{(C) \\ } \\frac{x}{2}+\\frac{y}{3}=2 \\qquad$\n$\\mathrm{(D) \\ } \\frac{4y}{y-6}=x \\qquad \\mathrm{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_748", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x\\not=0$ or $4$ and $y\\not=0$ or $6$, then $\\frac{2}{x}+\\frac{3}{y}=\\frac{1}{2}$ is equivalent to\n$\\mathrm{(A)\\ } 4x+3y=xy \\qquad \\mathrm{(B) \\ }y=\\frac{4x}{6-y} \\qquad \\mathrm{(C) \\ } \\frac{x}{2}+\\frac{y}{3}=2 \\qquad$\n$\\mathrm{(D) \\ } \\frac{4y}{y-6}=x \\qquad \\mathrm{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?\n$\\textbf{(A) }7 \\qquad \\textbf{(B) }8 \\qquad \\textbf{(C) }9 \\qquad \\textbf{(D) }10 \\qquad \\textbf{(E) }11 \\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_749", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?\n$\\textbf{(A) }7 \\qquad \\textbf{(B) }8 \\qquad \\textbf{(C) }9 \\qquad \\textbf{(D) }10 \\qquad \\textbf{(E) }11 \\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mary is $20\\%$ older than Sally, and Sally is $40\\%$ younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday?\n$\\mathrm{(A) \\ } 7\\qquad \\mathrm{(B) \\ } 8\\qquad \\mathrm{(C) \\ } 9\\qquad \\mathrm{(D) \\ } 10\\qquad \\mathrm{(E) \\ } 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_750", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mary is $20\\%$ older than Sally, and Sally is $40\\%$ younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday?\n$\\mathrm{(A) \\ } 7\\qquad \\mathrm{(B) \\ } 8\\qquad \\mathrm{(C) \\ } 9\\qquad \\mathrm{(D) \\ } 10\\qquad \\mathrm{(E) \\ } 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next $n$ shots are bullseyes she will be guaranteed victory. What is the minimum value for $n$?\n$\\textbf{(A)}\\ 38 \\qquad \\textbf{(B)}\\ 40 \\qquad \\textbf{(C)}\\ 42 \\qquad \\textbf{(D)}\\ 44 \\qquad \\textbf{(E)}\\ 46$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_751", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next $n$ shots are bullseyes she will be guaranteed victory. What is the minimum value for $n$?\n$\\textbf{(A)}\\ 38 \\qquad \\textbf{(B)}\\ 40 \\qquad \\textbf{(C)}\\ 42 \\qquad \\textbf{(D)}\\ 44 \\qquad \\textbf{(E)}\\ 46$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following [fractions](https://artofproblemsolving.com/wiki/index.php/Fraction) has the largest value?\n$\\text{(A)}\\ \\frac{3}{7} \\qquad \\text{(B)}\\ \\frac{4}{9} \\qquad \\text{(C)}\\ \\frac{17}{35} \\qquad \\text{(D)}\\ \\frac{100}{201} \\qquad \\text{(E)}\\ \\frac{151}{301}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_752", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following [fractions](https://artofproblemsolving.com/wiki/index.php/Fraction) has the largest value?\n$\\text{(A)}\\ \\frac{3}{7} \\qquad \\text{(B)}\\ \\frac{4}{9} \\qquad \\text{(C)}\\ \\frac{17}{35} \\qquad \\text{(D)}\\ \\frac{100}{201} \\qquad \\text{(E)}\\ \\frac{151}{301}$" + } + }, + { + "question": "Return your final response within \\boxed{}. To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, then the supply of medicine would last approximately\n$\\text{(A)}\\ 1\\text{ month} \\qquad \\text{(B)}\\ 4\\text{ months} \\qquad \\text{(C)}\\ 6\\text{ months} \\qquad \\text{(D)}\\ 8\\text{ months} \\qquad \\text{(E)}\\ 1\\text{ year}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_753", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, then the supply of medicine would last approximately\n$\\text{(A)}\\ 1\\text{ month} \\qquad \\text{(B)}\\ 4\\text{ months} \\qquad \\text{(C)}\\ 6\\text{ months} \\qquad \\text{(D)}\\ 8\\text{ months} \\qquad \\text{(E)}\\ 1\\text{ year}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\mathbb{Z}$ be the set of integers. Find all functions $f : \\mathbb{Z} \\rightarrow \\mathbb{Z}$ such that \\[xf(2f(y)-x)+y^2f(2x-f(y))=\\frac{f(x)^2}{x}+f(yf(y))\\] for all $x, y \\in \\mathbb{Z}$ with $x \\neq 0$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "0", + "index": "Sky-T1_10k_754", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\mathbb{Z}$ be the set of integers. Find all functions $f : \\mathbb{Z} \\rightarrow \\mathbb{Z}$ such that \\[xf(2f(y)-x)+y^2f(2x-f(y))=\\frac{f(x)^2}{x}+f(yf(y))\\] for all $x, y \\in \\mathbb{Z}$ with $x \\neq 0$." + } + }, + { + "question": "Return your final response within \\boxed{}. If the Highest Common Divisor of $6432$ and $132$ is diminished by $8$, it will equal: \n$\\textbf{(A)}\\ -6 \\qquad \\textbf{(B)}\\ 6 \\qquad \\textbf{(C)}\\ -2 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_755", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the Highest Common Divisor of $6432$ and $132$ is diminished by $8$, it will equal: \n$\\textbf{(A)}\\ -6 \\qquad \\textbf{(B)}\\ 6 \\qquad \\textbf{(C)}\\ -2 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $200\\leq a \\leq 400$ and $600\\leq b\\leq 1200$, then the largest value of the quotient $\\frac{b}{a}$ is\n$\\text{(A)}\\ \\frac{3}{2} \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 300 \\qquad \\text{(E)}\\ 600$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_756", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $200\\leq a \\leq 400$ and $600\\leq b\\leq 1200$, then the largest value of the quotient $\\frac{b}{a}$ is\n$\\text{(A)}\\ \\frac{3}{2} \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 300 \\qquad \\text{(E)}\\ 600$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the radius of a circle is increased by $1$ unit, the ratio of the new circumference to the new diameter is: \n$\\textbf{(A)}\\ \\pi+2\\qquad\\textbf{(B)}\\ \\frac{2\\pi+1}{2}\\qquad\\textbf{(C)}\\ \\pi\\qquad\\textbf{(D)}\\ \\frac{2\\pi-1}{2}\\qquad\\textbf{(E)}\\ \\pi-2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_757", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the radius of a circle is increased by $1$ unit, the ratio of the new circumference to the new diameter is: \n$\\textbf{(A)}\\ \\pi+2\\qquad\\textbf{(B)}\\ \\frac{2\\pi+1}{2}\\qquad\\textbf{(C)}\\ \\pi\\qquad\\textbf{(D)}\\ \\frac{2\\pi-1}{2}\\qquad\\textbf{(E)}\\ \\pi-2$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the smallest positive integer $n$ such that $\\sqrt{n}-\\sqrt{n-1}<.01$?\n$\\textbf{(A) }2499\\qquad \\textbf{(B) }2500\\qquad \\textbf{(C) }2501\\qquad \\textbf{(D) }10,000\\qquad \\textbf{(E) }\\text{There is no such integer}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_758", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the smallest positive integer $n$ such that $\\sqrt{n}-\\sqrt{n-1}<.01$?\n$\\textbf{(A) }2499\\qquad \\textbf{(B) }2500\\qquad \\textbf{(C) }2501\\qquad \\textbf{(D) }10,000\\qquad \\textbf{(E) }\\text{There is no such integer}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many of the twelve pentominoes pictured below have at least one line of reflectional symmetry?\n[asy] unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare); draw(shift(2,5)*unitsquare); draw(shift(2,6)*unitsquare); draw(shift(1,5)*unitsquare); draw(shift(0,5)*unitsquare); draw(shift(4,8)*unitsquare); draw(shift(3,8)*unitsquare); draw(shift(2,8)*unitsquare); draw(shift(1,8)*unitsquare); draw(shift(0,8)*unitsquare); draw(shift(6,8)*unitsquare); draw(shift(7,8)*unitsquare); draw(shift(8,8)*unitsquare); draw(shift(9,8)*unitsquare); draw(shift(9,9)*unitsquare); draw(shift(6,5)*unitsquare); draw(shift(7,5)*unitsquare); draw(shift(8,5)*unitsquare); draw(shift(7,6)*unitsquare); draw(shift(7,4)*unitsquare); draw(shift(6,1)*unitsquare); draw(shift(7,1)*unitsquare); draw(shift(8,1)*unitsquare); draw(shift(6,0)*unitsquare); draw(shift(7,2)*unitsquare); draw(shift(11,8)*unitsquare); draw(shift(12,8)*unitsquare); draw(shift(13,8)*unitsquare); draw(shift(14,8)*unitsquare); draw(shift(13,9)*unitsquare); draw(shift(11,5)*unitsquare); draw(shift(12,5)*unitsquare); draw(shift(13,5)*unitsquare); draw(shift(11,6)*unitsquare); draw(shift(13,4)*unitsquare); draw(shift(11,1)*unitsquare); draw(shift(12,1)*unitsquare); draw(shift(13,1)*unitsquare); draw(shift(13,2)*unitsquare); draw(shift(14,2)*unitsquare); draw(shift(16,8)*unitsquare); draw(shift(17,8)*unitsquare); draw(shift(18,8)*unitsquare); draw(shift(17,9)*unitsquare); draw(shift(18,9)*unitsquare); draw(shift(16,5)*unitsquare); draw(shift(17,6)*unitsquare); draw(shift(18,5)*unitsquare); draw(shift(16,6)*unitsquare); draw(shift(18,6)*unitsquare); draw(shift(16,0)*unitsquare); draw(shift(17,0)*unitsquare); draw(shift(17,1)*unitsquare); draw(shift(18,1)*unitsquare); draw(shift(18,2)*unitsquare);[/asy]\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } 6 \\qquad\\textbf{(E) } 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_759", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many of the twelve pentominoes pictured below have at least one line of reflectional symmetry?\n[asy] unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare); draw(shift(2,5)*unitsquare); draw(shift(2,6)*unitsquare); draw(shift(1,5)*unitsquare); draw(shift(0,5)*unitsquare); draw(shift(4,8)*unitsquare); draw(shift(3,8)*unitsquare); draw(shift(2,8)*unitsquare); draw(shift(1,8)*unitsquare); draw(shift(0,8)*unitsquare); draw(shift(6,8)*unitsquare); draw(shift(7,8)*unitsquare); draw(shift(8,8)*unitsquare); draw(shift(9,8)*unitsquare); draw(shift(9,9)*unitsquare); draw(shift(6,5)*unitsquare); draw(shift(7,5)*unitsquare); draw(shift(8,5)*unitsquare); draw(shift(7,6)*unitsquare); draw(shift(7,4)*unitsquare); draw(shift(6,1)*unitsquare); draw(shift(7,1)*unitsquare); draw(shift(8,1)*unitsquare); draw(shift(6,0)*unitsquare); draw(shift(7,2)*unitsquare); draw(shift(11,8)*unitsquare); draw(shift(12,8)*unitsquare); draw(shift(13,8)*unitsquare); draw(shift(14,8)*unitsquare); draw(shift(13,9)*unitsquare); draw(shift(11,5)*unitsquare); draw(shift(12,5)*unitsquare); draw(shift(13,5)*unitsquare); draw(shift(11,6)*unitsquare); draw(shift(13,4)*unitsquare); draw(shift(11,1)*unitsquare); draw(shift(12,1)*unitsquare); draw(shift(13,1)*unitsquare); draw(shift(13,2)*unitsquare); draw(shift(14,2)*unitsquare); draw(shift(16,8)*unitsquare); draw(shift(17,8)*unitsquare); draw(shift(18,8)*unitsquare); draw(shift(17,9)*unitsquare); draw(shift(18,9)*unitsquare); draw(shift(16,5)*unitsquare); draw(shift(17,6)*unitsquare); draw(shift(18,5)*unitsquare); draw(shift(16,6)*unitsquare); draw(shift(18,6)*unitsquare); draw(shift(16,0)*unitsquare); draw(shift(17,0)*unitsquare); draw(shift(17,1)*unitsquare); draw(shift(18,1)*unitsquare); draw(shift(18,2)*unitsquare);[/asy]\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } 6 \\qquad\\textbf{(E) } 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\\overline{AC}$ and points $D$ and $F$ are on $\\overline{AB}$ such that both $\\overline{DE}$ and $\\overline{FG}$ are parallel to $\\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$?\n\n$\\textbf{(A) }1\\qquad \\textbf{(B) }\\dfrac{3}{2}\\qquad \\textbf{(C) }\\dfrac{21}{13}\\qquad \\textbf{(D) }\\dfrac{13}{8}\\qquad \\textbf{(E) }\\dfrac{5}{3}\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_760", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\\overline{AC}$ and points $D$ and $F$ are on $\\overline{AB}$ such that both $\\overline{DE}$ and $\\overline{FG}$ are parallel to $\\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$?\n\n$\\textbf{(A) }1\\qquad \\textbf{(B) }\\dfrac{3}{2}\\qquad \\textbf{(C) }\\dfrac{21}{13}\\qquad \\textbf{(D) }\\dfrac{13}{8}\\qquad \\textbf{(E) }\\dfrac{5}{3}\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "53", + "index": "Sky-T1_10k_761", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is:\n$\\text{(A) } 15\\quad \\text{(B) } 6\\quad \\text{(C) } 5\\quad \\text{(D) } 3\\quad \\text{(E) } 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_762", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is:\n$\\text{(A) } 15\\quad \\text{(B) } 6\\quad \\text{(C) } 5\\quad \\text{(D) } 3\\quad \\text{(E) } 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?\n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 15\\qquad\\textbf{(C)}\\ 20\\qquad\\textbf{(D)}\\ 25\\qquad\\textbf{(E)}\\ 30$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_763", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?\n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 15\\qquad\\textbf{(C)}\\ 20\\qquad\\textbf{(D)}\\ 25\\qquad\\textbf{(E)}\\ 30$" + } + }, + { + "question": "Return your final response within \\boxed{}. A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?\n$\\textbf{(A)}\\ \\text{Friday}\\qquad\\textbf{(B)}\\ \\text{Saturday}\\qquad\\textbf{(C)}\\ \\text{Sunday}\\qquad\\textbf{(D)}\\ \\text{Monday}\\qquad\\textbf{(E)}\\ \\text{Tuesday}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_764", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?\n$\\textbf{(A)}\\ \\text{Friday}\\qquad\\textbf{(B)}\\ \\text{Saturday}\\qquad\\textbf{(C)}\\ \\text{Sunday}\\qquad\\textbf{(D)}\\ \\text{Monday}\\qquad\\textbf{(E)}\\ \\text{Tuesday}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\dfrac{a+b}{b+c}=\\dfrac{c+d}{d+a}$, then:\n$\\textbf{(A) }a \\text{ must equal }c\\qquad\\textbf{(B) }a+b+c+d\\text{ must equal zero}\\qquad$\n$\\textbf{(C) }\\text{either }a=c\\text{ or }a+b+c+d=0\\text{, or both}\\qquad$\n$\\textbf{(D) }a+b+c+d\\ne 0\\text{ if }a=c\\qquad$\n$\\textbf{(E) }a(b+c+d)=c(a+b+d)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_765", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\dfrac{a+b}{b+c}=\\dfrac{c+d}{d+a}$, then:\n$\\textbf{(A) }a \\text{ must equal }c\\qquad\\textbf{(B) }a+b+c+d\\text{ must equal zero}\\qquad$\n$\\textbf{(C) }\\text{either }a=c\\text{ or }a+b+c+d=0\\text{, or both}\\qquad$\n$\\textbf{(D) }a+b+c+d\\ne 0\\text{ if }a=c\\qquad$\n$\\textbf{(E) }a(b+c+d)=c(a+b+d)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,32,33,34,35,36,37,38,39,$ and $40.$ What is the value of $\\frac{N}{M}?$\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2 \\qquad\\textbf{(C)}\\ 37 \\qquad\\textbf{(D)}\\ 74 \\qquad\\textbf{(E)}\\ 2886$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_766", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,32,33,34,35,36,37,38,39,$ and $40.$ What is the value of $\\frac{N}{M}?$\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2 \\qquad\\textbf{(C)}\\ 37 \\qquad\\textbf{(D)}\\ 74 \\qquad\\textbf{(E)}\\ 2886$" + } + }, + { + "question": "Return your final response within \\boxed{}. The equation $x + \\sqrt{x-2} = 4$ has:\n$\\textbf{(A)}\\ 2\\text{ real roots }\\qquad\\textbf{(B)}\\ 1\\text{ real and}\\ 1\\text{ imaginary root}\\qquad\\textbf{(C)}\\ 2\\text{ imaginary roots}\\qquad\\textbf{(D)}\\ \\text{ no roots}\\qquad\\textbf{(E)}\\ 1\\text{ real root}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_767", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The equation $x + \\sqrt{x-2} = 4$ has:\n$\\textbf{(A)}\\ 2\\text{ real roots }\\qquad\\textbf{(B)}\\ 1\\text{ real and}\\ 1\\text{ imaginary root}\\qquad\\textbf{(C)}\\ 2\\text{ imaginary roots}\\qquad\\textbf{(D)}\\ \\text{ no roots}\\qquad\\textbf{(E)}\\ 1\\text{ real root}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB\\perp AC$, $AF\\perp BC$, and $BD=DC=FC=1$. Find $AC$.\n$\\mathrm{(A) \\ }\\sqrt{2} \\qquad \\mathrm{(B) \\ }\\sqrt{3} \\qquad \\mathrm{(C) \\ } \\sqrt[3]{2} \\qquad \\mathrm{(D) \\ }\\sqrt[3]{3} \\qquad \\mathrm{(E) \\ } \\sqrt[4]{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_768", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB\\perp AC$, $AF\\perp BC$, and $BD=DC=FC=1$. Find $AC$.\n$\\mathrm{(A) \\ }\\sqrt{2} \\qquad \\mathrm{(B) \\ }\\sqrt{3} \\qquad \\mathrm{(C) \\ } \\sqrt[3]{2} \\qquad \\mathrm{(D) \\ }\\sqrt[3]{3} \\qquad \\mathrm{(E) \\ } \\sqrt[4]{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is\n$\\textbf{(A)}\\ 333 \\qquad\\textbf{(B)}\\ 335 \\qquad\\textbf{(C)}\\ 337 \\qquad\\textbf{(D)}\\ 339 \\qquad\\textbf{(E)}\\ 341$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_769", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is\n$\\textbf{(A)}\\ 333 \\qquad\\textbf{(B)}\\ 335 \\qquad\\textbf{(C)}\\ 337 \\qquad\\textbf{(D)}\\ 339 \\qquad\\textbf{(E)}\\ 341$" + } + }, + { + "question": "Return your final response within \\boxed{}. A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at the point $5$ inches from the center $M$ is:\n$\\text{(A) } 1\\quad \\text{(B) } 15\\quad \\text{(C) } 15\\tfrac{1}{3}\\quad \\text{(D) } 15\\tfrac{1}{2}\\quad \\text{(E) } 15\\tfrac{3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_770", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at the point $5$ inches from the center $M$ is:\n$\\text{(A) } 1\\quad \\text{(B) } 15\\quad \\text{(C) } 15\\tfrac{1}{3}\\quad \\text{(D) } 15\\tfrac{1}{2}\\quad \\text{(E) } 15\\tfrac{3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The value of $x$ at the intersection of $y=\\frac{8}{x^2+4}$ and $x+y=2$ is: \n$\\textbf{(A)}\\ -2+\\sqrt{5} \\qquad \\textbf{(B)}\\ -2-\\sqrt{5} \\qquad \\textbf{(C)}\\ 0 \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_771", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The value of $x$ at the intersection of $y=\\frac{8}{x^2+4}$ and $x+y=2$ is: \n$\\textbf{(A)}\\ -2+\\sqrt{5} \\qquad \\textbf{(B)}\\ -2-\\sqrt{5} \\qquad \\textbf{(C)}\\ 0 \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?\n$\\textbf{(A) }2\\text{ and }19\\qquad\\textbf{(B) }20\\text{ and }39\\qquad\\textbf{(C) }40\\text{ and }59\\qquad\\textbf{(D) }60\\text{ and }79\\qquad\\textbf{(E) }80\\text{ and }124$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_772", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?\n$\\textbf{(A) }2\\text{ and }19\\qquad\\textbf{(B) }20\\text{ and }39\\qquad\\textbf{(C) }40\\text{ and }59\\qquad\\textbf{(D) }60\\text{ and }79\\qquad\\textbf{(E) }80\\text{ and }124$" + } + }, + { + "question": "Return your final response within \\boxed{}. On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \\ge 0$ lines. How many different values of $k$ are possible?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 4 \\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_773", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \\ge 0$ lines. How many different values of $k$ are possible?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 4 \\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $b$ men take $c$ days to lay $f$ bricks, then the number of days it will take $c$ men working at the same rate to lay $b$ bricks, is\n$\\textbf{(A) }fb^2\\qquad \\textbf{(B) }b/f^2\\qquad \\textbf{(C) }f^2/b\\qquad \\textbf{(D) }b^2/f\\qquad \\textbf{(E) }f/b^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_774", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $b$ men take $c$ days to lay $f$ bricks, then the number of days it will take $c$ men working at the same rate to lay $b$ bricks, is\n$\\textbf{(A) }fb^2\\qquad \\textbf{(B) }b/f^2\\qquad \\textbf{(C) }f^2/b\\qquad \\textbf{(D) }b^2/f\\qquad \\textbf{(E) }f/b^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Gilda has a bag of marbles. She gives $20\\%$ of them to her friend Pedro. Then Gilda gives $10\\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?\n$\\textbf{(A) }20\\qquad\\textbf{(B) }33\\frac{1}{3}\\qquad\\textbf{(C) }38\\qquad\\textbf{(D) }45\\qquad\\textbf{(E) }54$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_775", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Gilda has a bag of marbles. She gives $20\\%$ of them to her friend Pedro. Then Gilda gives $10\\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?\n$\\textbf{(A) }20\\qquad\\textbf{(B) }33\\frac{1}{3}\\qquad\\textbf{(C) }38\\qquad\\textbf{(D) }45\\qquad\\textbf{(E) }54$" + } + }, + { + "question": "Return your final response within \\boxed{}. Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?\n$\\textbf{(A)} ~24 \\qquad\\textbf{(B)} ~30 \\qquad\\textbf{(C)} ~48 \\qquad\\textbf{(D)} ~60 \\qquad\\textbf{(E)} ~64$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_776", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?\n$\\textbf{(A)} ~24 \\qquad\\textbf{(B)} ~30 \\qquad\\textbf{(C)} ~48 \\qquad\\textbf{(D)} ~60 \\qquad\\textbf{(E)} ~64$" + } + }, + { + "question": "Return your final response within \\boxed{}. Karl's rectangular vegetable garden is $20$ feet by $45$ feet, and Makenna's is $25$ feet by $40$ feet. Whose garden is larger in area?\n$\\textbf{(A)}\\ \\text{Karl's garden is larger by 100 square feet.}$\n$\\textbf{(B)}\\ \\text{Karl's garden is larger by 25 square feet.}$\n$\\textbf{(C)}\\ \\text{The gardens are the same size.}$\n$\\textbf{(D)}\\ \\text{Makenna's garden is larger by 25 square feet.}$\n$\\textbf{(E)}\\ \\text{Makenna's garden is larger by 100 square feet.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_777", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Karl's rectangular vegetable garden is $20$ feet by $45$ feet, and Makenna's is $25$ feet by $40$ feet. Whose garden is larger in area?\n$\\textbf{(A)}\\ \\text{Karl's garden is larger by 100 square feet.}$\n$\\textbf{(B)}\\ \\text{Karl's garden is larger by 25 square feet.}$\n$\\textbf{(C)}\\ \\text{The gardens are the same size.}$\n$\\textbf{(D)}\\ \\text{Makenna's garden is larger by 25 square feet.}$\n$\\textbf{(E)}\\ \\text{Makenna's garden is larger by 100 square feet.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is\n\n$\\text{(A)}\\ 16 \\qquad \\text{(B)}\\ 24 \\qquad \\text{(C)}\\ 32 \\qquad \\text{(D)}\\ 64 \\qquad \\text{(E)}\\ 128$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_778", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is\n\n$\\text{(A)}\\ 16 \\qquad \\text{(B)}\\ 24 \\qquad \\text{(C)}\\ 32 \\qquad \\text{(D)}\\ 64 \\qquad \\text{(E)}\\ 128$" + } + }, + { + "question": "Return your final response within \\boxed{}. Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\\%$ of the group are girls. How many girls were initially in the group?\n$\\textbf{(A) } 4 \\qquad\\textbf{(B) } 6 \\qquad\\textbf{(C) } 8 \\qquad\\textbf{(D) } 10 \\qquad\\textbf{(E) } 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_779", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\\%$ of the group are girls. How many girls were initially in the group?\n$\\textbf{(A) } 4 \\qquad\\textbf{(B) } 6 \\qquad\\textbf{(C) } 8 \\qquad\\textbf{(D) } 10 \\qquad\\textbf{(E) } 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. At Olympic High School, $\\frac{2}{5}$ of the freshmen and $\\frac{4}{5}$ of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true? \n$\\textbf{(A)}$ There are five times as many sophomores as freshmen. \n$\\textbf{(B)}$ There are twice as many sophomores as freshmen.\n$\\textbf{(C)}$ There are as many freshmen as sophomores.\n$\\textbf{(D)}$ There are twice as many freshmen as sophomores. \n$\\textbf{(E)}$ There are five times as many freshmen as sophomores.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_780", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. At Olympic High School, $\\frac{2}{5}$ of the freshmen and $\\frac{4}{5}$ of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true? \n$\\textbf{(A)}$ There are five times as many sophomores as freshmen. \n$\\textbf{(B)}$ There are twice as many sophomores as freshmen.\n$\\textbf{(C)}$ There are as many freshmen as sophomores.\n$\\textbf{(D)}$ There are twice as many freshmen as sophomores. \n$\\textbf{(E)}$ There are five times as many freshmen as sophomores." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$. What is $a + b + c + d$?\n$\\text{(A)}\\ -5 \\qquad \\text{(B)}\\ -10/3 \\qquad \\text{(C)}\\ -7/3 \\qquad \\text{(D)}\\ 5/3 \\qquad \\text{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_781", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$. What is $a + b + c + d$?\n$\\text{(A)}\\ -5 \\qquad \\text{(B)}\\ -10/3 \\qquad \\text{(C)}\\ -7/3 \\qquad \\text{(D)}\\ 5/3 \\qquad \\text{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\\frac{1}{2}$ inches. Then $QR$, in inches, is:\n$\\textbf{(A) }6\\qquad\\textbf{(B) }7\\qquad\\textbf{(C) }8\\qquad\\textbf{(D) }9\\qquad \\textbf{(E) }10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_782", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\\frac{1}{2}$ inches. Then $QR$, in inches, is:\n$\\textbf{(A) }6\\qquad\\textbf{(B) }7\\qquad\\textbf{(C) }8\\qquad\\textbf{(D) }9\\qquad \\textbf{(E) }10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let ($a_1$, $a_2$, ... $a_{10}$) be a list of the first 10 positive integers such that for each $2\\le$ $i$ $\\le10$ either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?\n\n$\\textbf{(A)}\\ \\ 120\\qquad\\textbf{(B)}\\ 512\\qquad\\textbf{(C)}\\ \\ 1024\\qquad\\textbf{(D)}\\ 181,440\\qquad\\textbf{(E)}\\ \\ 362,880$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_784", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let ($a_1$, $a_2$, ... $a_{10}$) be a list of the first 10 positive integers such that for each $2\\le$ $i$ $\\le10$ either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?\n\n$\\textbf{(A)}\\ \\ 120\\qquad\\textbf{(B)}\\ 512\\qquad\\textbf{(C)}\\ \\ 1024\\qquad\\textbf{(D)}\\ 181,440\\qquad\\textbf{(E)}\\ \\ 362,880$" + } + }, + { + "question": "Return your final response within \\boxed{}. Side $\\overline{AB}$ of $\\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$?\n$\\textbf{(A) }16 \\qquad \\textbf{(B) }17 \\qquad \\textbf{(C) }18 \\qquad \\textbf{(D) }19 \\qquad \\textbf{(E) }20 \\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_785", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Side $\\overline{AB}$ of $\\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$?\n$\\textbf{(A) }16 \\qquad \\textbf{(B) }17 \\qquad \\textbf{(C) }18 \\qquad \\textbf{(D) }19 \\qquad \\textbf{(E) }20 \\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket?\n$\\text{(A)}\\ 4\\% \\qquad \\text{(B)}\\ 25\\% \\qquad \\text{(C)}\\ 40\\% \\qquad \\text{(D)}\\ 41\\% \\qquad \\text{(E)}\\ 59\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_786", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket?\n$\\text{(A)}\\ 4\\% \\qquad \\text{(B)}\\ 25\\% \\qquad \\text{(C)}\\ 40\\% \\qquad \\text{(D)}\\ 41\\% \\qquad \\text{(E)}\\ 59\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9 1$,\n\\[\\frac{a_0+\\cdots+a_n}{n+1}\\cdot\\frac{a_1+\\cdots+a_{n-1}}{n-1}\\ge\\frac{a_0+\\cdots+a_{n-1}}{n}\\cdot\\frac{a_1+\\cdots+a_{n}}{n}.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The inequality holds as required.", + "index": "Sky-T1_10k_800", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a_0, a_1, a_2,\\cdots$ be a sequence of positive real numbers satisfying $a_{i-1}a_{i+1}\\le a^2_i$\nfor $i = 1, 2, 3,\\cdots$ . (Such a sequence is said to be log concave.) Show that for\neach $n > 1$,\n\\[\\frac{a_0+\\cdots+a_n}{n+1}\\cdot\\frac{a_1+\\cdots+a_{n-1}}{n-1}\\ge\\frac{a_0+\\cdots+a_{n-1}}{n}\\cdot\\frac{a_1+\\cdots+a_{n}}{n}.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$. Moreover $\\overline{BX}$ and $\\overline{CX}$ have integer lengths. What is $BC$?\n$\\textbf{(A)}\\ 11\\qquad\\textbf{(B)}\\ 28\\qquad\\textbf{(C)}\\ 33\\qquad\\textbf{(D)}\\ 61\\qquad\\textbf{(E)}\\ 72$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_801", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$. Moreover $\\overline{BX}$ and $\\overline{CX}$ have integer lengths. What is $BC$?\n$\\textbf{(A)}\\ 11\\qquad\\textbf{(B)}\\ 28\\qquad\\textbf{(C)}\\ 33\\qquad\\textbf{(D)}\\ 61\\qquad\\textbf{(E)}\\ 72$" + } + }, + { + "question": "Return your final response within \\boxed{}. For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$?\n$\\textbf{(A) }9 \\qquad \\textbf{(B) } 14 \\qquad \\textbf{(C) } 16 \\qquad \\textbf{(D) } 17 \\qquad \\textbf{(E) } 19$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_802", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$?\n$\\textbf{(A) }9 \\qquad \\textbf{(B) } 14 \\qquad \\textbf{(C) } 16 \\qquad \\textbf{(D) } 17 \\qquad \\textbf{(E) } 19$" + } + }, + { + "question": "Return your final response within \\boxed{}. The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third quartile $Q_3 = 43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartle ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3 - Q_1$. How many outliers does this data set have?\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_803", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third quartile $Q_3 = 43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartle ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3 - Q_1$. How many outliers does this data set have?\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. For the positive integer $n$, let $\\langle n\\rangle$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\\langle 4\\rangle=1+2=3$ and $\\langle 12 \\rangle =1+2+3+4+6=16$. What is $\\langle\\langle\\langle 6\\rangle\\rangle\\rangle$?\n$\\mathrm{(A)}\\ 6\\qquad\\mathrm{(B)}\\ 12\\qquad\\mathrm{(C)}\\ 24\\qquad\\mathrm{(D)}\\ 32\\qquad\\mathrm{(E)}\\ 36$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_804", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For the positive integer $n$, let $\\langle n\\rangle$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\\langle 4\\rangle=1+2=3$ and $\\langle 12 \\rangle =1+2+3+4+6=16$. What is $\\langle\\langle\\langle 6\\rangle\\rangle\\rangle$?\n$\\mathrm{(A)}\\ 6\\qquad\\mathrm{(B)}\\ 12\\qquad\\mathrm{(C)}\\ 24\\qquad\\mathrm{(D)}\\ 32\\qquad\\mathrm{(E)}\\ 36$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many non-congruent right triangles are there such that the perimeter in $\\text{cm}$ and the area in $\\text{cm}^2$ are numerically equal? \n$\\textbf{(A)} \\ \\text{none} \\qquad \\textbf{(B)} \\ 1 \\qquad \\textbf{(C)} \\ 2 \\qquad \\textbf{(D)} \\ 4 \\qquad \\textbf{(E)} \\ \\text{infinitely many}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_805", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many non-congruent right triangles are there such that the perimeter in $\\text{cm}$ and the area in $\\text{cm}^2$ are numerically equal? \n$\\textbf{(A)} \\ \\text{none} \\qquad \\textbf{(B)} \\ 1 \\qquad \\textbf{(C)} \\ 2 \\qquad \\textbf{(D)} \\ 4 \\qquad \\textbf{(E)} \\ \\text{infinitely many}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The limit of $\\frac {x^2-1}{x-1}$ as $x$ approaches $1$ as a limit is:\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ \\text{Indeterminate} \\qquad \\textbf{(C)}\\ x-1 \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_806", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The limit of $\\frac {x^2-1}{x-1}$ as $x$ approaches $1$ as a limit is:\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ \\text{Indeterminate} \\qquad \\textbf{(C)}\\ x-1 \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\\cdot MD = MC^2$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "MF = MC if and only if MB \\cdot MD = MC^2", + "index": "Sky-T1_10k_807", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\\cdot MD = MC^2$." + } + }, + { + "question": "Return your final response within \\boxed{}. How many different four-digit numbers can be formed by rearranging the four digits in $2004$?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 24\\qquad\\textbf{(E)}\\ 81$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_808", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many different four-digit numbers can be formed by rearranging the four digits in $2004$?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 24\\qquad\\textbf{(E)}\\ 81$" + } + }, + { + "question": "Return your final response within \\boxed{}. Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\\tfrac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?\n$\\textbf{(A)}\\; \\dfrac{1}{2} \\qquad\\textbf{(B)}\\; \\dfrac{3}{5} \\qquad\\textbf{(C)}\\; \\dfrac{2}{3} \\qquad\\textbf{(D)}\\; \\dfrac{3}{4} \\qquad\\textbf{(E)}\\; \\dfrac{4}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_809", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\\tfrac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?\n$\\textbf{(A)}\\; \\dfrac{1}{2} \\qquad\\textbf{(B)}\\; \\dfrac{3}{5} \\qquad\\textbf{(C)}\\; \\dfrac{2}{3} \\qquad\\textbf{(D)}\\; \\dfrac{3}{4} \\qquad\\textbf{(E)}\\; \\dfrac{4}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Gabriel Carroll) Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that\n\\[\\left|x\\right| + \\left|y + \\frac {1}{2}\\right| < n\\]\nA path is a sequence of distinct points $(x_1 , y_1 ), (x_2 , y_2 ), \\ldots , (x_\\ell, y_\\ell)$ in $S_n$ such that, for $i = 2, \\ldots , \\ell$, the distance between $(x_i , y_i )$ and $(x_{i - 1} , y_{i - 1} )$ is $1$ (in other words, the points $(x_i , y_i )$ and $(x_{i - 1} , y_{i - 1} )$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\\mathcal{P}$).", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "n", + "index": "Sky-T1_10k_810", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Gabriel Carroll) Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that\n\\[\\left|x\\right| + \\left|y + \\frac {1}{2}\\right| < n\\]\nA path is a sequence of distinct points $(x_1 , y_1 ), (x_2 , y_2 ), \\ldots , (x_\\ell, y_\\ell)$ in $S_n$ such that, for $i = 2, \\ldots , \\ell$, the distance between $(x_i , y_i )$ and $(x_{i - 1} , y_{i - 1} )$ is $1$ (in other words, the points $(x_i , y_i )$ and $(x_{i - 1} , y_{i - 1} )$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\\mathcal{P}$)." + } + }, + { + "question": "Return your final response within \\boxed{}. On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 13\\qquad\\textbf{(D)}\\ 19\\qquad\\textbf{(E)}\\ 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_811", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 13\\qquad\\textbf{(D)}\\ 19\\qquad\\textbf{(E)}\\ 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. Estimate the year in which the population of Nisos will be approximately 6,000.\n$\\text{(A)}\\ 2050 \\qquad \\text{(B)}\\ 2075 \\qquad \\text{(C)}\\ 2100 \\qquad \\text{(D)}\\ 2125 \\qquad \\text{(E)}\\ 2150$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_812", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Estimate the year in which the population of Nisos will be approximately 6,000.\n$\\text{(A)}\\ 2050 \\qquad \\text{(B)}\\ 2075 \\qquad \\text{(C)}\\ 2100 \\qquad \\text{(D)}\\ 2125 \\qquad \\text{(E)}\\ 2150$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let n be the number of real values of $p$ for which the roots of \n$x^2-px+p=0$\nare equal. Then n equals:\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ \\text{a finite number greater than 2}\\qquad \\textbf{(E)}\\ \\infty$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_813", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let n be the number of real values of $p$ for which the roots of \n$x^2-px+p=0$\nare equal. Then n equals:\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ \\text{a finite number greater than 2}\\qquad \\textbf{(E)}\\ \\infty$" + } + }, + { + "question": "Return your final response within \\boxed{}. A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?\n\n$\\textbf{(A)}\\ 2:3\\qquad\\textbf{(B)}\\ 3:2\\qquad\\textbf{(C)}\\ 6:\\pi\\qquad\\textbf{(D)}\\ 9:\\pi\\qquad\\textbf{(E)}\\ 30 :\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_814", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?\n\n$\\textbf{(A)}\\ 2:3\\qquad\\textbf{(B)}\\ 3:2\\qquad\\textbf{(C)}\\ 6:\\pi\\qquad\\textbf{(D)}\\ 9:\\pi\\qquad\\textbf{(E)}\\ 30 :\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that\n\nall of his happy snakes can add,\nnone of his purple snakes can subtract, and\nall of his snakes that can't subtract also can't add.\nWhich of these conclusions can be drawn about Tom's snakes?\n$\\textbf{(A) }$ Purple snakes can add.\n$\\textbf{(B) }$ Purple snakes are happy.\n$\\textbf{(C) }$ Snakes that can add are purple.\n$\\textbf{(D) }$ Happy snakes are not purple.\n$\\textbf{(E) }$ Happy snakes can't subtract.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_815", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that\n\nall of his happy snakes can add,\nnone of his purple snakes can subtract, and\nall of his snakes that can't subtract also can't add.\nWhich of these conclusions can be drawn about Tom's snakes?\n$\\textbf{(A) }$ Purple snakes can add.\n$\\textbf{(B) }$ Purple snakes are happy.\n$\\textbf{(C) }$ Snakes that can add are purple.\n$\\textbf{(D) }$ Happy snakes are not purple.\n$\\textbf{(E) }$ Happy snakes can't subtract." + } + }, + { + "question": "Return your final response within \\boxed{}. The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\n$\\textbf{(A)}\\; 20 \\qquad\\textbf{(B)}\\; \\dfrac{360}{17} \\qquad\\textbf{(C)}\\; \\dfrac{107}{5} \\qquad\\textbf{(D)}\\; \\dfrac{43}{2} \\qquad\\textbf{(E)}\\; \\dfrac{281}{13}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_816", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\n$\\textbf{(A)}\\; 20 \\qquad\\textbf{(B)}\\; \\dfrac{360}{17} \\qquad\\textbf{(C)}\\; \\dfrac{107}{5} \\qquad\\textbf{(D)}\\; \\dfrac{43}{2} \\qquad\\textbf{(E)}\\; \\dfrac{281}{13}$" + } + }, + { + "question": "Return your final response within \\boxed{}. An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?\n$\\textbf{(A)}\\ \\frac{31}{16}\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\frac{17}{8}\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ \\frac{65}{16}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_817", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?\n$\\textbf{(A)}\\ \\frac{31}{16}\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\frac{17}{8}\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ \\frac{65}{16}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $5y$ varies inversely as the square of $x$. When $y=16, x=1$. When $x=8, y$ equals: \n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 128 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ \\frac{1}{4} \\qquad \\textbf{(E)}\\ 1024$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_818", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $5y$ varies inversely as the square of $x$. When $y=16, x=1$. When $x=8, y$ equals: \n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 128 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ \\frac{1}{4} \\qquad \\textbf{(E)}\\ 1024$" + } + }, + { + "question": "Return your final response within \\boxed{}. A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $$17.71$. What was the cost of a pencil in cents?\n$\\textbf{(A)}\\ 7 \\qquad \\textbf{(B)}\\ 11 \\qquad \\textbf{(C)}\\ 17 \\qquad \\textbf{(D)}\\ 23 \\qquad \\textbf{(E)}\\ 77$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_819", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $$17.71$. What was the cost of a pencil in cents?\n$\\textbf{(A)}\\ 7 \\qquad \\textbf{(B)}\\ 11 \\qquad \\textbf{(C)}\\ 17 \\qquad \\textbf{(D)}\\ 23 \\qquad \\textbf{(E)}\\ 77$" + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 56\\qquad\\textbf{(C)}\\ 64\\qquad\\textbf{(D)}\\ 96\\qquad\\textbf{(E)}\\ 144$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_820", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?\n$\\textbf{(A)}\\ 48\\qquad\\textbf{(B)}\\ 56\\qquad\\textbf{(C)}\\ 64\\qquad\\textbf{(D)}\\ 96\\qquad\\textbf{(E)}\\ 144$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the sum $1 + 2 + 3 + \\cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are: \n$\\textbf{(A)}\\ \\text{only }1\\qquad \\textbf{(B)}\\ 1\\text{ and }8\\qquad \\textbf{(C)}\\ \\text{only }8\\qquad \\textbf{(D)}\\ 8\\text{ and }49\\qquad \\textbf{(E)}\\ 1,8,\\text{ and }49$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_821", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the sum $1 + 2 + 3 + \\cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are: \n$\\textbf{(A)}\\ \\text{only }1\\qquad \\textbf{(B)}\\ 1\\text{ and }8\\qquad \\textbf{(C)}\\ \\text{only }8\\qquad \\textbf{(D)}\\ 8\\text{ and }49\\qquad \\textbf{(E)}\\ 1,8,\\text{ and }49$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $y=f(x)=\\frac{x+2}{x-1}$, then it is incorrect to say:\n$\\textbf{(A)\\ }x=\\frac{y+2}{y-1}\\qquad\\textbf{(B)\\ }f(0)=-2\\qquad\\textbf{(C)\\ }f(1)=0\\qquad$\n$\\textbf{(D)\\ }f(-2)=0\\qquad\\textbf{(E)\\ }f(y)=x$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_822", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $y=f(x)=\\frac{x+2}{x-1}$, then it is incorrect to say:\n$\\textbf{(A)\\ }x=\\frac{y+2}{y-1}\\qquad\\textbf{(B)\\ }f(0)=-2\\qquad\\textbf{(C)\\ }f(1)=0\\qquad$\n$\\textbf{(D)\\ }f(-2)=0\\qquad\\textbf{(E)\\ }f(y)=x$" + } + }, + { + "question": "Return your final response within \\boxed{}. A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "803", + "index": "Sky-T1_10k_823", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. The real roots of $x^2+4$ are:\n$\\textbf{(A)}\\ (x^{2}+2)(x^{2}+2)\\qquad\\textbf{(B)}\\ (x^{2}+2)(x^{2}-2)\\qquad\\textbf{(C)}\\ x^{2}(x^{2}+4)\\qquad\\\\ \\textbf{(D)}\\ (x^{2}-2x+2)(x^{2}+2x+2)\\qquad\\textbf{(E)}\\ \\text{Non-existent}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_824", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The real roots of $x^2+4$ are:\n$\\textbf{(A)}\\ (x^{2}+2)(x^{2}+2)\\qquad\\textbf{(B)}\\ (x^{2}+2)(x^{2}-2)\\qquad\\textbf{(C)}\\ x^{2}(x^{2}+4)\\qquad\\\\ \\textbf{(D)}\\ (x^{2}-2x+2)(x^{2}+2x+2)\\qquad\\textbf{(E)}\\ \\text{Non-existent}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $N = 34 \\cdot 34 \\cdot 63 \\cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?\n$\\textbf{(A)} ~1 : 16 \\qquad\\textbf{(B)} ~1 : 15 \\qquad\\textbf{(C)} ~1 : 14 \\qquad\\textbf{(D)} ~1 : 8 \\qquad\\textbf{(E)} ~1 : 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_825", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $N = 34 \\cdot 34 \\cdot 63 \\cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?\n$\\textbf{(A)} ~1 : 16 \\qquad\\textbf{(B)} ~1 : 15 \\qquad\\textbf{(C)} ~1 : 14 \\qquad\\textbf{(D)} ~1 : 8 \\qquad\\textbf{(E)} ~1 : 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \\cdots + c_n$ be a [polynomial](https://artofproblemsolving.com/wiki/index.php/Polynomial) in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "(a^2 + b^2 + 1)^2 < 4 b^2 + 1", + "index": "Sky-T1_10k_826", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \\cdots + c_n$ be a [polynomial](https://artofproblemsolving.com/wiki/index.php/Polynomial) in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$." + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ varies as the cube of $y$, and $y$ varies as the fifth root of $z$, then $x$ varies as the nth power of $z$, where n is:\n$\\textbf{(A)}\\ \\frac{1}{15} \\qquad\\textbf{(B)}\\ \\frac{5}{3} \\qquad\\textbf{(C)}\\ \\frac{3}{5} \\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_827", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ varies as the cube of $y$, and $y$ varies as the fifth root of $z$, then $x$ varies as the nth power of $z$, where n is:\n$\\textbf{(A)}\\ \\frac{1}{15} \\qquad\\textbf{(B)}\\ \\frac{5}{3} \\qquad\\textbf{(C)}\\ \\frac{3}{5} \\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 30\\qquad\\textbf{(C)}\\ 40\\qquad\\textbf{(D)}\\ 60\\qquad\\textbf{(E)}\\ 70$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_828", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 30\\qquad\\textbf{(C)}\\ 40\\qquad\\textbf{(D)}\\ 60\\qquad\\textbf{(E)}\\ 70$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the value of $20$ quarters and $10$ dimes equals the value of $10$ quarters and $n$ dimes, then $n=$\n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 20 \\qquad \\text{(C)}\\ 30 \\qquad \\text{(D)}\\ 35 \\qquad \\text{(E)}\\ 45$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_829", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the value of $20$ quarters and $10$ dimes equals the value of $10$ quarters and $n$ dimes, then $n=$\n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 20 \\qquad \\text{(C)}\\ 30 \\qquad \\text{(D)}\\ 35 \\qquad \\text{(E)}\\ 45$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many different integers can be expressed as the sum of three distinct members of the set $\\{1,4,7,10,13,16,19\\}$?\n$\\text{(A)}\\ 13 \\qquad \\text{(B)}\\ 16 \\qquad \\text{(C)}\\ 24 \\qquad \\text{(D)}\\ 30 \\qquad \\text{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_830", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many different integers can be expressed as the sum of three distinct members of the set $\\{1,4,7,10,13,16,19\\}$?\n$\\text{(A)}\\ 13 \\qquad \\text{(B)}\\ 16 \\qquad \\text{(C)}\\ 24 \\qquad \\text{(D)}\\ 30 \\qquad \\text{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the progression $10^{\\dfrac{1}{11}}, 10^{\\dfrac{2}{11}}, 10^{\\dfrac{3}{11}}, 10^{\\dfrac{4}{11}},\\dots , 10^{\\dfrac{n}{11}}$. \nThe least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$ is\n$\\textbf{(A) }7\\qquad \\textbf{(B) }8\\qquad \\textbf{(C) }9\\qquad \\textbf{(D) }10\\qquad \\textbf{(E) }11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_831", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the progression $10^{\\dfrac{1}{11}}, 10^{\\dfrac{2}{11}}, 10^{\\dfrac{3}{11}}, 10^{\\dfrac{4}{11}},\\dots , 10^{\\dfrac{n}{11}}$. \nThe least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$ is\n$\\textbf{(A) }7\\qquad \\textbf{(B) }8\\qquad \\textbf{(C) }9\\qquad \\textbf{(D) }10\\qquad \\textbf{(E) }11$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$ is:\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B) }\\ 1 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 3 \\qquad \\textbf{(E) }\\ \\text{more than 4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_832", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$ is:\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B) }\\ 1 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 3 \\qquad \\textbf{(E) }\\ \\text{more than 4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of triples $(a, b, c)$ of [positive integers](https://artofproblemsolving.com/wiki/index.php/Natural_numbers) which satisfy the simultaneous [equations](https://artofproblemsolving.com/wiki/index.php/Equation)\n$ab+bc=44$\n$ac+bc=23$\nis\n$\\mathrm{(A) \\ }0 \\qquad \\mathrm{(B) \\ }1 \\qquad \\mathrm{(C) \\ } 2 \\qquad \\mathrm{(D) \\ }3 \\qquad \\mathrm{(E) \\ } 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_833", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of triples $(a, b, c)$ of [positive integers](https://artofproblemsolving.com/wiki/index.php/Natural_numbers) which satisfy the simultaneous [equations](https://artofproblemsolving.com/wiki/index.php/Equation)\n$ab+bc=44$\n$ac+bc=23$\nis\n$\\mathrm{(A) \\ }0 \\qquad \\mathrm{(B) \\ }1 \\qquad \\mathrm{(C) \\ } 2 \\qquad \\mathrm{(D) \\ }3 \\qquad \\mathrm{(E) \\ } 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?\n$\\textbf{(A) } 1\\qquad\\textbf{(B) } 2\\qquad\\textbf{(C) } 3\\qquad\\textbf{(D) } 4\\qquad\\textbf{(E) } 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_834", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?\n$\\textbf{(A) } 1\\qquad\\textbf{(B) } 2\\qquad\\textbf{(C) } 3\\qquad\\textbf{(D) } 4\\qquad\\textbf{(E) } 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear.\nA legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "ab is even", + "index": "Sky-T1_10k_835", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear.\nA legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even." + } + }, + { + "question": "Return your final response within \\boxed{}. The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.\n$\\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\\\ + & 8 & 2 & 9 & 4 & 3 & 0 \\\\ \\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \\end{tabular}$\n$\\mathrm{(A) \\ 4 } \\qquad \\mathrm{(B) \\ 6 } \\qquad \\mathrm{(C) \\ 8 } \\qquad \\mathrm{(D) \\ 10 } \\qquad \\mathrm{(E) \\ \\text{more than 10} }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_836", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.\n$\\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\\\ + & 8 & 2 & 9 & 4 & 3 & 0 \\\\ \\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \\end{tabular}$\n$\\mathrm{(A) \\ 4 } \\qquad \\mathrm{(B) \\ 6 } \\qquad \\mathrm{(C) \\ 8 } \\qquad \\mathrm{(D) \\ 10 } \\qquad \\mathrm{(E) \\ \\text{more than 10} }$" + } + }, + { + "question": "Return your final response within \\boxed{}. Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?\n$(\\textbf{A})\\: 1.6\\qquad(\\textbf{B}) \\: 1.8\\qquad(\\textbf{C}) \\: 2.0\\qquad(\\textbf{D}) \\: 2.2\\qquad(\\textbf{E}) \\: 2.4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_837", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?\n$(\\textbf{A})\\: 1.6\\qquad(\\textbf{B}) \\: 1.8\\qquad(\\textbf{C}) \\: 2.0\\qquad(\\textbf{D}) \\: 2.2\\qquad(\\textbf{E}) \\: 2.4$" + } + }, + { + "question": "Return your final response within \\boxed{}. An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?\n$\\textbf{(A) } \\frac16 \\qquad \\textbf{(B) }\\frac15 \\qquad \\textbf{(C) } \\frac14 \\qquad \\textbf{(D) } \\frac13 \\qquad \\textbf{(E) } \\frac12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_838", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?\n$\\textbf{(A) } \\frac16 \\qquad \\textbf{(B) }\\frac15 \\qquad \\textbf{(C) } \\frac14 \\qquad \\textbf{(D) } \\frac13 \\qquad \\textbf{(E) } \\frac12$" + } + }, + { + "question": "Return your final response within \\boxed{}. The remainder can be defined for all real numbers $x$ and $y$ with $y \\neq 0$ by \\[\\text{rem} (x ,y)=x-y\\left \\lfloor \\frac{x}{y} \\right \\rfloor\\]where $\\left \\lfloor \\tfrac{x}{y} \\right \\rfloor$ denotes the greatest integer less than or equal to $\\tfrac{x}{y}$. What is the value of $\\text{rem} (\\tfrac{3}{8}, -\\tfrac{2}{5} )$?\n$\\textbf{(A) } -\\frac{3}{8} \\qquad \\textbf{(B) } -\\frac{1}{40} \\qquad \\textbf{(C) } 0 \\qquad \\textbf{(D) } \\frac{3}{8} \\qquad \\textbf{(E) } \\frac{31}{40}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_839", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The remainder can be defined for all real numbers $x$ and $y$ with $y \\neq 0$ by \\[\\text{rem} (x ,y)=x-y\\left \\lfloor \\frac{x}{y} \\right \\rfloor\\]where $\\left \\lfloor \\tfrac{x}{y} \\right \\rfloor$ denotes the greatest integer less than or equal to $\\tfrac{x}{y}$. What is the value of $\\text{rem} (\\tfrac{3}{8}, -\\tfrac{2}{5} )$?\n$\\textbf{(A) } -\\frac{3}{8} \\qquad \\textbf{(B) } -\\frac{1}{40} \\qquad \\textbf{(C) } 0 \\qquad \\textbf{(D) } \\frac{3}{8} \\qquad \\textbf{(E) } \\frac{31}{40}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? \n\n$\\textbf{(A) }70 \\qquad \\textbf{(B) }84 \\qquad \\textbf{(C) }100 \\qquad \\textbf{(D) }106 \\qquad \\textbf{(E) }120$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_840", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? \n\n$\\textbf{(A) }70 \\qquad \\textbf{(B) }84 \\qquad \\textbf{(C) }100 \\qquad \\textbf{(D) }106 \\qquad \\textbf{(E) }120$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \\cdots + n?$\n$\\textbf{(A) } 8 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 16 \\qquad \\textbf{(D) } 17 \\qquad \\textbf{(E) } 21$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_841", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \\cdots + n?$\n$\\textbf{(A) } 8 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 16 \\qquad \\textbf{(D) } 17 \\qquad \\textbf{(E) } 21$" + } + }, + { + "question": "Return your final response within \\boxed{}. Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?\n$\\text{(A)}\\ 90 \\qquad \\text{(B)}\\ 100 \\qquad \\text{(C)}\\ 125 \\qquad \\text{(D)}\\ 180 \\qquad \\text{(E)}\\ 250$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_842", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?\n$\\text{(A)}\\ 90 \\qquad \\text{(B)}\\ 100 \\qquad \\text{(C)}\\ 125 \\qquad \\text{(D)}\\ 180 \\qquad \\text{(E)}\\ 250$" + } + }, + { + "question": "Return your final response within \\boxed{}. Quadrilateral $ABCD$ is inscribed in a circle with $\\angle BAC=70^{\\circ}, \\angle ADB=40^{\\circ}, AD=4,$ and $BC=6$. What is $AC$?\n$\\textbf{(A)}\\; 3+\\sqrt{5} \\qquad\\textbf{(B)}\\; 6 \\qquad\\textbf{(C)}\\; \\dfrac{9}{2}\\sqrt{2} \\qquad\\textbf{(D)}\\; 8-\\sqrt{2} \\qquad\\textbf{(E)}\\; 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_843", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Quadrilateral $ABCD$ is inscribed in a circle with $\\angle BAC=70^{\\circ}, \\angle ADB=40^{\\circ}, AD=4,$ and $BC=6$. What is $AC$?\n$\\textbf{(A)}\\; 3+\\sqrt{5} \\qquad\\textbf{(B)}\\; 6 \\qquad\\textbf{(C)}\\; \\dfrac{9}{2}\\sqrt{2} \\qquad\\textbf{(D)}\\; 8-\\sqrt{2} \\qquad\\textbf{(E)}\\; 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean), \n$\\text{I. Averaging is associative }$\n$\\text{II. Averaging is commutative }$\n$\\text{III. Averaging distributes over addition }$\n$\\text{IV. Addition distributes over averaging }$\n$\\text{V. Averaging has an identity element }$\nthose which are always true are \n$\\textbf{(A)}\\ \\text{All}\\qquad\\textbf{(B)}\\ \\text{I and II only}\\qquad\\textbf{(C)}\\ \\text{II and III only}$\n$\\textbf{(D)}\\ \\text{II and IV only}\\qquad\\textbf{(E)}\\ \\text{II and V only}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_844", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean), \n$\\text{I. Averaging is associative }$\n$\\text{II. Averaging is commutative }$\n$\\text{III. Averaging distributes over addition }$\n$\\text{IV. Addition distributes over averaging }$\n$\\text{V. Averaging has an identity element }$\nthose which are always true are \n$\\textbf{(A)}\\ \\text{All}\\qquad\\textbf{(B)}\\ \\text{I and II only}\\qquad\\textbf{(C)}\\ \\text{II and III only}$\n$\\textbf{(D)}\\ \\text{II and IV only}\\qquad\\textbf{(E)}\\ \\text{II and V only}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?\n$\\textbf{(A)}\\ 17\\qquad\\textbf{(B)}\\ 19\\qquad\\textbf{(C)}\\ 21\\qquad\\textbf{(D)}\\ 22\\qquad\\textbf{(E)}\\ 23$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_845", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?\n$\\textbf{(A)}\\ 17\\qquad\\textbf{(B)}\\ 19\\qquad\\textbf{(C)}\\ 21\\qquad\\textbf{(D)}\\ 22\\qquad\\textbf{(E)}\\ 23$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\omega=-\\tfrac{1}{2}+\\tfrac{1}{2}i\\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\\omega+c\\omega^2,$ where $0\\leq a \\leq 1,0\\leq b\\leq 1,$ and $0\\leq c\\leq 1.$ What is the area of $S$?\n$\\textbf{(A) } \\frac{1}{2}\\sqrt3 \\qquad\\textbf{(B) } \\frac{3}{4}\\sqrt3 \\qquad\\textbf{(C) } \\frac{3}{2}\\sqrt3\\qquad\\textbf{(D) } \\frac{1}{2}\\pi\\sqrt3 \\qquad\\textbf{(E) } \\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_846", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\omega=-\\tfrac{1}{2}+\\tfrac{1}{2}i\\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\\omega+c\\omega^2,$ where $0\\leq a \\leq 1,0\\leq b\\leq 1,$ and $0\\leq c\\leq 1.$ What is the area of $S$?\n$\\textbf{(A) } \\frac{1}{2}\\sqrt3 \\qquad\\textbf{(B) } \\frac{3}{4}\\sqrt3 \\qquad\\textbf{(C) } \\frac{3}{2}\\sqrt3\\qquad\\textbf{(D) } \\frac{1}{2}\\pi\\sqrt3 \\qquad\\textbf{(E) } \\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal $97$ to does $D(n) = 2$?\n$\\textbf{(A)}\\ 16\\qquad\\textbf{(B)}\\ 20\\qquad\\textbf{(C)}\\ 26\\qquad\\textbf{(D)}\\ 30\\qquad\\textbf{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_847", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal $97$ to does $D(n) = 2$?\n$\\textbf{(A)}\\ 16\\qquad\\textbf{(B)}\\ 20\\qquad\\textbf{(C)}\\ 26\\qquad\\textbf{(D)}\\ 30\\qquad\\textbf{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\\circ$ from Alice's position and $60^\\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?\n$\\textbf{(A)}\\ 3.5 \\qquad\\textbf{(B)}\\ 4 \\qquad\\textbf{(C)}\\ 4.5 \\qquad\\textbf{(D)}\\ 5 \\qquad\\textbf{(E)}\\ 5.5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_848", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\\circ$ from Alice's position and $60^\\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?\n$\\textbf{(A)}\\ 3.5 \\qquad\\textbf{(B)}\\ 4 \\qquad\\textbf{(C)}\\ 4.5 \\qquad\\textbf{(D)}\\ 5 \\qquad\\textbf{(E)}\\ 5.5$" + } + }, + { + "question": "Return your final response within \\boxed{}. A deck of $n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $(n+1)/2$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\frac{n+1}{2}", + "index": "Sky-T1_10k_849", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A deck of $n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $(n+1)/2$." + } + }, + { + "question": "Return your final response within \\boxed{}. Circle $O$ has diameters $AB$ and $CD$ perpendicular to each other. $AM$ is any chord intersecting $CD$ at $P$. \nThen $AP\\cdot AM$ is equal to: \n\n$\\textbf{(A)}\\ AO\\cdot OB \\qquad \\textbf{(B)}\\ AO\\cdot AB\\qquad \\\\ \\textbf{(C)}\\ CP\\cdot CD \\qquad \\textbf{(D)}\\ CP\\cdot PD\\qquad \\textbf{(E)}\\ CO\\cdot OP$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_850", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Circle $O$ has diameters $AB$ and $CD$ perpendicular to each other. $AM$ is any chord intersecting $CD$ at $P$. \nThen $AP\\cdot AM$ is equal to: \n\n$\\textbf{(A)}\\ AO\\cdot OB \\qquad \\textbf{(B)}\\ AO\\cdot AB\\qquad \\\\ \\textbf{(C)}\\ CP\\cdot CD \\qquad \\textbf{(D)}\\ CP\\cdot PD\\qquad \\textbf{(E)}\\ CO\\cdot OP$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\\log_b a)^{2017}=\\log_b(a^{2017})?$\n$\\textbf{(A)}\\ 198\\qquad\\textbf{(B)}\\ 199\\qquad\\textbf{(C)}\\ 398\\qquad\\textbf{(D)}\\ 399\\qquad\\textbf{(E)}\\ 597$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_851", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\\log_b a)^{2017}=\\log_b(a^{2017})?$\n$\\textbf{(A)}\\ 198\\qquad\\textbf{(B)}\\ 199\\qquad\\textbf{(C)}\\ 398\\qquad\\textbf{(D)}\\ 399\\qquad\\textbf{(E)}\\ 597$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABC$ be a triangle where $M$ is the midpoint of $\\overline{AC}$, and $\\overline{CN}$ is the angle bisector of $\\angle{ACB}$ with $N$ on $\\overline{AB}$. Let $X$ be the intersection of the median $\\overline{BM}$ and the bisector $\\overline{CN}$. In addition $\\triangle BXN$ is equilateral with $AC=2$. What is $BX^2$?\n$\\textbf{(A)}\\ \\frac{10-6\\sqrt{2}}{7} \\qquad \\textbf{(B)}\\ \\frac{2}{9} \\qquad \\textbf{(C)}\\ \\frac{5\\sqrt{2}-3\\sqrt{3}}{8} \\qquad \\textbf{(D)}\\ \\frac{\\sqrt{2}}{6} \\qquad \\textbf{(E)}\\ \\frac{3\\sqrt{3}-4}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_852", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABC$ be a triangle where $M$ is the midpoint of $\\overline{AC}$, and $\\overline{CN}$ is the angle bisector of $\\angle{ACB}$ with $N$ on $\\overline{AB}$. Let $X$ be the intersection of the median $\\overline{BM}$ and the bisector $\\overline{CN}$. In addition $\\triangle BXN$ is equilateral with $AC=2$. What is $BX^2$?\n$\\textbf{(A)}\\ \\frac{10-6\\sqrt{2}}{7} \\qquad \\textbf{(B)}\\ \\frac{2}{9} \\qquad \\textbf{(C)}\\ \\frac{5\\sqrt{2}-3\\sqrt{3}}{8} \\qquad \\textbf{(D)}\\ \\frac{\\sqrt{2}}{6} \\qquad \\textbf{(E)}\\ \\frac{3\\sqrt{3}-4}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a_1, b_1, a_2, b_2, \\dots , a_n, b_n$ be nonnegative real numbers. Prove that\n\\[\\sum_{i, j = 1}^{n} \\min\\{a_ia_j, b_ib_j\\} \\le \\sum_{i, j = 1}^{n} \\min\\{a_ib_j, a_jb_i\\}.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\sum_{i, j = 1}^{n} \\min\\{a_ia_j, b_ib_j\\} \\le \\sum_{i, j = 1}^{n} \\min\\{a_ib_j, a_jb_i\\}", + "index": "Sky-T1_10k_853", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a_1, b_1, a_2, b_2, \\dots , a_n, b_n$ be nonnegative real numbers. Prove that\n\\[\\sum_{i, j = 1}^{n} \\min\\{a_ia_j, b_ib_j\\} \\le \\sum_{i, j = 1}^{n} \\min\\{a_ib_j, a_jb_i\\}.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. The region consisting of all points in three-dimensional space within $3$ units of line segment $\\overline{AB}$ has volume $216\\pi$. What is the length $\\textit{AB}$?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 18\\qquad\\textbf{(D)}\\ 20\\qquad\\textbf{(E)}\\ 24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_854", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The region consisting of all points in three-dimensional space within $3$ units of line segment $\\overline{AB}$ has volume $216\\pi$. What is the length $\\textit{AB}$?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 18\\qquad\\textbf{(D)}\\ 20\\qquad\\textbf{(E)}\\ 24$" + } + }, + { + "question": "Return your final response within \\boxed{}. The equation $\\sqrt {x + 4} - \\sqrt {x - 3} + 1 = 0$ has: \n$\\textbf{(A)}\\ \\text{no root} \\qquad \\textbf{(B)}\\ \\text{one real root} \\\\ \\textbf{(C)}\\ \\text{one real root and one imaginary root} \\\\ \\textbf{(D)}\\ \\text{two imaginary roots} \\qquad \\qquad\\textbf{(E)}\\ \\text{two real roots}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_855", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The equation $\\sqrt {x + 4} - \\sqrt {x - 3} + 1 = 0$ has: \n$\\textbf{(A)}\\ \\text{no root} \\qquad \\textbf{(B)}\\ \\text{one real root} \\\\ \\textbf{(C)}\\ \\text{one real root and one imaginary root} \\\\ \\textbf{(D)}\\ \\text{two imaginary roots} \\qquad \\qquad\\textbf{(E)}\\ \\text{two real roots}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. \nLine segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$, respectively. \nIf line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, \nso that points $A$ and $D$ coincide, to form a triangle with positive area, \nthen which of the following three inequalities must be satisfied?\n$\\textbf{I. }x<\\frac{z}{2}\\qquad\\\\ \\textbf{II. }yy$ and $y=z$\n$\\textbf{(B)} \\: x=y-1$ and $y=z-1$\n$\\textbf{(C)} \\: x=z+1$ and $y=x+1$\n$\\textbf{(D)} \\: x=z$ and $y-1=x$\n$\\textbf{(E)} \\: x+y+z=1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_864", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation\n\\[x(x-y)+y(y-z)+z(z-x) = 1?\\]\n$\\textbf{(A)} \\: x>y$ and $y=z$\n$\\textbf{(B)} \\: x=y-1$ and $y=z-1$\n$\\textbf{(C)} \\: x=z+1$ and $y=x+1$\n$\\textbf{(D)} \\: x=z$ and $y-1=x$\n$\\textbf{(E)} \\: x+y+z=1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? \n$\\textbf{(A) }\\ \\frac15 \\qquad\\textbf{(B) }\\ \\frac13 \\qquad\\textbf{(C) }\\ \\frac25 \\qquad\\textbf{(D) }\\ \\frac23 \\qquad\\textbf{(E) }\\ \\frac45$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_865", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? \n$\\textbf{(A) }\\ \\frac15 \\qquad\\textbf{(B) }\\ \\frac13 \\qquad\\textbf{(C) }\\ \\frac25 \\qquad\\textbf{(D) }\\ \\frac23 \\qquad\\textbf{(E) }\\ \\frac45$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P_1$, $P_2$, $\\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$, and so on, until we have labeled all of the blue points $B_1, \\dots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \\to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \\dots, R_n$ of the red points.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The number of such arcs containing (1,0) is independent of the ordering of the red points.", + "index": "Sky-T1_10k_866", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P_1$, $P_2$, $\\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$, and so on, until we have labeled all of the blue points $B_1, \\dots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \\to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \\dots, R_n$ of the red points." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABC$ be an equilateral triangle inscribed in circle $O$. $M$ is a point on arc $BC$. \nLines $\\overline{AM}$, $\\overline{BM}$, and $\\overline{CM}$ are drawn. Then $AM$ is: \n\n$\\textbf{(A)}\\ \\text{equal to }{BM + CM}\\qquad \\textbf{(B)}\\ \\text{less than }{BM + CM}\\qquad \\\\ \\textbf{(C)}\\ \\text{greater than }{BM+CM}\\qquad \\\\ \\textbf{(D)}\\ \\text{equal, less than, or greater than }{BM + CM}\\text{, depending upon the position of } {M}\\qquad \\\\ \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_867", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABC$ be an equilateral triangle inscribed in circle $O$. $M$ is a point on arc $BC$. \nLines $\\overline{AM}$, $\\overline{BM}$, and $\\overline{CM}$ are drawn. Then $AM$ is: \n\n$\\textbf{(A)}\\ \\text{equal to }{BM + CM}\\qquad \\textbf{(B)}\\ \\text{less than }{BM + CM}\\qquad \\\\ \\textbf{(C)}\\ \\text{greater than }{BM+CM}\\qquad \\\\ \\textbf{(D)}\\ \\text{equal, less than, or greater than }{BM + CM}\\text{, depending upon the position of } {M}\\qquad \\\\ \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? \n$\\mathrm{(A) \\ } 8\\qquad \\mathrm{(B) \\ } 9\\qquad \\mathrm{(C) \\ } 10\\qquad \\mathrm{(D) \\ } 11\\qquad \\mathrm{(E) \\ } 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_868", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? \n$\\mathrm{(A) \\ } 8\\qquad \\mathrm{(B) \\ } 9\\qquad \\mathrm{(C) \\ } 10\\qquad \\mathrm{(D) \\ } 11\\qquad \\mathrm{(E) \\ } 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. A bakery owner turns on his doughnut machine at $\\text{8:30}\\ {\\small\\text{AM}}$. At $\\text{11:10}\\ {\\small\\text{AM}}$ the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?\n$\\mathrm{(A)}\\ \\text{1:50}\\ {\\small\\text{PM}}\\qquad\\mathrm{(B)}\\ \\text{3:00}\\ {\\small\\text{PM}}\\qquad\\mathrm{(C)}\\ \\text{3:30}\\ {\\small\\text{PM}}\\qquad\\mathrm{(D)}\\ \\text{4:30}\\ {\\small\\text{PM}}\\qquad\\mathrm{(E)}\\ \\text{5:50}\\ {\\small\\text{PM}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_869", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A bakery owner turns on his doughnut machine at $\\text{8:30}\\ {\\small\\text{AM}}$. At $\\text{11:10}\\ {\\small\\text{AM}}$ the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?\n$\\mathrm{(A)}\\ \\text{1:50}\\ {\\small\\text{PM}}\\qquad\\mathrm{(B)}\\ \\text{3:00}\\ {\\small\\text{PM}}\\qquad\\mathrm{(C)}\\ \\text{3:30}\\ {\\small\\text{PM}}\\qquad\\mathrm{(D)}\\ \\text{4:30}\\ {\\small\\text{PM}}\\qquad\\mathrm{(E)}\\ \\text{5:50}\\ {\\small\\text{PM}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$?\n$\\mathrm{(A)}\\ 6\\qquad\\mathrm{(B)}\\ 7\\qquad\\mathrm{(C)}\\ 8\\qquad\\mathrm{(D)}\\ 9\\qquad\\mathrm{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_870", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$?\n$\\mathrm{(A)}\\ 6\\qquad\\mathrm{(B)}\\ 7\\qquad\\mathrm{(C)}\\ 8\\qquad\\mathrm{(D)}\\ 9\\qquad\\mathrm{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many $3$-digit positive integers have digits whose product equals $24$?\n\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 15 \\qquad \\textbf{(C)}\\ 18 \\qquad \\textbf{(D)}\\ 21 \\qquad \\textbf{(E)}\\ 24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_871", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many $3$-digit positive integers have digits whose product equals $24$?\n\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 15 \\qquad \\textbf{(C)}\\ 18 \\qquad \\textbf{(D)}\\ 21 \\qquad \\textbf{(E)}\\ 24$" + } + }, + { + "question": "Return your final response within \\boxed{}. A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?\n$\\text{(A) }8! \\qquad \\text{(B) }2^8 \\cdot 8! \\qquad \\text{(C) }(8!)^2 \\qquad \\text{(D) }\\frac {16!}{2^8} \\qquad \\text{(E) }16!$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_872", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?\n$\\text{(A) }8! \\qquad \\text{(B) }2^8 \\cdot 8! \\qquad \\text{(C) }(8!)^2 \\qquad \\text{(D) }\\frac {16!}{2^8} \\qquad \\text{(E) }16!$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a @ b = \\frac{a\\times b}{a+b}$ for $a,b$ positive integers, then what is $5 @10$? \n$\\textbf{(A)}\\ \\frac{3}{10} \\qquad\\textbf{(B)}\\ 1 \\qquad\\textbf{(C)}\\ 2 \\qquad\\textbf{(D)}\\ \\frac{10}{3} \\qquad\\textbf{(E)}\\ 50$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_873", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a @ b = \\frac{a\\times b}{a+b}$ for $a,b$ positive integers, then what is $5 @10$? \n$\\textbf{(A)}\\ \\frac{3}{10} \\qquad\\textbf{(B)}\\ 1 \\qquad\\textbf{(C)}\\ 2 \\qquad\\textbf{(D)}\\ \\frac{10}{3} \\qquad\\textbf{(E)}\\ 50$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?\n$\\textbf{(A) } \\text{All Dramps are Brafs and are Crups.}\\\\ \\textbf{(B) } \\text{All Brafs are Crups and are Dramps.}\\\\ \\textbf{(C) } \\text{All Arogs are Crups and are Dramps.}\\\\ \\textbf{(D) } \\text{All Crups are Arogs and are Brafs.}\\\\ \\textbf{(E) } \\text{All Arogs are Dramps and some Arogs may not be Crups.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_874", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?\n$\\textbf{(A) } \\text{All Dramps are Brafs and are Crups.}\\\\ \\textbf{(B) } \\text{All Brafs are Crups and are Dramps.}\\\\ \\textbf{(C) } \\text{All Arogs are Crups and are Dramps.}\\\\ \\textbf{(D) } \\text{All Crups are Arogs and are Brafs.}\\\\ \\textbf{(E) } \\text{All Arogs are Dramps and some Arogs may not be Crups.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define $n_a!$ for $n$ and $a$ positive to be \n\\[n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)\\]\nwhere $k$ is the greatest integer for which $n>ka$. Then the quotient $72_8!/18_2!$ is equal to \n$\\textbf{(A)}\\ 4^5 \\qquad \\textbf{(B)}\\ 4^6 \\qquad \\textbf{(C)}\\ 4^8 \\qquad \\textbf{(D)}\\ 4^9 \\qquad \\textbf{(E)}\\ 4^{12}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_875", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define $n_a!$ for $n$ and $a$ positive to be \n\\[n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)\\]\nwhere $k$ is the greatest integer for which $n>ka$. Then the quotient $72_8!/18_2!$ is equal to \n$\\textbf{(A)}\\ 4^5 \\qquad \\textbf{(B)}\\ 4^6 \\qquad \\textbf{(C)}\\ 4^8 \\qquad \\textbf{(D)}\\ 4^9 \\qquad \\textbf{(E)}\\ 4^{12}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 6 \\qquad \\textbf{(C) } 8 \\qquad \\textbf{(D) } 9 \\qquad \\textbf{(E) } 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_876", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 6 \\qquad \\textbf{(C) } 8 \\qquad \\textbf{(D) } 9 \\qquad \\textbf{(E) } 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $10,\\!444$ and $3,\\!245$, and LeRoy obtains the sum $S = 13,\\!689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$?\n$\\textbf{(A)}\\ 5 \\qquad\\textbf{(B)}\\ 10 \\qquad\\textbf{(C)}\\ 15 \\qquad\\textbf{(D)}\\ 20 \\qquad\\textbf{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_877", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $10,\\!444$ and $3,\\!245$, and LeRoy obtains the sum $S = 13,\\!689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$?\n$\\textbf{(A)}\\ 5 \\qquad\\textbf{(B)}\\ 10 \\qquad\\textbf{(C)}\\ 15 \\qquad\\textbf{(D)}\\ 20 \\qquad\\textbf{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the sum of the squares of all real numbers satisfying the equation $x^{256}-256^{32}=0$.\n$\\textbf{(A) }8\\qquad \\textbf{(B) }128\\qquad \\textbf{(C) }512\\qquad \\textbf{(D) }65,536\\qquad \\textbf{(E) }2(256^{32})$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_878", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the sum of the squares of all real numbers satisfying the equation $x^{256}-256^{32}=0$.\n$\\textbf{(A) }8\\qquad \\textbf{(B) }128\\qquad \\textbf{(C) }512\\qquad \\textbf{(D) }65,536\\qquad \\textbf{(E) }2(256^{32})$" + } + }, + { + "question": "Return your final response within \\boxed{}. Circle $X$ has a radius of $\\pi$. Circle $Y$ has a circumference of $8 \\pi$. Circle $Z$ has an area of $9 \\pi$. List the circles in order from smallest to the largest radius. \n$\\textbf{(A)}\\ X, Y, Z\\qquad\\textbf{(B)}\\ Z, X, Y\\qquad\\textbf{(C)}\\ Y, X, Z\\qquad\\textbf{(D)}\\ Z, Y, X\\qquad\\textbf{(E)}\\ X, Z, Y$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_879", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Circle $X$ has a radius of $\\pi$. Circle $Y$ has a circumference of $8 \\pi$. Circle $Z$ has an area of $9 \\pi$. List the circles in order from smallest to the largest radius. \n$\\textbf{(A)}\\ X, Y, Z\\qquad\\textbf{(B)}\\ Z, X, Y\\qquad\\textbf{(C)}\\ Y, X, Z\\qquad\\textbf{(D)}\\ Z, Y, X\\qquad\\textbf{(E)}\\ X, Z, Y$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?\n\n$\\textbf{(A)}\\ 50 \\qquad \\textbf{(B)}\\ 62 \\qquad \\textbf{(C)}\\ 70 \\qquad \\textbf{(D)}\\ 89 \\qquad \\textbf{(E)}\\ 100$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_880", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?\n\n$\\textbf{(A)}\\ 50 \\qquad \\textbf{(B)}\\ 62 \\qquad \\textbf{(C)}\\ 70 \\qquad \\textbf{(D)}\\ 89 \\qquad \\textbf{(E)}\\ 100$" + } + }, + { + "question": "Return your final response within \\boxed{}. Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "108", + "index": "Sky-T1_10k_881", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially." + } + }, + { + "question": "Return your final response within \\boxed{}. Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing?\n$\\textbf{(A)}\\ 7 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 9 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_882", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing?\n$\\textbf{(A)}\\ 7 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 9 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice sells an item at $$10$ less than the list price and receives $10\\%$ of her selling price as her commission. \nBob sells the same item at $$20$ less than the list price and receives $20\\%$ of his selling price as his commission. \nIf they both get the same commission, then the list price is\n$\\textbf{(A) } $20\\qquad \\textbf{(B) } $30\\qquad \\textbf{(C) } $50\\qquad \\textbf{(D) } $70\\qquad \\textbf{(E) } $100$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_883", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice sells an item at $$10$ less than the list price and receives $10\\%$ of her selling price as her commission. \nBob sells the same item at $$20$ less than the list price and receives $20\\%$ of his selling price as his commission. \nIf they both get the same commission, then the list price is\n$\\textbf{(A) } $20\\qquad \\textbf{(B) } $30\\qquad \\textbf{(C) } $50\\qquad \\textbf{(D) } $70\\qquad \\textbf{(E) } $100$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n > 1$ be an integer. Find, with proof, all sequences\n$x_1, x_2, \\ldots, x_{n-1}$ of positive integers with the following\nthree properties:\n\n\n (a). $x_1 < x_2 < \\cdots 1$ be an integer. Find, with proof, all sequences\n$x_1, x_2, \\ldots, x_{n-1}$ of positive integers with the following\nthree properties:\n\n\n (a). $x_1 < x_2 < \\cdots 2\\qquad\\textbf{(E)}\\ \\text{no value of }b$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_909", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number $121_b$, written in the integral base $b$, is the square of an integer, for \n$\\textbf{(A)}\\ b = 10,\\text{ only}\\qquad\\textbf{(B)}\\ b = 10\\text{ and }b = 5,\\text{ only}\\qquad$\n$\\textbf{(C)}\\ 2\\leq b\\leq 10\\qquad\\textbf{(D)}\\ b > 2\\qquad\\textbf{(E)}\\ \\text{no value of }b$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $AP = AQ$. Let $S$ and $R$ be distinct points on segment $\\overline{BC}$ such that $S$ lies between $B$ and $R$, $\\angle{BPS} = \\angle{PRS}$, and $\\angle{CQR} = \\angle{QSR}$. Prove that $P$, $Q$, $R$, $S$ are concyclic (in other words, these four points lie on a circle).", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "P, Q, R, S are concyclic", + "index": "Sky-T1_10k_910", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $AP = AQ$. Let $S$ and $R$ be distinct points on segment $\\overline{BC}$ such that $S$ lies between $B$ and $R$, $\\angle{BPS} = \\angle{PRS}$, and $\\angle{CQR} = \\angle{QSR}$. Prove that $P$, $Q$, $R$, $S$ are concyclic (in other words, these four points lie on a circle)." + } + }, + { + "question": "Return your final response within \\boxed{}. Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can have?\n$\\textbf{(A) }\\text{806} \\qquad \\textbf{(B) }\\text{8068} \\qquad \\textbf{(C) }\\text{8072} \\qquad \\textbf{(D) }\\text{8076}\\qquad \\textbf{(E) }\\text{8082}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_911", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can have?\n$\\textbf{(A) }\\text{806} \\qquad \\textbf{(B) }\\text{8068} \\qquad \\textbf{(C) }\\text{8072} \\qquad \\textbf{(D) }\\text{8076}\\qquad \\textbf{(E) }\\text{8082}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?\n$\\textbf{(A) }78\\qquad\\textbf{(B) }80\\qquad\\textbf{(C) }144\\qquad\\textbf{(D) }146\\qquad\\textbf{(E) }152$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_912", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?\n$\\textbf{(A) }78\\qquad\\textbf{(B) }80\\qquad\\textbf{(C) }144\\qquad\\textbf{(D) }146\\qquad\\textbf{(E) }152$" + } + }, + { + "question": "Return your final response within \\boxed{}. Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 14 \\qquad \\textbf{(C)}\\ 26 \\qquad \\textbf{(D)}\\ 38 \\qquad \\textbf{(E)}\\ 50$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_913", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 14 \\qquad \\textbf{(C)}\\ 26 \\qquad \\textbf{(D)}\\ 38 \\qquad \\textbf{(E)}\\ 50$" + } + }, + { + "question": "Return your final response within \\boxed{}. A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?\n$(\\mathrm {A}) \\ 1404 \\qquad (\\mathrm {B}) \\ 1462 \\qquad (\\mathrm {C})\\ 1604 \\qquad (\\mathrm {D}) \\ 1605 \\qquad (\\mathrm {E})\\ 1804$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_914", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?\n$(\\mathrm {A}) \\ 1404 \\qquad (\\mathrm {B}) \\ 1462 \\qquad (\\mathrm {C})\\ 1604 \\qquad (\\mathrm {D}) \\ 1605 \\qquad (\\mathrm {E})\\ 1804$" + } + }, + { + "question": "Return your final response within \\boxed{}. For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \\ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \\le 50$ is $f_{50}(n) = 12?$\n$\\textbf{(A) }7\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }9\\qquad\\textbf{(D) }10\\qquad\\textbf{(E) }11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_915", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \\ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \\le 50$ is $f_{50}(n) = 12?$\n$\\textbf{(A) }7\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }9\\qquad\\textbf{(D) }10\\qquad\\textbf{(E) }11$" + } + }, + { + "question": "Return your final response within \\boxed{}. $|3-\\pi|=$\n$\\textbf{(A)\\ }\\frac{1}{7}\\qquad\\textbf{(B)\\ }0.14\\qquad\\textbf{(C)\\ }3-\\pi\\qquad\\textbf{(D)\\ }3+\\pi\\qquad\\textbf{(E)\\ }\\pi-3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_916", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $|3-\\pi|=$\n$\\textbf{(A)\\ }\\frac{1}{7}\\qquad\\textbf{(B)\\ }0.14\\qquad\\textbf{(C)\\ }3-\\pi\\qquad\\textbf{(D)\\ }3+\\pi\\qquad\\textbf{(E)\\ }\\pi-3$" + } + }, + { + "question": "Return your final response within \\boxed{}. In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\\pm 0.00312$. \nThe experimenter wishes to announce a value for $C$ in which every digit is significant. \nThat is, whatever $C$ is, the announced value must be the correct result when $C$ is rounded to that number of digits. \nThe most accurate value the experimenter can announce for $C$ is\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 2.4\\qquad \\textbf{(C)}\\ 2.43\\qquad \\textbf{(D)}\\ 2.44\\qquad \\textbf{(E)}\\ 2.439$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_917", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\\pm 0.00312$. \nThe experimenter wishes to announce a value for $C$ in which every digit is significant. \nThat is, whatever $C$ is, the announced value must be the correct result when $C$ is rounded to that number of digits. \nThe most accurate value the experimenter can announce for $C$ is\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 2.4\\qquad \\textbf{(C)}\\ 2.43\\qquad \\textbf{(D)}\\ 2.44\\qquad \\textbf{(E)}\\ 2.439$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is:\n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 12\\qquad \\textbf{(C)}\\ 24\\qquad \\textbf{(D)}\\ 48\\qquad \\textbf{(E)}\\ 96$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_918", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is:\n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 12\\qquad \\textbf{(C)}\\ 24\\qquad \\textbf{(D)}\\ 48\\qquad \\textbf{(E)}\\ 96$" + } + }, + { + "question": "Return your final response within \\boxed{}. The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. The number of males in the band who are NOT in the orchestra is\n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 20 \\qquad \\text{(C)}\\ 30 \\qquad \\text{(D)}\\ 50 \\qquad \\text{(E)}\\ 70$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_919", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. The number of males in the band who are NOT in the orchestra is\n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 20 \\qquad \\text{(C)}\\ 30 \\qquad \\text{(D)}\\ 50 \\qquad \\text{(E)}\\ 70$" + } + }, + { + "question": "Return your final response within \\boxed{}. The two circles pictured have the same center $C$. Chord $\\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\\overline{AD}$ has length $16$. What is the area between the two circles?\n\n\n$\\textbf{(A)}\\ 36 \\pi \\qquad\\textbf{(B)}\\ 49 \\pi\\qquad\\textbf{(C)}\\ 64 \\pi\\qquad\\textbf{(D)}\\ 81 \\pi\\qquad\\textbf{(E)}\\ 100 \\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_920", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The two circles pictured have the same center $C$. Chord $\\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\\overline{AD}$ has length $16$. What is the area between the two circles?\n\n\n$\\textbf{(A)}\\ 36 \\pi \\qquad\\textbf{(B)}\\ 49 \\pi\\qquad\\textbf{(C)}\\ 64 \\pi\\qquad\\textbf{(D)}\\ 81 \\pi\\qquad\\textbf{(E)}\\ 100 \\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which one of the following integers can be expressed as the sum of $100$ consecutive positive integers?\n$\\textbf{(A)}\\ 1,\\!627,\\!384,\\!950\\qquad\\textbf{(B)}\\ 2,\\!345,\\!678,\\!910\\qquad\\textbf{(C)}\\ 3,\\!579,\\!111,\\!300\\qquad\\textbf{(D)}\\ 4,\\!692,\\!581,\\!470\\qquad\\textbf{(E)}\\ 5,\\!815,\\!937,\\!260$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_921", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which one of the following integers can be expressed as the sum of $100$ consecutive positive integers?\n$\\textbf{(A)}\\ 1,\\!627,\\!384,\\!950\\qquad\\textbf{(B)}\\ 2,\\!345,\\!678,\\!910\\qquad\\textbf{(C)}\\ 3,\\!579,\\!111,\\!300\\qquad\\textbf{(D)}\\ 4,\\!692,\\!581,\\!470\\qquad\\textbf{(E)}\\ 5,\\!815,\\!937,\\!260$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "1", + "index": "Sky-T1_10k_922", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square." + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of\n\\[\\left(\\left((2+1)^{-1}+1\\right)^{-1}+1\\right)^{-1}+1?\\]\n$\\textbf{(A) } \\frac58 \\qquad \\textbf{(B) }\\frac{11}7 \\qquad \\textbf{(C) } \\frac85 \\qquad \\textbf{(D) } \\frac{18}{11} \\qquad \\textbf{(E) } \\frac{15}8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_923", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of\n\\[\\left(\\left((2+1)^{-1}+1\\right)^{-1}+1\\right)^{-1}+1?\\]\n$\\textbf{(A) } \\frac58 \\qquad \\textbf{(B) }\\frac{11}7 \\qquad \\textbf{(C) } \\frac85 \\qquad \\textbf{(D) } \\frac{18}{11} \\qquad \\textbf{(E) } \\frac{15}8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?\n$\\textbf{(A) }\\dfrac{3}{10}\\qquad\\textbf{(B) }\\dfrac{2}{5}\\qquad\\textbf{(C) }\\dfrac{1}{2}\\qquad\\textbf{(D) }\\dfrac{3}{5}\\qquad \\textbf{(E) }\\dfrac{7}{10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_924", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?\n$\\textbf{(A) }\\dfrac{3}{10}\\qquad\\textbf{(B) }\\dfrac{2}{5}\\qquad\\textbf{(C) }\\dfrac{1}{2}\\qquad\\textbf{(D) }\\dfrac{3}{5}\\qquad \\textbf{(E) }\\dfrac{7}{10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km all downhill, and then back to town $A$, a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$-km ride than it takes Penny?\n$\\textbf{(A)}\\ 45\\qquad\\textbf{(B)}\\ 60\\qquad\\textbf{(C)}\\ 65\\qquad\\textbf{(D)}\\ 90\\qquad\\textbf{(E)}\\ 95$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_925", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km all downhill, and then back to town $A$, a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$-km ride than it takes Penny?\n$\\textbf{(A)}\\ 45\\qquad\\textbf{(B)}\\ 60\\qquad\\textbf{(C)}\\ 65\\qquad\\textbf{(D)}\\ 90\\qquad\\textbf{(E)}\\ 95$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\frac{xy}{x+y}= a,\\frac{xz}{x+z}= b,\\frac{yz}{y+z}= c$, where $a, b, c$ are other than zero, then $x$ equals: \n$\\textbf{(A)}\\ \\frac{abc}{ab+ac+bc}\\qquad\\textbf{(B)}\\ \\frac{2abc}{ab+bc+ac}\\qquad\\textbf{(C)}\\ \\frac{2abc}{ab+ac-bc}$\n$\\textbf{(D)}\\ \\frac{2abc}{ab+bc-ac}\\qquad\\textbf{(E)}\\ \\frac{2abc}{ac+bc-ab}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_926", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\frac{xy}{x+y}= a,\\frac{xz}{x+z}= b,\\frac{yz}{y+z}= c$, where $a, b, c$ are other than zero, then $x$ equals: \n$\\textbf{(A)}\\ \\frac{abc}{ab+ac+bc}\\qquad\\textbf{(B)}\\ \\frac{2abc}{ab+bc+ac}\\qquad\\textbf{(C)}\\ \\frac{2abc}{ab+ac-bc}$\n$\\textbf{(D)}\\ \\frac{2abc}{ab+bc-ac}\\qquad\\textbf{(E)}\\ \\frac{2abc}{ac+bc-ab}$" + } + }, + { + "question": "Return your final response within \\boxed{}. At a certain beach if it is at least $80^{\\circ} F$ and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?\n$\\textbf{(A)}\\ \\text{The temperature was cooler than } 80^{\\circ} \\text{F and it was not sunny.}$\n$\\textbf{(B)}\\ \\text{The temperature was cooler than } 80^{\\circ} \\text{F or it was not sunny.}$\n$\\textbf{(C)}\\ \\text{If the temperature was at least } 80^{\\circ} \\text{F, then it was sunny.}$\n$\\textbf{(D)}\\ \\text{If the temperature was cooler than } 80^{\\circ} \\text{F, then it was sunny.}$\n$\\textbf{(E)}\\ \\text{If the temperature was cooler than } 80^{\\circ} \\text{F, then it was not sunny.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_927", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. At a certain beach if it is at least $80^{\\circ} F$ and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?\n$\\textbf{(A)}\\ \\text{The temperature was cooler than } 80^{\\circ} \\text{F and it was not sunny.}$\n$\\textbf{(B)}\\ \\text{The temperature was cooler than } 80^{\\circ} \\text{F or it was not sunny.}$\n$\\textbf{(C)}\\ \\text{If the temperature was at least } 80^{\\circ} \\text{F, then it was sunny.}$\n$\\textbf{(D)}\\ \\text{If the temperature was cooler than } 80^{\\circ} \\text{F, then it was sunny.}$\n$\\textbf{(E)}\\ \\text{If the temperature was cooler than } 80^{\\circ} \\text{F, then it was not sunny.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. When the [mean](https://artofproblemsolving.com/wiki/index.php/Mean), [median](https://artofproblemsolving.com/wiki/index.php/Median), and [mode](https://artofproblemsolving.com/wiki/index.php/Mode) of the list\n\\[10,2,5,2,4,2,x\\]\nare arranged in increasing order, they form a non-constant [arithmetic progression](https://artofproblemsolving.com/wiki/index.php/Arithmetic_progression). What is the sum of all possible real values of $x$?\n$\\text {(A)}\\ 3 \\qquad \\text {(B)}\\ 6 \\qquad \\text {(C)}\\ 9 \\qquad \\text {(D)}\\ 17 \\qquad \\text {(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_928", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When the [mean](https://artofproblemsolving.com/wiki/index.php/Mean), [median](https://artofproblemsolving.com/wiki/index.php/Median), and [mode](https://artofproblemsolving.com/wiki/index.php/Mode) of the list\n\\[10,2,5,2,4,2,x\\]\nare arranged in increasing order, they form a non-constant [arithmetic progression](https://artofproblemsolving.com/wiki/index.php/Arithmetic_progression). What is the sum of all possible real values of $x$?\n$\\text {(A)}\\ 3 \\qquad \\text {(B)}\\ 6 \\qquad \\text {(C)}\\ 9 \\qquad \\text {(D)}\\ 17 \\qquad \\text {(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$\n\n[asy] defaultpen(fontsize(8)+0.8); size(150); pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4; real x=45, y=90, z=60; O=origin; A1=dir(x); A2=dir(x+y); A3=dir(x+2y); A4=dir(x+3y); B1=dir(x-z); B2=dir(x+y-z); B3=dir(x+2y-z); B4=dir(x+3y-z); C1=dir(x-2z); C2=dir(x+y-2z); C3=dir(x+2y-2z); C4=dir(x+3y-2z); draw(A1--A2--A3--A4--A1, gray+0.25+dashed); filldraw(B1--B2--B3--B4--cycle, white, gray+dashed+linewidth(0.25)); filldraw(C1--C2--C3--C4--cycle, white, gray+dashed+linewidth(0.25)); dot(O); pair P1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4; P1=extension(A1,A2,B1,B2); Q1=extension(A1,A2,C3,C4); P2=extension(A2,A3,B2,B3); Q2=extension(A2,A3,C4,C1); P3=extension(A3,A4,B3,B4); Q3=extension(A3,A4,C1,C2); P4=extension(A4,A1,B4,B1); Q4=extension(A4,A1,C2,C3); R1=extension(C2,C3,B2,B3); R2=extension(C3,C4,B3,B4); R3=extension(C4,C1,B4,B1); R4=extension(C1,C2,B1,B2); draw(A1--P1--B2--R1--C3--Q1--A2); draw(A2--P2--B3--R2--C4--Q2--A3); draw(A3--P3--B4--R3--C1--Q3--A4); draw(A4--P4--B1--R4--C2--Q4--A1); [/asy]\n$(\\textbf{A})\\: 75\\qquad(\\textbf{B}) \\: 93\\qquad(\\textbf{C}) \\: 96\\qquad(\\textbf{D}) \\: 129\\qquad(\\textbf{E}) \\: 147$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_929", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$\n\n[asy] defaultpen(fontsize(8)+0.8); size(150); pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4; real x=45, y=90, z=60; O=origin; A1=dir(x); A2=dir(x+y); A3=dir(x+2y); A4=dir(x+3y); B1=dir(x-z); B2=dir(x+y-z); B3=dir(x+2y-z); B4=dir(x+3y-z); C1=dir(x-2z); C2=dir(x+y-2z); C3=dir(x+2y-2z); C4=dir(x+3y-2z); draw(A1--A2--A3--A4--A1, gray+0.25+dashed); filldraw(B1--B2--B3--B4--cycle, white, gray+dashed+linewidth(0.25)); filldraw(C1--C2--C3--C4--cycle, white, gray+dashed+linewidth(0.25)); dot(O); pair P1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4; P1=extension(A1,A2,B1,B2); Q1=extension(A1,A2,C3,C4); P2=extension(A2,A3,B2,B3); Q2=extension(A2,A3,C4,C1); P3=extension(A3,A4,B3,B4); Q3=extension(A3,A4,C1,C2); P4=extension(A4,A1,B4,B1); Q4=extension(A4,A1,C2,C3); R1=extension(C2,C3,B2,B3); R2=extension(C3,C4,B3,B4); R3=extension(C4,C1,B4,B1); R4=extension(C1,C2,B1,B2); draw(A1--P1--B2--R1--C3--Q1--A2); draw(A2--P2--B3--R2--C4--Q2--A3); draw(A3--P3--B4--R3--C1--Q3--A4); draw(A4--P4--B1--R4--C2--Q4--A1); [/asy]\n$(\\textbf{A})\\: 75\\qquad(\\textbf{B}) \\: 93\\qquad(\\textbf{C}) \\: 96\\qquad(\\textbf{D}) \\: 129\\qquad(\\textbf{E}) \\: 147$" + } + }, + { + "question": "Return your final response within \\boxed{}. In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x > y > 0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? \n\n$\\textbf{(A) } \\text{Her estimate is larger than } x-y \\qquad \\textbf{(B) } \\text{Her estimate is smaller than } x-y \\qquad \\textbf{(C) } \\text{Her estimate equals } x-y \\\\ \\qquad \\textbf{(D) } \\text{Her estimate equals } x-y \\qquad \\textbf{(E) } \\text{Her estimate is } 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_930", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x > y > 0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? \n\n$\\textbf{(A) } \\text{Her estimate is larger than } x-y \\qquad \\textbf{(B) } \\text{Her estimate is smaller than } x-y \\qquad \\textbf{(C) } \\text{Her estimate equals } x-y \\\\ \\qquad \\textbf{(D) } \\text{Her estimate equals } x-y \\qquad \\textbf{(E) } \\text{Her estimate is } 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. For the consumer, a single discount of $n\\%$ is more advantageous than any of the following discounts:\n(1) two successive $15\\%$ discounts\n(2) three successive $10\\%$ discounts\n(3) a $25\\%$ discount followed by a $5\\%$ discount\nWhat is the smallest possible positive integer value of $n$?\n$\\textbf{(A)}\\ \\ 27\\qquad\\textbf{(B)}\\ 28\\qquad\\textbf{(C)}\\ 29\\qquad\\textbf{(D)}\\ 31\\qquad\\textbf{(E)}\\ 33$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_931", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For the consumer, a single discount of $n\\%$ is more advantageous than any of the following discounts:\n(1) two successive $15\\%$ discounts\n(2) three successive $10\\%$ discounts\n(3) a $25\\%$ discount followed by a $5\\%$ discount\nWhat is the smallest possible positive integer value of $n$?\n$\\textbf{(A)}\\ \\ 27\\qquad\\textbf{(B)}\\ 28\\qquad\\textbf{(C)}\\ 29\\qquad\\textbf{(D)}\\ 31\\qquad\\textbf{(E)}\\ 33$" + } + }, + { + "question": "Return your final response within \\boxed{}. Orvin went to the store with just enough money to buy $30$ balloons. When he arrived he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?\n$\\textbf{(A)}\\ 33\\qquad\\textbf{(B)}\\ 34\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 38\\qquad\\textbf{(E)}\\ 39$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_932", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Orvin went to the store with just enough money to buy $30$ balloons. When he arrived he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?\n$\\textbf{(A)}\\ 33\\qquad\\textbf{(B)}\\ 34\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 38\\qquad\\textbf{(E)}\\ 39$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the maximum value of $\\frac{(2^t-3t)t}{4^t}$ for real values of $t?$\n$\\textbf{(A)}\\ \\frac{1}{16} \\qquad\\textbf{(B)}\\ \\frac{1}{15} \\qquad\\textbf{(C)}\\ \\frac{1}{12} \\qquad\\textbf{(D)}\\ \\frac{1}{10} \\qquad\\textbf{(E)}\\ \\frac{1}{9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_933", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the maximum value of $\\frac{(2^t-3t)t}{4^t}$ for real values of $t?$\n$\\textbf{(A)}\\ \\frac{1}{16} \\qquad\\textbf{(B)}\\ \\frac{1}{15} \\qquad\\textbf{(C)}\\ \\frac{1}{12} \\qquad\\textbf{(D)}\\ \\frac{1}{10} \\qquad\\textbf{(E)}\\ \\frac{1}{9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Kiran Kedlaya) Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.\nProve that there exist at least two balancing lines.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "At least two balancing lines exist.", + "index": "Sky-T1_10k_934", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Kiran Kedlaya) Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.\nProve that there exist at least two balancing lines." + } + }, + { + "question": "Return your final response within \\boxed{}. An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?\n$\\textbf{(A) }\\frac{14}{75}\\qquad\\textbf{(B) }\\frac{56}{225}\\qquad\\textbf{(C) }\\frac{107}{400}\\qquad\\textbf{(D) }\\frac{7}{25}\\qquad\\textbf{(E) }\\frac{9}{25}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_935", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?\n$\\textbf{(A) }\\frac{14}{75}\\qquad\\textbf{(B) }\\frac{56}{225}\\qquad\\textbf{(C) }\\frac{107}{400}\\qquad\\textbf{(D) }\\frac{7}{25}\\qquad\\textbf{(E) }\\frac{9}{25}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$?\n$\\textbf{(A)}\\ 60^\\circ \\qquad \\textbf{(B)}\\ 75^\\circ \\qquad \\textbf{(C)}\\ 90^\\circ \\qquad \\textbf{(D)}\\ 105^\\circ \\qquad \\textbf{(E)}\\ 120^\\circ$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_936", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$?\n$\\textbf{(A)}\\ 60^\\circ \\qquad \\textbf{(B)}\\ 75^\\circ \\qquad \\textbf{(C)}\\ 90^\\circ \\qquad \\textbf{(D)}\\ 105^\\circ \\qquad \\textbf{(E)}\\ 120^\\circ$" + } + }, + { + "question": "Return your final response within \\boxed{}. When the number $2^{1000}$ is divided by $13$, the remainder in the division is\n$\\textbf{(A) }1\\qquad \\textbf{(B) }2\\qquad \\textbf{(C) }3\\qquad \\textbf{(D) }7\\qquad \\textbf{(E) }11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_937", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When the number $2^{1000}$ is divided by $13$, the remainder in the division is\n$\\textbf{(A) }1\\qquad \\textbf{(B) }2\\qquad \\textbf{(C) }3\\qquad \\textbf{(D) }7\\qquad \\textbf{(E) }11$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "34", + "index": "Sky-T1_10k_938", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$." + } + }, + { + "question": "Return your final response within \\boxed{}. A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?\n\n$\\textbf{(A) } 1+\\frac12 \\sqrt2 \\qquad \\textbf{(B) } \\sqrt3 \\qquad \\textbf{(C) } \\frac74 \\qquad \\textbf{(D) } \\frac{15}{8} \\qquad \\textbf{(E) } 2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_939", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?\n\n$\\textbf{(A) } 1+\\frac12 \\sqrt2 \\qquad \\textbf{(B) } \\sqrt3 \\qquad \\textbf{(C) } \\frac74 \\qquad \\textbf{(D) } \\frac{15}{8} \\qquad \\textbf{(E) } 2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is: \n$\\text{(A) } \\frac {r + t}{r - t} \\qquad \\text{(B) } \\frac {r}{r - t} \\qquad \\text{(C) } \\frac {r + t}{r} \\qquad \\text{(D) } \\frac{r}{t}\\qquad \\text{(E) } \\frac{r+k}{t-k}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_940", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is: \n$\\text{(A) } \\frac {r + t}{r - t} \\qquad \\text{(B) } \\frac {r}{r - t} \\qquad \\text{(C) } \\frac {r + t}{r} \\qquad \\text{(D) } \\frac{r}{t}\\qquad \\text{(E) } \\frac{r+k}{t-k}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The hypotenuse $c$ and one arm $a$ of a right triangle are consecutive integers. The square of the second arm is: \n$\\textbf{(A)}\\ ca\\qquad \\textbf{(B)}\\ \\frac{c}{a}\\qquad \\textbf{(C)}\\ c+a\\qquad \\textbf{(D)}\\ c-a\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_941", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The hypotenuse $c$ and one arm $a$ of a right triangle are consecutive integers. The square of the second arm is: \n$\\textbf{(A)}\\ ca\\qquad \\textbf{(B)}\\ \\frac{c}{a}\\qquad \\textbf{(C)}\\ c+a\\qquad \\textbf{(D)}\\ c-a\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The Dunbar family consists of a mother, a father, and some children. The [average](https://artofproblemsolving.com/wiki/index.php/Average) age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_942", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The Dunbar family consists of a mother, a father, and some children. The [average](https://artofproblemsolving.com/wiki/index.php/Average) age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following could not be the units digit [ones digit] of the square of a whole number?\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_943", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following could not be the units digit [ones digit] of the square of a whole number?\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list?\n$\\mathrm{(A) \\ } 4\\qquad \\mathrm{(B) \\ } 5\\qquad \\mathrm{(C) \\ } 6\\qquad \\mathrm{(D) \\ } 7\\qquad \\mathrm{(E) \\ } 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_944", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list?\n$\\mathrm{(A) \\ } 4\\qquad \\mathrm{(B) \\ } 5\\qquad \\mathrm{(C) \\ } 6\\qquad \\mathrm{(D) \\ } 7\\qquad \\mathrm{(E) \\ } 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?\n$\\textbf{(A)}\\ \\frac{1}{10} \\qquad\\textbf{(B)}\\ \\frac{1}{6} \\qquad\\textbf{(C)}\\ \\frac{1}{5} \\qquad\\textbf{(D)}\\ \\frac{1}{3} \\qquad\\textbf{(E)}\\ \\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_945", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?\n$\\textbf{(A)}\\ \\frac{1}{10} \\qquad\\textbf{(B)}\\ \\frac{1}{6} \\qquad\\textbf{(C)}\\ \\frac{1}{5} \\qquad\\textbf{(D)}\\ \\frac{1}{3} \\qquad\\textbf{(E)}\\ \\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is:\n$\\text{(A) } \\frac{2}{n-1}\\quad \\text{(B) } \\frac{n-1}{2}\\quad \\text{(C) } \\frac{2}{n}\\quad \\text{(D) } 2n\\quad \\text{(E) } 2(n-1)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_946", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is:\n$\\text{(A) } \\frac{2}{n-1}\\quad \\text{(B) } \\frac{n-1}{2}\\quad \\text{(C) } \\frac{2}{n}\\quad \\text{(D) } 2n\\quad \\text{(E) } 2(n-1)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Raashan, Sylvia, and Ted play the following game. Each starts with $$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $$1$? (For example, Raashan and Ted may each decide to give $$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $$0$, Sylvia will have $$2$, and Ted will have $$1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $$1$ to, and the holdings will be the same at the end of the second round.)\n$\\textbf{(A) } \\frac{1}{7} \\qquad\\textbf{(B) } \\frac{1}{4} \\qquad\\textbf{(C) } \\frac{1}{3} \\qquad\\textbf{(D) } \\frac{1}{2} \\qquad\\textbf{(E) } \\frac{2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_947", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Raashan, Sylvia, and Ted play the following game. Each starts with $$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $$1$? (For example, Raashan and Ted may each decide to give $$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $$0$, Sylvia will have $$2$, and Ted will have $$1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $$1$ to, and the holdings will be the same at the end of the second round.)\n$\\textbf{(A) } \\frac{1}{7} \\qquad\\textbf{(B) } \\frac{1}{4} \\qquad\\textbf{(C) } \\frac{1}{3} \\qquad\\textbf{(D) } \\frac{1}{2} \\qquad\\textbf{(E) } \\frac{2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol. \nIf the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded \nthe sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol. \nDetermine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative.\n$\\textbf{(A)}\\ \\text{Dick, Ann, Carol, Bill} \\qquad \\textbf{(B)}\\ \\text{Dick, Ann, Bill, Carol} \\qquad \\textbf{(C)}\\ \\text{Dick, Carol, Bill, Ann}\\\\ \\qquad \\textbf{(D)}\\ \\text{Ann, Dick, Carol, Bill}\\qquad \\textbf{(E)}\\ \\text{Ann, Dick, Bill, Carol}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_948", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol. \nIf the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded \nthe sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol. \nDetermine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative.\n$\\textbf{(A)}\\ \\text{Dick, Ann, Carol, Bill} \\qquad \\textbf{(B)}\\ \\text{Dick, Ann, Bill, Carol} \\qquad \\textbf{(C)}\\ \\text{Dick, Carol, Bill, Ann}\\\\ \\qquad \\textbf{(D)}\\ \\text{Ann, Dick, Carol, Bill}\\qquad \\textbf{(E)}\\ \\text{Ann, Dick, Bill, Carol}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find a positive integral solution to the equation\n$\\frac{1+3+5+\\dots+(2n-1)}{2+4+6+\\dots+2n}=\\frac{115}{116}$\n$\\textbf{(A) }110\\qquad \\textbf{(B) }115\\qquad \\textbf{(C) }116\\qquad \\textbf{(D) }231\\qquad\\\\ \\textbf{(E) }\\text{The equation has no positive integral solutions.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_949", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find a positive integral solution to the equation\n$\\frac{1+3+5+\\dots+(2n-1)}{2+4+6+\\dots+2n}=\\frac{115}{116}$\n$\\textbf{(A) }110\\qquad \\textbf{(B) }115\\qquad \\textbf{(C) }116\\qquad \\textbf{(D) }231\\qquad\\\\ \\textbf{(E) }\\text{The equation has no positive integral solutions.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ and $y$ are non-zero numbers such that $x=1+\\frac{1}{y}$ and $y=1+\\frac{1}{x}$, then $y$ equals\n$\\text{(A) } x-1\\quad \\text{(B) } 1-x\\quad \\text{(C) } 1+x\\quad \\text{(D) } -x\\quad \\text{(E) } x$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_950", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ and $y$ are non-zero numbers such that $x=1+\\frac{1}{y}$ and $y=1+\\frac{1}{x}$, then $y$ equals\n$\\text{(A) } x-1\\quad \\text{(B) } 1-x\\quad \\text{(C) } 1+x\\quad \\text{(D) } -x\\quad \\text{(E) } x$" + } + }, + { + "question": "Return your final response within \\boxed{}. Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3$ cm from $P$?\n$\\textbf{(A)} \\ 1 \\qquad \\textbf{(B)} \\ 2 \\qquad \\textbf{(C)} \\ 3 \\qquad \\textbf{(D)} \\ 4 \\qquad \\textbf{(E)} \\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_951", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3$ cm from $P$?\n$\\textbf{(A)} \\ 1 \\qquad \\textbf{(B)} \\ 2 \\qquad \\textbf{(C)} \\ 3 \\qquad \\textbf{(D)} \\ 4 \\qquad \\textbf{(E)} \\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?\n\n$\\mathrm{(A)}\\ \\frac 12\\qquad \\mathrm{(B)}\\ \\frac 58\\qquad \\mathrm{(C)}\\ \\frac 34\\qquad \\mathrm{(D)}\\ \\frac 56\\qquad \\mathrm{(E)}\\ \\frac {9}{10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_952", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?\n\n$\\mathrm{(A)}\\ \\frac 12\\qquad \\mathrm{(B)}\\ \\frac 58\\qquad \\mathrm{(C)}\\ \\frac 34\\qquad \\mathrm{(D)}\\ \\frac 56\\qquad \\mathrm{(E)}\\ \\frac {9}{10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $(2\\times 3\\times 4)\\left(\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}\\right) =$\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 24 \\qquad \\text{(E)}\\ 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_953", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $(2\\times 3\\times 4)\\left(\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}\\right) =$\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 24 \\qquad \\text{(E)}\\ 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two subsets of the set $S=\\lbrace a,b,c,d,e\\rbrace$ are to be chosen so that their union is $S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?\n$\\mathrm{(A)}\\ 20\\qquad\\mathrm{(B)}\\ 40\\qquad\\mathrm{(C)}\\ 60\\qquad\\mathrm{(D)}\\ 160\\qquad\\mathrm{(E)}\\ 320$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_954", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two subsets of the set $S=\\lbrace a,b,c,d,e\\rbrace$ are to be chosen so that their union is $S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?\n$\\mathrm{(A)}\\ 20\\qquad\\mathrm{(B)}\\ 40\\qquad\\mathrm{(C)}\\ 60\\qquad\\mathrm{(D)}\\ 160\\qquad\\mathrm{(E)}\\ 320$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\\[n = f_1\\cdot f_2\\cdots f_k,\\]where $k\\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\\cdot 3$, and $3\\cdot2$, so $D(6) = 3$. What is $D(96)$?\n$\\textbf{(A) } 112 \\qquad\\textbf{(B) } 128 \\qquad\\textbf{(C) } 144 \\qquad\\textbf{(D) } 172 \\qquad\\textbf{(E) } 184$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_955", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\\[n = f_1\\cdot f_2\\cdots f_k,\\]where $k\\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\\cdot 3$, and $3\\cdot2$, so $D(6) = 3$. What is $D(96)$?\n$\\textbf{(A) } 112 \\qquad\\textbf{(B) } 128 \\qquad\\textbf{(C) } 144 \\qquad\\textbf{(D) } 172 \\qquad\\textbf{(E) } 184$" + } + }, + { + "question": "Return your final response within \\boxed{}. The $16$ squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:\n(1) fold the top half over the bottom half\n(2) fold the bottom half over the top half\n(3) fold the right half over the left half\n(4) fold the left half over the right half.\nWhich numbered square is on top after step $4$?\n\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 9 \\qquad \\text{(C)}\\ 10 \\qquad \\text{(D)}\\ 14 \\qquad \\text{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_956", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The $16$ squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:\n(1) fold the top half over the bottom half\n(2) fold the bottom half over the top half\n(3) fold the right half over the left half\n(4) fold the left half over the right half.\nWhich numbered square is on top after step $4$?\n\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 9 \\qquad \\text{(C)}\\ 10 \\qquad \\text{(D)}\\ 14 \\qquad \\text{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes?\n$\\textbf{(A) } 64 \\qquad \\textbf{(B) } 65 \\qquad \\textbf{(C) } 66 \\qquad \\textbf{(D) } 67 \\qquad \\textbf{(E) } 68$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_957", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes?\n$\\textbf{(A) } 64 \\qquad \\textbf{(B) } 65 \\qquad \\textbf{(C) } 66 \\qquad \\textbf{(D) } 67 \\qquad \\textbf{(E) } 68$" + } + }, + { + "question": "Return your final response within \\boxed{}. The product of the two $99$-digit numbers\n$303,030,303,...,030,303$ and $505,050,505,...,050,505$\nhas thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?\n$\\mathrm{(A)}\\ 3 \\qquad \\mathrm{(B)}\\ 5 \\qquad \\mathrm{(C)}\\ 6 \\qquad \\mathrm{(D)}\\ 8 \\qquad \\mathrm{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_958", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The product of the two $99$-digit numbers\n$303,030,303,...,030,303$ and $505,050,505,...,050,505$\nhas thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?\n$\\mathrm{(A)}\\ 3 \\qquad \\mathrm{(B)}\\ 5 \\qquad \\mathrm{(C)}\\ 6 \\qquad \\mathrm{(D)}\\ 8 \\qquad \\mathrm{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. One hundred students at Century High School participated in the AHSME last year, and their mean score was 100. The number of non-seniors taking the AHSME was $50\\%$ more than the number of seniors, and the mean score of the seniors was $50\\%$ higher than that of the non-seniors. What was the mean score of the seniors?\n(A) $100$ (B) $112.5$ (C) $120$ (D) $125$ (E) $150$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_959", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One hundred students at Century High School participated in the AHSME last year, and their mean score was 100. The number of non-seniors taking the AHSME was $50\\%$ more than the number of seniors, and the mean score of the seniors was $50\\%$ higher than that of the non-seniors. What was the mean score of the seniors?\n(A) $100$ (B) $112.5$ (C) $120$ (D) $125$ (E) $150$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. \nJack picks one of these slips at random and puts it back. Then Jill picks a slip at random. \nWhich digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer?\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 8\\qquad \\textbf{(D)}\\ 9\\qquad \\textbf{(E)}\\ \\text{each digit is equally likely}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_960", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. \nJack picks one of these slips at random and puts it back. Then Jill picks a slip at random. \nWhich digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer?\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 8\\qquad \\textbf{(D)}\\ 9\\qquad \\textbf{(E)}\\ \\text{each digit is equally likely}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\\times5$ grid. What fraction of the grid is covered by the triangle?\n$\\textbf{(A) }\\frac{1}{6} \\qquad \\textbf{(B) }\\frac{1}{5} \\qquad \\textbf{(C) }\\frac{1}{4} \\qquad \\textbf{(D) }\\frac{1}{3} \\qquad \\textbf{(E) }\\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_961", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\\times5$ grid. What fraction of the grid is covered by the triangle?\n$\\textbf{(A) }\\frac{1}{6} \\qquad \\textbf{(B) }\\frac{1}{5} \\qquad \\textbf{(C) }\\frac{1}{4} \\qquad \\textbf{(D) }\\frac{1}{3} \\qquad \\textbf{(E) }\\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A \"domino\" is made up of two small squares:\n[asy] unitsize(12); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((1,1)--(2,1)--(2,0)--(1,0)); [/asy]\nWhich of the \"checkerboards\" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes?\n[asy] unitsize(12); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black); fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,black); fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,black); fill((2,2)--(2,3)--(3,3)--(3,2)--cycle,black); draw((0,0)--(0,3)--(4,3)--(4,0)--cycle); draw((6,0)--(11,0)--(11,3)--(6,3)--cycle); fill((6,0)--(7,0)--(7,1)--(6,1)--cycle,black); fill((8,0)--(9,0)--(9,1)--(8,1)--cycle,black); fill((10,0)--(11,0)--(11,1)--(10,1)--cycle,black); fill((7,1)--(7,2)--(8,2)--(8,1)--cycle,black); fill((9,1)--(9,2)--(10,2)--(10,1)--cycle,black); fill((6,2)--(6,3)--(7,3)--(7,2)--cycle,black); fill((8,2)--(8,3)--(9,3)--(9,2)--cycle,black); fill((10,2)--(10,3)--(11,3)--(11,2)--cycle,black); draw((13,-1)--(13,3)--(17,3)--(17,-1)--cycle); fill((13,3)--(14,3)--(14,2)--(13,2)--cycle,black); fill((15,3)--(16,3)--(16,2)--(15,2)--cycle,black); fill((14,2)--(15,2)--(15,1)--(14,1)--cycle,black); fill((16,2)--(17,2)--(17,1)--(16,1)--cycle,black); fill((13,1)--(14,1)--(14,0)--(13,0)--cycle,black); fill((15,1)--(16,1)--(16,0)--(15,0)--cycle,black); fill((14,0)--(15,0)--(15,-1)--(14,-1)--cycle,black); fill((16,0)--(17,0)--(17,-1)--(16,-1)--cycle,black); draw((19,3)--(24,3)--(24,-1)--(19,-1)--cycle,black); fill((19,3)--(20,3)--(20,2)--(19,2)--cycle,black); fill((21,3)--(22,3)--(22,2)--(21,2)--cycle,black); fill((23,3)--(24,3)--(24,2)--(23,2)--cycle,black); fill((20,2)--(21,2)--(21,1)--(20,1)--cycle,black); fill((22,2)--(23,2)--(23,1)--(22,1)--cycle,black); fill((19,1)--(20,1)--(20,0)--(19,0)--cycle,black); fill((21,1)--(22,1)--(22,0)--(21,0)--cycle,black); fill((23,1)--(24,1)--(24,0)--(23,0)--cycle,black); fill((20,0)--(21,0)--(21,-1)--(20,-1)--cycle,black); fill((22,0)--(23,0)--(23,-1)--(22,-1)--cycle,black); draw((26,3)--(29,3)--(29,-3)--(26,-3)--cycle); fill((26,3)--(27,3)--(27,2)--(26,2)--cycle,black); fill((28,3)--(29,3)--(29,2)--(28,2)--cycle,black); fill((27,2)--(28,2)--(28,1)--(27,1)--cycle,black); fill((26,1)--(27,1)--(27,0)--(26,0)--cycle,black); fill((28,1)--(29,1)--(29,0)--(28,0)--cycle,black); fill((27,0)--(28,0)--(28,-1)--(27,-1)--cycle,black); fill((26,-1)--(27,-1)--(27,-2)--(26,-2)--cycle,black); fill((28,-1)--(29,-1)--(29,-2)--(28,-2)--cycle,black); fill((27,-2)--(28,-2)--(28,-3)--(27,-3)--cycle,black); [/asy]\n$\\text{(A)}\\ 3\\times 4 \\qquad \\text{(B)}\\ 3\\times 5 \\qquad \\text{(C)}\\ 4\\times 4 \\qquad \\text{(D)}\\ 4\\times 5 \\qquad \\text{(E)}\\ 6\\times 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_962", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A \"domino\" is made up of two small squares:\n[asy] unitsize(12); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((1,1)--(2,1)--(2,0)--(1,0)); [/asy]\nWhich of the \"checkerboards\" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes?\n[asy] unitsize(12); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black); fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,black); fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,black); fill((2,2)--(2,3)--(3,3)--(3,2)--cycle,black); draw((0,0)--(0,3)--(4,3)--(4,0)--cycle); draw((6,0)--(11,0)--(11,3)--(6,3)--cycle); fill((6,0)--(7,0)--(7,1)--(6,1)--cycle,black); fill((8,0)--(9,0)--(9,1)--(8,1)--cycle,black); fill((10,0)--(11,0)--(11,1)--(10,1)--cycle,black); fill((7,1)--(7,2)--(8,2)--(8,1)--cycle,black); fill((9,1)--(9,2)--(10,2)--(10,1)--cycle,black); fill((6,2)--(6,3)--(7,3)--(7,2)--cycle,black); fill((8,2)--(8,3)--(9,3)--(9,2)--cycle,black); fill((10,2)--(10,3)--(11,3)--(11,2)--cycle,black); draw((13,-1)--(13,3)--(17,3)--(17,-1)--cycle); fill((13,3)--(14,3)--(14,2)--(13,2)--cycle,black); fill((15,3)--(16,3)--(16,2)--(15,2)--cycle,black); fill((14,2)--(15,2)--(15,1)--(14,1)--cycle,black); fill((16,2)--(17,2)--(17,1)--(16,1)--cycle,black); fill((13,1)--(14,1)--(14,0)--(13,0)--cycle,black); fill((15,1)--(16,1)--(16,0)--(15,0)--cycle,black); fill((14,0)--(15,0)--(15,-1)--(14,-1)--cycle,black); fill((16,0)--(17,0)--(17,-1)--(16,-1)--cycle,black); draw((19,3)--(24,3)--(24,-1)--(19,-1)--cycle,black); fill((19,3)--(20,3)--(20,2)--(19,2)--cycle,black); fill((21,3)--(22,3)--(22,2)--(21,2)--cycle,black); fill((23,3)--(24,3)--(24,2)--(23,2)--cycle,black); fill((20,2)--(21,2)--(21,1)--(20,1)--cycle,black); fill((22,2)--(23,2)--(23,1)--(22,1)--cycle,black); fill((19,1)--(20,1)--(20,0)--(19,0)--cycle,black); fill((21,1)--(22,1)--(22,0)--(21,0)--cycle,black); fill((23,1)--(24,1)--(24,0)--(23,0)--cycle,black); fill((20,0)--(21,0)--(21,-1)--(20,-1)--cycle,black); fill((22,0)--(23,0)--(23,-1)--(22,-1)--cycle,black); draw((26,3)--(29,3)--(29,-3)--(26,-3)--cycle); fill((26,3)--(27,3)--(27,2)--(26,2)--cycle,black); fill((28,3)--(29,3)--(29,2)--(28,2)--cycle,black); fill((27,2)--(28,2)--(28,1)--(27,1)--cycle,black); fill((26,1)--(27,1)--(27,0)--(26,0)--cycle,black); fill((28,1)--(29,1)--(29,0)--(28,0)--cycle,black); fill((27,0)--(28,0)--(28,-1)--(27,-1)--cycle,black); fill((26,-1)--(27,-1)--(27,-2)--(26,-2)--cycle,black); fill((28,-1)--(29,-1)--(29,-2)--(28,-2)--cycle,black); fill((27,-2)--(28,-2)--(28,-3)--(27,-3)--cycle,black); [/asy]\n$\\text{(A)}\\ 3\\times 4 \\qquad \\text{(B)}\\ 3\\times 5 \\qquad \\text{(C)}\\ 4\\times 4 \\qquad \\text{(D)}\\ 4\\times 5 \\qquad \\text{(E)}\\ 6\\times 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the graphs of $y=2\\log{x}$ and $y=\\log{2x}$. We may say that:\n$\\textbf{(A)}\\ \\text{They do not intersect}\\qquad \\\\ \\textbf{(B)}\\ \\text{They intersect at 1 point only}\\qquad \\\\ \\textbf{(C)}\\ \\text{They intersect at 2 points only} \\qquad \\\\ \\textbf{(D)}\\ \\text{They intersect at a finite number of points but greater than 2} \\qquad \\\\ \\textbf{(E)}\\ \\text{They coincide}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_963", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the graphs of $y=2\\log{x}$ and $y=\\log{2x}$. We may say that:\n$\\textbf{(A)}\\ \\text{They do not intersect}\\qquad \\\\ \\textbf{(B)}\\ \\text{They intersect at 1 point only}\\qquad \\\\ \\textbf{(C)}\\ \\text{They intersect at 2 points only} \\qquad \\\\ \\textbf{(D)}\\ \\text{They intersect at a finite number of points but greater than 2} \\qquad \\\\ \\textbf{(E)}\\ \\text{They coincide}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $(a,b,c)$ be the [real](https://artofproblemsolving.com/wiki/index.php/Real_number) solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\\frac {m}{n}$, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find $m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "158", + "index": "Sky-T1_10k_964", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $(a,b,c)$ be the [real](https://artofproblemsolving.com/wiki/index.php/Real_number) solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\\frac {m}{n}$, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find $m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.\n\nFor example, starting with an input of $N=7,$ the machine will output $3 \\cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$\\[7 \\to 22 \\to 11 \\to 34 \\to 17 \\to 52 \\to 26\\]When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$\\[N \\to \\rule{0.5cm}{0.15mm} \\to \\rule{0.5cm}{0.15mm} \\to \\rule{0.5cm}{0.15mm} \\to \\rule{0.5cm}{0.15mm} \\to \\rule{0.5cm}{0.15mm} \\to 1\\]\n$\\textbf{(A) }73 \\qquad \\textbf{(B) }74 \\qquad \\textbf{(C) }75 \\qquad \\textbf{(D) }82 \\qquad \\textbf{(E) }83$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_965", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.\n\nFor example, starting with an input of $N=7,$ the machine will output $3 \\cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$\\[7 \\to 22 \\to 11 \\to 34 \\to 17 \\to 52 \\to 26\\]When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$\\[N \\to \\rule{0.5cm}{0.15mm} \\to \\rule{0.5cm}{0.15mm} \\to \\rule{0.5cm}{0.15mm} \\to \\rule{0.5cm}{0.15mm} \\to \\rule{0.5cm}{0.15mm} \\to 1\\]\n$\\textbf{(A) }73 \\qquad \\textbf{(B) }74 \\qquad \\textbf{(C) }75 \\qquad \\textbf{(D) }82 \\qquad \\textbf{(E) }83$" + } + }, + { + "question": "Return your final response within \\boxed{}. Estimate to determine which of the following numbers is closest to $\\frac{401}{.205}$.\n$\\text{(A)}\\ .2 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 20 \\qquad \\text{(D)}\\ 200 \\qquad \\text{(E)}\\ 2000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_966", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Estimate to determine which of the following numbers is closest to $\\frac{401}{.205}$.\n$\\text{(A)}\\ .2 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 20 \\qquad \\text{(D)}\\ 200 \\qquad \\text{(E)}\\ 2000$" + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m?\n\n$\\text{(A)}\\ 11 \\qquad \\text{(B)}\\ 12 \\qquad \\text{(C)}\\ 13 \\qquad \\text{(D)}\\ 14 \\qquad \\text{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_967", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m?\n\n$\\text{(A)}\\ 11 \\qquad \\text{(B)}\\ 12 \\qquad \\text{(C)}\\ 13 \\qquad \\text{(D)}\\ 14 \\qquad \\text{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. What number is one third of the way from $\\frac14$ to $\\frac34$?\n$\\textbf{(A)}\\ \\frac {1}{3} \\qquad \\textbf{(B)}\\ \\frac {5}{12} \\qquad \\textbf{(C)}\\ \\frac {1}{2} \\qquad \\textbf{(D)}\\ \\frac {7}{12} \\qquad \\textbf{(E)}\\ \\frac {2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_968", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What number is one third of the way from $\\frac14$ to $\\frac34$?\n$\\textbf{(A)}\\ \\frac {1}{3} \\qquad \\textbf{(B)}\\ \\frac {5}{12} \\qquad \\textbf{(C)}\\ \\frac {1}{2} \\qquad \\textbf{(D)}\\ \\frac {7}{12} \\qquad \\textbf{(E)}\\ \\frac {2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?\n$\\textbf{(A) } -5 \\qquad\\textbf{(B) } 0 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } \\frac{15}{4} \\qquad\\textbf{(E) } \\frac{35}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_969", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?\n$\\textbf{(A) } -5 \\qquad\\textbf{(B) } 0 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } \\frac{15}{4} \\qquad\\textbf{(E) } \\frac{35}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $f(x) = 5x^2 - 2x - 1$, then $f(x + h) - f(x)$ equals: \n$\\textbf{(A)}\\ 5h^2 - 2h \\qquad \\textbf{(B)}\\ 10xh - 4x + 2 \\qquad \\textbf{(C)}\\ 10xh - 2x - 2 \\\\ \\textbf{(D)}\\ h(10x+5h-2)\\qquad\\textbf{(E)}\\ 3h$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_970", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $f(x) = 5x^2 - 2x - 1$, then $f(x + h) - f(x)$ equals: \n$\\textbf{(A)}\\ 5h^2 - 2h \\qquad \\textbf{(B)}\\ 10xh - 4x + 2 \\qquad \\textbf{(C)}\\ 10xh - 2x - 2 \\\\ \\textbf{(D)}\\ h(10x+5h-2)\\qquad\\textbf{(E)}\\ 3h$" + } + }, + { + "question": "Return your final response within \\boxed{}. A permutation of the set of positive integers $[n] = \\{1, 2, \\ldots, n\\}$ is a sequence $(a_1, a_2, \\ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P(n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1\\leq k\\leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "4489", + "index": "Sky-T1_10k_971", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A permutation of the set of positive integers $[n] = \\{1, 2, \\ldots, n\\}$ is a sequence $(a_1, a_2, \\ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P(n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1\\leq k\\leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$." + } + }, + { + "question": "Return your final response within \\boxed{}. This line graph represents the price of a trading card during the first $6$ months of $1993$.\n\nThe greatest monthly drop in price occurred during\n$\\text{(A)}\\ \\text{January} \\qquad \\text{(B)}\\ \\text{March} \\qquad \\text{(C)}\\ \\text{April} \\qquad \\text{(D)}\\ \\text{May} \\qquad \\text{(E)}\\ \\text{June}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_972", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. This line graph represents the price of a trading card during the first $6$ months of $1993$.\n\nThe greatest monthly drop in price occurred during\n$\\text{(A)}\\ \\text{January} \\qquad \\text{(B)}\\ \\text{March} \\qquad \\text{(C)}\\ \\text{April} \\qquad \\text{(D)}\\ \\text{May} \\qquad \\text{(E)}\\ \\text{June}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?\n$\\textbf{(A) }44\\qquad \\textbf{(B) }55\\qquad \\textbf{(C) }77\\qquad \\textbf{(D) }99\\qquad \\textbf{(E) }110$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_973", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?\n$\\textbf{(A) }44\\qquad \\textbf{(B) }55\\qquad \\textbf{(C) }77\\qquad \\textbf{(D) }99\\qquad \\textbf{(E) }110$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$\n$\\textbf{(A)}\\ 4:3 \\qquad\\textbf{(B)}\\ 3:2 \\qquad\\textbf{(C)}\\ 8:3 \\qquad\\textbf{(D)}\\ 4:1 \\qquad\\textbf{(E)}\\ 16:3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_974", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$\n$\\textbf{(A)}\\ 4:3 \\qquad\\textbf{(B)}\\ 3:2 \\qquad\\textbf{(C)}\\ 8:3 \\qquad\\textbf{(D)}\\ 4:1 \\qquad\\textbf{(E)}\\ 16:3$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $r$ be the result of doubling both the base and exponent of $a^b$, and $b$ does not equal to $0$. \nIf $r$ equals the product of $a^b$ by $x^b$, then $x$ equals:\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ 2a \\qquad \\textbf{(C)}\\ 4a \\qquad \\textbf{(D)}\\ 2\\qquad \\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_975", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $r$ be the result of doubling both the base and exponent of $a^b$, and $b$ does not equal to $0$. \nIf $r$ equals the product of $a^b$ by $x^b$, then $x$ equals:\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ 2a \\qquad \\textbf{(C)}\\ 4a \\qquad \\textbf{(D)}\\ 2\\qquad \\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define $x \\heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$. Which of the following statements is not true? \n$\\mathrm{(A) \\ } x \\heartsuit y = y \\heartsuit x$ for all $x$ and $y$\n$\\mathrm{(B) \\ } 2(x \\heartsuit y) = (2x) \\heartsuit (2y)$ for all $x$ and $y$\n$\\mathrm{(C) \\ } x \\heartsuit 0 = x$ for all $x$\n$\\mathrm{(D) \\ } x \\heartsuit x = 0$ for all $x$\n$\\mathrm{(E) \\ } x \\heartsuit y > 0$ if $x \\neq y$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_976", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define $x \\heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$. Which of the following statements is not true? \n$\\mathrm{(A) \\ } x \\heartsuit y = y \\heartsuit x$ for all $x$ and $y$\n$\\mathrm{(B) \\ } 2(x \\heartsuit y) = (2x) \\heartsuit (2y)$ for all $x$ and $y$\n$\\mathrm{(C) \\ } x \\heartsuit 0 = x$ for all $x$\n$\\mathrm{(D) \\ } x \\heartsuit x = 0$ for all $x$\n$\\mathrm{(E) \\ } x \\heartsuit y > 0$ if $x \\neq y$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S$ be a set of $6$ integers taken from $\\{1,2,\\dots,12\\}$ with the property that if $a$ and $b$ are elements of $S$ with $aBK\\quad\\\\ \\text{(C) sometimes have } MH=MK \\text{ but not always}\\quad\\\\ \\text{(D) always have } MH>MB\\quad\\\\ \\text{(E) always have } BHBK\\quad\\\\ \\text{(C) sometimes have } MH=MK \\text{ but not always}\\quad\\\\ \\text{(D) always have } MH>MB\\quad\\\\ \\text{(E) always have } BH 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$. Find the remainder when $\\sum\\limits_{n=20}^{100} F(n)$ is divided by $1000$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "512", + "index": "Sky-T1_10k_1035", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$. Find the remainder when $\\sum\\limits_{n=20}^{100} F(n)$ is divided by $1000$." + } + }, + { + "question": "Return your final response within \\boxed{}. If $y$ varies directly as $x$, and if $y=8$ when $x=4$, the value of $y$ when $x=-8$ is:\n$\\textbf{(A)}\\ -16 \\qquad \\textbf{(B)}\\ -4 \\qquad \\textbf{(C)}\\ -2 \\qquad \\textbf{(D)}\\ 4k, k= \\pm1, \\pm2, \\dots \\qquad \\\\ \\textbf{(E)}\\ 16k, k=\\pm1,\\pm2,\\dots$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1036", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $y$ varies directly as $x$, and if $y=8$ when $x=4$, the value of $y$ when $x=-8$ is:\n$\\textbf{(A)}\\ -16 \\qquad \\textbf{(B)}\\ -4 \\qquad \\textbf{(C)}\\ -2 \\qquad \\textbf{(D)}\\ 4k, k= \\pm1, \\pm2, \\dots \\qquad \\\\ \\textbf{(E)}\\ 16k, k=\\pm1,\\pm2,\\dots$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the statement, \"If $n$ is not prime, then $n-2$ is prime.\" Which of the following values of $n$ is a counterexample to this statement?\n$\\textbf{(A) } 11 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 19 \\qquad \\textbf{(D) } 21 \\qquad \\textbf{(E) } 27$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1037", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the statement, \"If $n$ is not prime, then $n-2$ is prime.\" Which of the following values of $n$ is a counterexample to this statement?\n$\\textbf{(A) } 11 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 19 \\qquad \\textbf{(D) } 21 \\qquad \\textbf{(E) } 27$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $x=-2016$. What is the value of $\\Bigg\\vert\\Big\\vert |x|-x\\Big\\vert-|x|\\Bigg\\vert-x$ ?\n$\\textbf{(A)}\\ -2016\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2016\\qquad\\textbf{(D)}\\ 4032\\qquad\\textbf{(E)}\\ 6048$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1038", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $x=-2016$. What is the value of $\\Bigg\\vert\\Big\\vert |x|-x\\Big\\vert-|x|\\Bigg\\vert-x$ ?\n$\\textbf{(A)}\\ -2016\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2016\\qquad\\textbf{(D)}\\ 4032\\qquad\\textbf{(E)}\\ 6048$" + } + }, + { + "question": "Return your final response within \\boxed{}. A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?\n$\\textbf{(A)}\\ \\dfrac{1}{10} \\qquad \\textbf{(B)}\\ \\dfrac{1}{9} \\qquad \\textbf{(C)}\\ \\dfrac{1}{7} \\qquad \\textbf{(D)}\\ \\dfrac{1}{6} \\qquad \\textbf{(E)}\\ \\dfrac{1}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1039", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?\n$\\textbf{(A)}\\ \\dfrac{1}{10} \\qquad \\textbf{(B)}\\ \\dfrac{1}{9} \\qquad \\textbf{(C)}\\ \\dfrac{1}{7} \\qquad \\textbf{(D)}\\ \\dfrac{1}{6} \\qquad \\textbf{(E)}\\ \\dfrac{1}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is\n[asy] draw((0,0)--(3,0)--(3,1)--(0,1)--cycle); draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle); [/asy]\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 21 \\qquad \\text{(C)}\\ 22 \\qquad \\text{(D)}\\ 24 \\qquad \\text{(E)}\\ 30$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1040", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is\n[asy] draw((0,0)--(3,0)--(3,1)--(0,1)--cycle); draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle); [/asy]\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 21 \\qquad \\text{(C)}\\ 22 \\qquad \\text{(D)}\\ 24 \\qquad \\text{(E)}\\ 30$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$\n$\\textbf{(A) } 80 \\qquad \\textbf{(B) } 100 \\qquad \\textbf{(C) } 125 \\qquad \\textbf{(D) } 200 \\qquad \\textbf{(E) } 500$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1041", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$\n$\\textbf{(A) } 80 \\qquad \\textbf{(B) } 100 \\qquad \\textbf{(C) } 125 \\qquad \\textbf{(D) } 200 \\qquad \\textbf{(E) } 500$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is:\n$\\textbf{(A)}\\ 33\\qquad \\textbf{(B)}\\ 34\\qquad \\textbf{(C)}\\ 35\\qquad \\textbf{(D)}\\ 100\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1042", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is:\n$\\textbf{(A)}\\ 33\\qquad \\textbf{(B)}\\ 34\\qquad \\textbf{(C)}\\ 35\\qquad \\textbf{(D)}\\ 100\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two permutations $a_1, a_2, \\ldots, a_{2010}$ and\n$b_1, b_2, \\ldots, b_{2010}$ of the numbers $1, 2, \\ldots, 2010$\nare said to intersect if $a_k = b_k$ for some value of $k$ in the\nrange $1 \\le k\\le 2010$. Show that there exist $1006$ permutations\nof the numbers $1, 2, \\ldots, 2010$ such that any other such\npermutation is guaranteed to intersect at least one of these $1006$\npermutations.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "1006", + "index": "Sky-T1_10k_1043", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two permutations $a_1, a_2, \\ldots, a_{2010}$ and\n$b_1, b_2, \\ldots, b_{2010}$ of the numbers $1, 2, \\ldots, 2010$\nare said to intersect if $a_k = b_k$ for some value of $k$ in the\nrange $1 \\le k\\le 2010$. Show that there exist $1006$ permutations\nof the numbers $1, 2, \\ldots, 2010$ such that any other such\npermutation is guaranteed to intersect at least one of these $1006$\npermutations." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $c = \\frac{2\\pi}{11}.$ What is the value of\n\\[\\frac{\\sin 3c \\cdot \\sin 6c \\cdot \\sin 9c \\cdot \\sin 12c \\cdot \\sin 15c}{\\sin c \\cdot \\sin 2c \\cdot \\sin 3c \\cdot \\sin 4c \\cdot \\sin 5c}?\\]\n$\\textbf{(A)}\\ {-}1 \\qquad\\textbf{(B)}\\ {-}\\frac{\\sqrt{11}}{5} \\qquad\\textbf{(C)}\\ \\frac{\\sqrt{11}}{5} \\qquad\\textbf{(D)}\\ \\frac{10}{11} \\qquad\\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1044", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $c = \\frac{2\\pi}{11}.$ What is the value of\n\\[\\frac{\\sin 3c \\cdot \\sin 6c \\cdot \\sin 9c \\cdot \\sin 12c \\cdot \\sin 15c}{\\sin c \\cdot \\sin 2c \\cdot \\sin 3c \\cdot \\sin 4c \\cdot \\sin 5c}?\\]\n$\\textbf{(A)}\\ {-}1 \\qquad\\textbf{(B)}\\ {-}\\frac{\\sqrt{11}}{5} \\qquad\\textbf{(C)}\\ \\frac{\\sqrt{11}}{5} \\qquad\\textbf{(D)}\\ \\frac{10}{11} \\qquad\\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $2x+1=8$, then $4x+1=$\n$\\mathrm{(A)\\ } 15 \\qquad \\mathrm{(B) \\ }16 \\qquad \\mathrm{(C) \\ } 17 \\qquad \\mathrm{(D) \\ } 18 \\qquad \\mathrm{(E) \\ }19$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1045", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $2x+1=8$, then $4x+1=$\n$\\mathrm{(A)\\ } 15 \\qquad \\mathrm{(B) \\ }16 \\qquad \\mathrm{(C) \\ } 17 \\qquad \\mathrm{(D) \\ } 18 \\qquad \\mathrm{(E) \\ }19$" + } + }, + { + "question": "Return your final response within \\boxed{}. A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m+10$, the mean and median of the new list would be $24$ and $m+10$, respectively. If were $m$ instead replaced by $m-8$, the median of the new list would be $m-4$. What is $m$?\n$\\textbf{(A)}\\ 16\\qquad\\textbf{(B)}\\ 17\\qquad\\textbf{(C)}\\ 18\\qquad\\textbf{(D)}\\ 19\\qquad\\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1046", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m+10$, the mean and median of the new list would be $24$ and $m+10$, respectively. If were $m$ instead replaced by $m-8$, the median of the new list would be $m-4$. What is $m$?\n$\\textbf{(A)}\\ 16\\qquad\\textbf{(B)}\\ 17\\qquad\\textbf{(C)}\\ 18\\qquad\\textbf{(D)}\\ 19\\qquad\\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$. What is $c$?\n$\\textbf{(A)}\\ -13\\qquad\\textbf{(B)}\\ -8\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 13$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1047", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$. What is $c$?\n$\\textbf{(A)}\\ -13\\qquad\\textbf{(B)}\\ -8\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 13$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $x$ be the number\n\\[0.\\underbrace{0000...0000}_{1996\\text{ zeros}}1,\\]\nwhere there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?\n$\\text{(A)}\\ 3+x \\qquad \\text{(B)}\\ 3-x \\qquad \\text{(C)}\\ 3\\cdot x \\qquad \\text{(D)}\\ 3/x \\qquad \\text{(E)}\\ x/3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1048", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $x$ be the number\n\\[0.\\underbrace{0000...0000}_{1996\\text{ zeros}}1,\\]\nwhere there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?\n$\\text{(A)}\\ 3+x \\qquad \\text{(B)}\\ 3-x \\qquad \\text{(C)}\\ 3\\cdot x \\qquad \\text{(D)}\\ 3/x \\qquad \\text{(E)}\\ x/3$" + } + }, + { + "question": "Return your final response within \\boxed{}. In rectangle $ABCD$, $\\overline{AB}=20$ and $\\overline{BC}=10$. Let $E$ be a point on $\\overline{CD}$ such that $\\angle CBE=15^\\circ$. What is $\\overline{AE}$?\n$\\textbf{(A)}\\ \\dfrac{20\\sqrt3}3\\qquad\\textbf{(B)}\\ 10\\sqrt3\\qquad\\textbf{(C)}\\ 18\\qquad\\textbf{(D)}\\ 11\\sqrt3\\qquad\\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1049", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In rectangle $ABCD$, $\\overline{AB}=20$ and $\\overline{BC}=10$. Let $E$ be a point on $\\overline{CD}$ such that $\\angle CBE=15^\\circ$. What is $\\overline{AE}$?\n$\\textbf{(A)}\\ \\dfrac{20\\sqrt3}3\\qquad\\textbf{(B)}\\ 10\\sqrt3\\qquad\\textbf{(C)}\\ 18\\qquad\\textbf{(D)}\\ 11\\sqrt3\\qquad\\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $\\frac{(2112-2021)^2}{169}$?\n$\\textbf{(A) } 7 \\qquad\\textbf{(B) } 21 \\qquad\\textbf{(C) } 49 \\qquad\\textbf{(D) } 64 \\qquad\\textbf{(E) } 91$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1050", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $\\frac{(2112-2021)^2}{169}$?\n$\\textbf{(A) } 7 \\qquad\\textbf{(B) } 21 \\qquad\\textbf{(C) } 49 \\qquad\\textbf{(D) } 64 \\qquad\\textbf{(E) } 91$" + } + }, + { + "question": "Return your final response within \\boxed{}. The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?\n$\\textbf{ (A) }5:50\\qquad\\textbf{(B) }6:00\\qquad\\textbf{(C) }6:30\\qquad\\textbf{(D) }6:55\\qquad \\textbf{(E) }8:10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1051", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?\n$\\textbf{ (A) }5:50\\qquad\\textbf{(B) }6:00\\qquad\\textbf{(C) }6:30\\qquad\\textbf{(D) }6:55\\qquad \\textbf{(E) }8:10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mientka Publishing Company prices its bestseller Where's Walter? as follows: \n$C(n) =\\left\\{\\begin{matrix}12n, &\\text{if }1\\le n\\le 24\\\\ 11n, &\\text{if }25\\le n\\le 48\\\\ 10n, &\\text{if }49\\le n\\end{matrix}\\right.$\nwhere $n$ is the number of books ordered, and $C(n)$ is the cost in dollars of $n$ books. Notice that $25$ books cost less than $24$ books. For how many values of $n$ is it cheaper to buy more than $n$ books than to buy exactly $n$ books?\n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1052", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mientka Publishing Company prices its bestseller Where's Walter? as follows: \n$C(n) =\\left\\{\\begin{matrix}12n, &\\text{if }1\\le n\\le 24\\\\ 11n, &\\text{if }25\\le n\\le 48\\\\ 10n, &\\text{if }49\\le n\\end{matrix}\\right.$\nwhere $n$ is the number of books ordered, and $C(n)$ is the cost in dollars of $n$ books. Notice that $25$ books cost less than $24$ books. For how many values of $n$ is it cheaper to buy more than $n$ books than to buy exactly $n$ books?\n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that necklace $\\, A \\,$ has 14 beads and necklace $\\, B \\,$ has \n19. Prove that for any odd integer $n \\geq 1$, there is a way to number \neach of the 33 beads with an integer from the sequence\n\\[\\{ n, n+1, n+2, \\dots, n+32 \\}\\]\nso that each integer is used once, and adjacent beads correspond to \nrelatively prime integers. (Here a \"necklace\" is viewed as a circle in \nwhich each bead is adjacent to two other beads.)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "Such a numbering is always possible for any odd integer n \\geq 1.", + "index": "Sky-T1_10k_1053", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that necklace $\\, A \\,$ has 14 beads and necklace $\\, B \\,$ has \n19. Prove that for any odd integer $n \\geq 1$, there is a way to number \neach of the 33 beads with an integer from the sequence\n\\[\\{ n, n+1, n+2, \\dots, n+32 \\}\\]\nso that each integer is used once, and adjacent beads correspond to \nrelatively prime integers. (Here a \"necklace\" is viewed as a circle in \nwhich each bead is adjacent to two other beads.)" + } + }, + { + "question": "Return your final response within \\boxed{}. For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\\frac{1}{n}$ is $0.\\overline{abcdef}$, a repeating decimal of period of $6$, and the decimal equivalent of $\\frac{1}{n+6}$ is $0.\\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie?\n$\\textbf{(A)}\\ [1,200]\\qquad\\textbf{(B)}\\ [201,400]\\qquad\\textbf{(C)}\\ [401,600]\\qquad\\textbf{(D)}\\ [601,800]\\qquad\\textbf{(E)}\\ [801,999]$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1054", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\\frac{1}{n}$ is $0.\\overline{abcdef}$, a repeating decimal of period of $6$, and the decimal equivalent of $\\frac{1}{n+6}$ is $0.\\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie?\n$\\textbf{(A)}\\ [1,200]\\qquad\\textbf{(B)}\\ [201,400]\\qquad\\textbf{(C)}\\ [401,600]\\qquad\\textbf{(D)}\\ [601,800]\\qquad\\textbf{(E)}\\ [801,999]$" + } + }, + { + "question": "Return your final response within \\boxed{}. The radius $R$ of a cylindrical box is $8$ inches, the height $H$ is $3$ inches. \nThe volume $V = \\pi R^2H$ is to be increased by the same fixed positive amount when $R$ \nis increased by $x$ inches as when $H$ is increased by $x$ inches. This condition is satisfied by:\n$\\textbf{(A)}\\ \\text{no real value of x} \\qquad \\\\ \\textbf{(B)}\\ \\text{one integral value of x} \\qquad \\\\ \\textbf{(C)}\\ \\text{one rational, but not integral, value of x} \\qquad \\\\ \\textbf{(D)}\\ \\text{one irrational value of x}\\qquad \\\\ \\textbf{(E)}\\ \\text{two real values of x}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1055", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The radius $R$ of a cylindrical box is $8$ inches, the height $H$ is $3$ inches. \nThe volume $V = \\pi R^2H$ is to be increased by the same fixed positive amount when $R$ \nis increased by $x$ inches as when $H$ is increased by $x$ inches. This condition is satisfied by:\n$\\textbf{(A)}\\ \\text{no real value of x} \\qquad \\\\ \\textbf{(B)}\\ \\text{one integral value of x} \\qquad \\\\ \\textbf{(C)}\\ \\text{one rational, but not integral, value of x} \\qquad \\\\ \\textbf{(D)}\\ \\text{one irrational value of x}\\qquad \\\\ \\textbf{(E)}\\ \\text{two real values of x}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a sign pyramid a cell gets a \"+\" if the two cells below it have the same sign, and it gets a \"-\" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a \"+\" at the top of the pyramid?\n\n$\\textbf{(A) } 2 \\qquad \\textbf{(B) } 4 \\qquad \\textbf{(C) } 8 \\qquad \\textbf{(D) } 12 \\qquad \\textbf{(E) } 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1056", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a sign pyramid a cell gets a \"+\" if the two cells below it have the same sign, and it gets a \"-\" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a \"+\" at the top of the pyramid?\n\n$\\textbf{(A) } 2 \\qquad \\textbf{(B) } 4 \\qquad \\textbf{(C) } 8 \\qquad \\textbf{(D) } 12 \\qquad \\textbf{(E) } 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number $25^{64}\\cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, the sum of the digits of $N$ is\n$\\mathrm{(A) \\ } 7\\qquad \\mathrm{(B) \\ } 14\\qquad \\mathrm{(C) \\ } 21\\qquad \\mathrm{(D) \\ } 28\\qquad \\mathrm{(E) \\ } 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1057", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number $25^{64}\\cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, the sum of the digits of $N$ is\n$\\mathrm{(A) \\ } 7\\qquad \\mathrm{(B) \\ } 14\\qquad \\mathrm{(C) \\ } 21\\qquad \\mathrm{(D) \\ } 28\\qquad \\mathrm{(E) \\ } 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals\n$\\mathrm{(A)\\ } -1024 \\qquad \\mathrm{(B) \\ }-1024i \\qquad \\mathrm{(C) \\ } 0 \\qquad \\mathrm{(D) \\ } 1024 \\qquad \\mathrm{(E) \\ }1024i$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1058", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals\n$\\mathrm{(A)\\ } -1024 \\qquad \\mathrm{(B) \\ }-1024i \\qquad \\mathrm{(C) \\ } 0 \\qquad \\mathrm{(D) \\ } 1024 \\qquad \\mathrm{(E) \\ }1024i$" + } + }, + { + "question": "Return your final response within \\boxed{}. One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has $\\{1,2,3,6,9\\}$ as its [combination](https://artofproblemsolving.com/wiki/index.php/Combination). Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?\n[1988-1.png](https://artofproblemsolving.com/wiki/index.php/File:1988-1.png)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "770", + "index": "Sky-T1_10k_1059", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has $\\{1,2,3,6,9\\}$ as its [combination](https://artofproblemsolving.com/wiki/index.php/Combination). Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?\n[1988-1.png](https://artofproblemsolving.com/wiki/index.php/File:1988-1.png)" + } + }, + { + "question": "Return your final response within \\boxed{}. The negation of the statement \"all men are honest,\" is:\n$\\textbf{(A)}\\ \\text{no men are honest} \\qquad \\textbf{(B)}\\ \\text{all men are dishonest} \\\\ \\textbf{(C)}\\ \\text{some men are dishonest}\\qquad \\textbf{(D)}\\ \\text{no men are dishonest}\\\\ \\textbf{(E)}\\ \\text{some men are honest}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1060", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The negation of the statement \"all men are honest,\" is:\n$\\textbf{(A)}\\ \\text{no men are honest} \\qquad \\textbf{(B)}\\ \\text{all men are dishonest} \\\\ \\textbf{(C)}\\ \\text{some men are dishonest}\\qquad \\textbf{(D)}\\ \\text{no men are dishonest}\\\\ \\textbf{(E)}\\ \\text{some men are honest}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "71", + "index": "Sky-T1_10k_1061", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player?\n$\\mathrm{(A)}\\ 270,000\\qquad\\mathrm{(B)}\\ 385,000\\qquad\\mathrm{(C)}\\ 400,000\\qquad\\mathrm{(D)}\\ 430,000\\qquad\\mathrm{(E)}\\ 700,000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1062", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player?\n$\\mathrm{(A)}\\ 270,000\\qquad\\mathrm{(B)}\\ 385,000\\qquad\\mathrm{(C)}\\ 400,000\\qquad\\mathrm{(D)}\\ 430,000\\qquad\\mathrm{(E)}\\ 700,000$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jenn randomly chooses a number $J$ from $1, 2, 3,\\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "29", + "index": "Sky-T1_10k_1063", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jenn randomly chooses a number $J$ from $1, 2, 3,\\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?\n[asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy]\n$\\textbf{(A)}\\ 50\\qquad\\textbf{(B)}\\ 100\\qquad\\textbf{(C)}\\ 125\\qquad\\textbf{(D)}\\ 150\\qquad\\textbf{(E)}\\ 200$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1064", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?\n[asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy]\n$\\textbf{(A)}\\ 50\\qquad\\textbf{(B)}\\ 100\\qquad\\textbf{(C)}\\ 125\\qquad\\textbf{(D)}\\ 150\\qquad\\textbf{(E)}\\ 200$" + } + }, + { + "question": "Return your final response within \\boxed{}. For each integer $n\\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\\leq j 3$. Let $x$ be the largest number such that the magnitude, \nin degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals:\n$\\textbf{(A)}\\ 150^{\\circ} \\qquad \\textbf{(B)}\\ 120^{\\circ}\\qquad \\textbf{(C)}\\ 105^{\\circ} \\qquad \\textbf{(D)}\\ 90^{\\circ} \\qquad \\textbf{(E)}\\ 60^{\\circ}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1092", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$, side $a = \\sqrt{3}$, side $b = \\sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, \nin degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals:\n$\\textbf{(A)}\\ 150^{\\circ} \\qquad \\textbf{(B)}\\ 120^{\\circ}\\qquad \\textbf{(C)}\\ 105^{\\circ} \\qquad \\textbf{(D)}\\ 90^{\\circ} \\qquad \\textbf{(E)}\\ 60^{\\circ}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The price of an article is cut $10 \\%.$ To restore it to its former value, the new price must be increased by:\n$\\textbf{(A) \\ } 10 \\% \\qquad\\textbf{(B) \\ } 9 \\% \\qquad \\textbf{(C) \\ } 11\\frac{1}{9} \\% \\qquad\\textbf{(D) \\ } 11 \\% \\qquad\\textbf{(E) \\ } \\text{none of these answers}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1093", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The price of an article is cut $10 \\%.$ To restore it to its former value, the new price must be increased by:\n$\\textbf{(A) \\ } 10 \\% \\qquad\\textbf{(B) \\ } 9 \\% \\qquad \\textbf{(C) \\ } 11\\frac{1}{9} \\% \\qquad\\textbf{(D) \\ } 11 \\% \\qquad\\textbf{(E) \\ } \\text{none of these answers}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For any real number a and positive integer k, define\n${a \\choose k} = \\frac{a(a-1)(a-2)\\cdots(a-(k-1))}{k(k-1)(k-2)\\cdots(2)(1)}$\nWhat is\n${-\\frac{1}{2} \\choose 100} \\div {\\frac{1}{2} \\choose 100}$?\n$\\textbf{(A)}\\ -199\\qquad \\textbf{(B)}\\ -197\\qquad \\textbf{(C)}\\ -1\\qquad \\textbf{(D)}\\ 197\\qquad \\textbf{(E)}\\ 199$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1094", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For any real number a and positive integer k, define\n${a \\choose k} = \\frac{a(a-1)(a-2)\\cdots(a-(k-1))}{k(k-1)(k-2)\\cdots(2)(1)}$\nWhat is\n${-\\frac{1}{2} \\choose 100} \\div {\\frac{1}{2} \\choose 100}$?\n$\\textbf{(A)}\\ -199\\qquad \\textbf{(B)}\\ -197\\qquad \\textbf{(C)}\\ -1\\qquad \\textbf{(D)}\\ 197\\qquad \\textbf{(E)}\\ 199$" + } + }, + { + "question": "Return your final response within \\boxed{}. Josanna's test scores to date are $90, 80, 70, 60,$ and $85.$ Her goal is to raise her test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal?\n$\\textbf{(A)}\\ 80 \\qquad \\textbf{(B)}\\ 82 \\qquad \\textbf{(C)}\\ 85 \\qquad \\textbf{(D)}\\ 90 \\qquad \\textbf{(E)}\\ 95$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1095", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Josanna's test scores to date are $90, 80, 70, 60,$ and $85.$ Her goal is to raise her test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal?\n$\\textbf{(A)}\\ 80 \\qquad \\textbf{(B)}\\ 82 \\qquad \\textbf{(C)}\\ 85 \\qquad \\textbf{(D)}\\ 90 \\qquad \\textbf{(E)}\\ 95$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:\n$\\textbf{(A)}\\ x^2-bx-ac=0\\qquad \\textbf{(B)}\\ x^2-bx+ac=0 \\qquad\\\\ \\textbf{(C)}\\ x^2+3bx+ca+2b^2=0 \\qquad \\textbf{(D)}\\ x^2+3bx-ca+2b^2=0 \\qquad\\\\ \\textbf{(E)}\\ x^2+bx(2-a)+a^2c+b^2(a+1)=0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1096", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:\n$\\textbf{(A)}\\ x^2-bx-ac=0\\qquad \\textbf{(B)}\\ x^2-bx+ac=0 \\qquad\\\\ \\textbf{(C)}\\ x^2+3bx+ca+2b^2=0 \\qquad \\textbf{(D)}\\ x^2+3bx-ca+2b^2=0 \\qquad\\\\ \\textbf{(E)}\\ x^2+bx(2-a)+a^2c+b^2(a+1)=0$" + } + }, + { + "question": "Return your final response within \\boxed{}. A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed \"9\" and you pressed [+1], it would display \"10.\" If you then pressed [x2], it would display \"20.\" Starting with the display \"1,\" what is the fewest number of keystrokes you would need to reach \"200\"?\n$\\textbf{(A)}\\ 8\\qquad\\textbf{(B)}\\ 9\\qquad\\textbf{(C)}\\ 10\\qquad\\textbf{(D)}\\ 11\\qquad\\textbf{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1097", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed \"9\" and you pressed [+1], it would display \"10.\" If you then pressed [x2], it would display \"20.\" Starting with the display \"1,\" what is the fewest number of keystrokes you would need to reach \"200\"?\n$\\textbf{(A)}\\ 8\\qquad\\textbf{(B)}\\ 9\\qquad\\textbf{(C)}\\ 10\\qquad\\textbf{(D)}\\ 11\\qquad\\textbf{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \\tfrac{1}{2}$?\n$\\textbf{(A) } \\frac{1}{3} \\qquad \\textbf{(B) } \\frac{7}{16} \\qquad \\textbf{(C) } \\frac{1}{2} \\qquad \\textbf{(D) } \\frac{9}{16} \\qquad \\textbf{(E) } \\frac{2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1098", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \\tfrac{1}{2}$?\n$\\textbf{(A) } \\frac{1}{3} \\qquad \\textbf{(B) } \\frac{7}{16} \\qquad \\textbf{(C) } \\frac{1}{2} \\qquad \\textbf{(D) } \\frac{9}{16} \\qquad \\textbf{(E) } \\frac{2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\\frac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. Then $r$ is:\n$\\text{(A) } 9 \\quad \\text{(B) } 10 \\quad \\text{(C) } 10\\frac{1}{2} \\quad \\text{(D) } 11 \\quad \\text{(E) } 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1099", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\\frac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. Then $r$ is:\n$\\text{(A) } 9 \\quad \\text{(B) } 10 \\quad \\text{(C) } 10\\frac{1}{2} \\quad \\text{(D) } 11 \\quad \\text{(E) } 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur within the group?\n$\\textbf{(A)}\\ 240\\qquad\\textbf{(B)}\\ 245\\qquad\\textbf{(C)}\\ 290\\qquad\\textbf{(D)}\\ 480\\qquad\\textbf{(E)}\\ 490$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1100", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur within the group?\n$\\textbf{(A)}\\ 240\\qquad\\textbf{(B)}\\ 245\\qquad\\textbf{(C)}\\ 290\\qquad\\textbf{(D)}\\ 480\\qquad\\textbf{(E)}\\ 490$" + } + }, + { + "question": "Return your final response within \\boxed{}. Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? \n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 7\\qquad\\textbf{(D)}\\ 11\\qquad\\textbf{(E)}\\ 14$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1101", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? \n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 7\\qquad\\textbf{(D)}\\ 11\\qquad\\textbf{(E)}\\ 14$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be the least positive integer greater than $1000$ for which\n\\[\\gcd(63, n+120) =21\\quad \\text{and} \\quad \\gcd(n+63, 120)=60.\\]\nWhat is the sum of the digits of $n$?\n$\\textbf{(A) } 12 \\qquad\\textbf{(B) } 15 \\qquad\\textbf{(C) } 18 \\qquad\\textbf{(D) } 21\\qquad\\textbf{(E) } 24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1102", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $n$ be the least positive integer greater than $1000$ for which\n\\[\\gcd(63, n+120) =21\\quad \\text{and} \\quad \\gcd(n+63, 120)=60.\\]\nWhat is the sum of the digits of $n$?\n$\\textbf{(A) } 12 \\qquad\\textbf{(B) } 15 \\qquad\\textbf{(C) } 18 \\qquad\\textbf{(D) } 21\\qquad\\textbf{(E) } 24$" + } + }, + { + "question": "Return your final response within \\boxed{}. LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally?\n$\\textbf{(A)}\\ \\frac{A + B}{2} \\qquad\\textbf{(B)}\\ \\dfrac{A - B}{2}\\qquad\\textbf{(C)}\\ \\dfrac{B - A}{2}\\qquad\\textbf{(D)}\\ B - A \\qquad\\textbf{(E)}\\ A + B$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1103", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally?\n$\\textbf{(A)}\\ \\frac{A + B}{2} \\qquad\\textbf{(B)}\\ \\dfrac{A - B}{2}\\qquad\\textbf{(C)}\\ \\dfrac{B - A}{2}\\qquad\\textbf{(D)}\\ B - A \\qquad\\textbf{(E)}\\ A + B$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the five \"T-like shapes\" would be [symmetric](https://artofproblemsolving.com/wiki/index.php/Symmetric) to the one shown with respect to the dashed line?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1104", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the five \"T-like shapes\" would be [symmetric](https://artofproblemsolving.com/wiki/index.php/Symmetric) to the one shown with respect to the dashed line?" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the statements:\n$\\textbf{(1)}\\ p\\text{ }\\wedge\\sim q\\wedge r\\qquad\\textbf{(2)}\\ \\sim p\\text{ }\\wedge\\sim q\\wedge r\\qquad\\textbf{(3)}\\ p\\text{ }\\wedge\\sim q\\text{ }\\wedge\\sim r\\qquad\\textbf{(4)}\\ \\sim p\\text{ }\\wedge q\\text{ }\\wedge r$\nwhere $p,q$, and $r$ are propositions. How many of these imply the truth of $(p\\rightarrow q)\\rightarrow r$?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1105", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the statements:\n$\\textbf{(1)}\\ p\\text{ }\\wedge\\sim q\\wedge r\\qquad\\textbf{(2)}\\ \\sim p\\text{ }\\wedge\\sim q\\wedge r\\qquad\\textbf{(3)}\\ p\\text{ }\\wedge\\sim q\\text{ }\\wedge\\sim r\\qquad\\textbf{(4)}\\ \\sim p\\text{ }\\wedge q\\text{ }\\wedge r$\nwhere $p,q$, and $r$ are propositions. How many of these imply the truth of $(p\\rightarrow q)\\rightarrow r$?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter\n$AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto\nlines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle\nformed by lines $PQ$ and $RS$ is half the size of $\\angle XOZ$, where\n$O$ is the midpoint of segment $AB$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\frac{1}{2} \\angle XOZ", + "index": "Sky-T1_10k_1106", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter\n$AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto\nlines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle\nformed by lines $PQ$ and $RS$ is half the size of $\\angle XOZ$, where\n$O$ is the midpoint of segment $AB$." + } + }, + { + "question": "Return your final response within \\boxed{}. Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?\n$\\mathrm{(A)}\\ 10\\qquad \\mathrm{(B)}\\ 12\\qquad \\mathrm{(C)}\\ 16\\qquad \\mathrm{(D)}\\ 18\\qquad \\mathrm{(E)}\\ 24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1107", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?\n$\\mathrm{(A)}\\ 10\\qquad \\mathrm{(B)}\\ 12\\qquad \\mathrm{(C)}\\ 16\\qquad \\mathrm{(D)}\\ 18\\qquad \\mathrm{(E)}\\ 24$" + } + }, + { + "question": "Return your final response within \\boxed{}. A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?\n\\[\\begingroup \\setlength{\\tabcolsep}{10pt} \\renewcommand{\\arraystretch}{1.5} \\begin{tabular}{|c|c|} \\hline Tree 1 & \\rule{0.4cm}{0.15mm} meters \\\\ Tree 2 & 11 meters \\\\ Tree 3 & \\rule{0.5cm}{0.15mm} meters \\\\ Tree 4 & \\rule{0.5cm}{0.15mm} meters \\\\ Tree 5 & \\rule{0.5cm}{0.15mm} meters \\\\ \\hline Average height & \\rule{0.5cm}{0.15mm}\\text{ .}2 meters \\\\ \\hline \\end{tabular} \\endgroup\\]\n$\\textbf{(A) }22.2 \\qquad \\textbf{(B) }24.2 \\qquad \\textbf{(C) }33.2 \\qquad \\textbf{(D) }35.2 \\qquad \\textbf{(E) }37.2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1108", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?\n\\[\\begingroup \\setlength{\\tabcolsep}{10pt} \\renewcommand{\\arraystretch}{1.5} \\begin{tabular}{|c|c|} \\hline Tree 1 & \\rule{0.4cm}{0.15mm} meters \\\\ Tree 2 & 11 meters \\\\ Tree 3 & \\rule{0.5cm}{0.15mm} meters \\\\ Tree 4 & \\rule{0.5cm}{0.15mm} meters \\\\ Tree 5 & \\rule{0.5cm}{0.15mm} meters \\\\ \\hline Average height & \\rule{0.5cm}{0.15mm}\\text{ .}2 meters \\\\ \\hline \\end{tabular} \\endgroup\\]\n$\\textbf{(A) }22.2 \\qquad \\textbf{(B) }24.2 \\qquad \\textbf{(C) }33.2 \\qquad \\textbf{(D) }35.2 \\qquad \\textbf{(E) }37.2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $$$1$ more than a pink pill, and Al's pills cost a total of $\\textdollar 546$ for the two weeks. How much does one green pill cost?\n$\\textbf{(A)}\\ \\textdollar 7 \\qquad\\textbf{(B) }\\textdollar 14 \\qquad\\textbf{(C) }\\textdollar 19\\qquad\\textbf{(D) }\\textdollar 20\\qquad\\textbf{(E) }\\textdollar 39$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1109", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $$$1$ more than a pink pill, and Al's pills cost a total of $\\textdollar 546$ for the two weeks. How much does one green pill cost?\n$\\textbf{(A)}\\ \\textdollar 7 \\qquad\\textbf{(B) }\\textdollar 14 \\qquad\\textbf{(C) }\\textdollar 19\\qquad\\textbf{(D) }\\textdollar 20\\qquad\\textbf{(E) }\\textdollar 39$" + } + }, + { + "question": "Return your final response within \\boxed{}. The simplest form of $1 - \\frac{1}{1 + \\frac{a}{1 - a}}$ is: \n$\\textbf{(A)}\\ {a}\\text{ if }{a\\not= 0} \\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ {a}\\text{ if }{a\\not=-1}\\qquad \\textbf{(D)}\\ {1-a}\\text{ with not restriction on }{a}\\qquad \\textbf{(E)}\\ {a}\\text{ if }{a\\not= 1}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1110", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The simplest form of $1 - \\frac{1}{1 + \\frac{a}{1 - a}}$ is: \n$\\textbf{(A)}\\ {a}\\text{ if }{a\\not= 0} \\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ {a}\\text{ if }{a\\not=-1}\\qquad \\textbf{(D)}\\ {1-a}\\text{ with not restriction on }{a}\\qquad \\textbf{(E)}\\ {a}\\text{ if }{a\\not= 1}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?\n$\\textbf{(A)}\\hspace{.05in}6\\qquad\\textbf{(B)}\\hspace{.05in}8\\qquad\\textbf{(C)}\\hspace{.05in}9\\qquad\\textbf{(D)}\\hspace{.05in}10\\qquad\\textbf{(E)}\\hspace{.05in}12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1111", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?\n$\\textbf{(A)}\\hspace{.05in}6\\qquad\\textbf{(B)}\\hspace{.05in}8\\qquad\\textbf{(C)}\\hspace{.05in}9\\qquad\\textbf{(D)}\\hspace{.05in}10\\qquad\\textbf{(E)}\\hspace{.05in}12$" + } + }, + { + "question": "Return your final response within \\boxed{}. The slope of the line $\\frac{x}{3} + \\frac{y}{2} = 1$ is\n$\\textbf{(A)}\\ -\\frac{3}{2}\\qquad \\textbf{(B)}\\ -\\frac{2}{3}\\qquad \\textbf{(C)}\\ \\frac{1}{3}\\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ \\frac{3}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1112", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The slope of the line $\\frac{x}{3} + \\frac{y}{2} = 1$ is\n$\\textbf{(A)}\\ -\\frac{3}{2}\\qquad \\textbf{(B)}\\ -\\frac{2}{3}\\qquad \\textbf{(C)}\\ \\frac{1}{3}\\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ \\frac{3}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The statement is proven.", + "index": "Sky-T1_10k_1113", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a convex quadrilateral such that diagonals $AC$ and $BD$ intersect at right angles, and let $E$ be their intersection. Prove that the reflections of $E$ across $AB$, $BC$, $CD$, $DA$ are concyclic.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The reflections of E across AB, BC, CD, DA are concyclic.", + "index": "Sky-T1_10k_1114", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a convex quadrilateral such that diagonals $AC$ and $BD$ intersect at right angles, and let $E$ be their intersection. Prove that the reflections of $E$ across $AB$, $BC$, $CD$, $DA$ are concyclic." + } + }, + { + "question": "Return your final response within \\boxed{}. Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "800", + "index": "Sky-T1_10k_1115", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$." + } + }, + { + "question": "Return your final response within \\boxed{}. Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?\n$\\textbf{(A) } \\frac{7}{36} \\qquad\\textbf{(B) } \\frac{5}{24} \\qquad\\textbf{(C) } \\frac{2}{9} \\qquad\\textbf{(D) } \\frac{17}{72} \\qquad\\textbf{(E) } \\frac{1}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1116", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?\n$\\textbf{(A) } \\frac{7}{36} \\qquad\\textbf{(B) } \\frac{5}{24} \\qquad\\textbf{(C) } \\frac{2}{9} \\qquad\\textbf{(D) } \\frac{17}{72} \\qquad\\textbf{(E) } \\frac{1}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\\frac{3}{4}$ of its former rate and arrives $3\\tfrac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3$ hours late. The length of the trip in miles was: \n$\\textbf{(A)}\\ 400 \\qquad \\textbf{(B)}\\ 465 \\qquad \\textbf{(C)}\\ 600 \\qquad \\textbf{(D)}\\ 640 \\qquad \\textbf{(E)}\\ 550$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1117", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\\frac{3}{4}$ of its former rate and arrives $3\\tfrac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3$ hours late. The length of the trip in miles was: \n$\\textbf{(A)}\\ 400 \\qquad \\textbf{(B)}\\ 465 \\qquad \\textbf{(C)}\\ 600 \\qquad \\textbf{(D)}\\ 640 \\qquad \\textbf{(E)}\\ 550$" + } + }, + { + "question": "Return your final response within \\boxed{}. The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?\n$\\textbf {(A) } 1 \\qquad \\textbf {(B) } 2 \\qquad \\textbf {(C) } 3 \\qquad \\textbf {(D) } 4 \\qquad \\textbf {(E) } 5$\n.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1118", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?\n$\\textbf {(A) } 1 \\qquad \\textbf {(B) } 2 \\qquad \\textbf {(C) } 3 \\qquad \\textbf {(D) } 4 \\qquad \\textbf {(E) } 5$\n." + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose $a$ is $150\\%$ of $b$. What percent of $a$ is $3b$?\n$\\textbf{(A) } 50 \\qquad \\textbf{(B) } 66+\\frac{2}{3} \\qquad \\textbf{(C) } 150 \\qquad \\textbf{(D) } 200 \\qquad \\textbf{(E) } 450$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1119", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose $a$ is $150\\%$ of $b$. What percent of $a$ is $3b$?\n$\\textbf{(A) } 50 \\qquad \\textbf{(B) } 66+\\frac{2}{3} \\qquad \\textbf{(C) } 150 \\qquad \\textbf{(D) } 200 \\qquad \\textbf{(E) } 450$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of distinct pairs of [integers](https://artofproblemsolving.com/wiki/index.php/Integers) $(x, y)$ such that $00$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area 6, then $ab=$\n$\\mathrm{(A) \\ 3 } \\qquad \\mathrm{(B) \\ 6 } \\qquad \\mathrm{(C) \\ 12 } \\qquad \\mathrm{(D) \\ 108 } \\qquad \\mathrm{(E) \\ 432 }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1218", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area 6, then $ab=$\n$\\mathrm{(A) \\ 3 } \\qquad \\mathrm{(B) \\ 6 } \\qquad \\mathrm{(C) \\ 12 } \\qquad \\mathrm{(D) \\ 108 } \\qquad \\mathrm{(E) \\ 432 }$" + } + }, + { + "question": "Return your final response within \\boxed{}. The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that\n$\\textbf{(A) }\\textstyle -2-\\frac{4}{3}\\qquad \\textbf{(E) }x<-\\frac{4}{3}\\text{ or }x>\\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1219", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that\n$\\textbf{(A) }\\textstyle -2-\\frac{4}{3}\\qquad \\textbf{(E) }x<-\\frac{4}{3}\\text{ or }x>\\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A list of five positive integers has [mean](https://artofproblemsolving.com/wiki/index.php/Mean) $12$ and [range](https://artofproblemsolving.com/wiki/index.php/Range) $18$. The [mode](https://artofproblemsolving.com/wiki/index.php/Mode) and [median](https://artofproblemsolving.com/wiki/index.php/Median) are both $8$. How many different values are possible for the second largest element of the list? \n$\\mathrm{(A) \\ 4 } \\qquad \\mathrm{(B) \\ 6 } \\qquad \\mathrm{(C) \\ 8 } \\qquad \\mathrm{(D) \\ 10 } \\qquad \\mathrm{(E) \\ 12 }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1220", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A list of five positive integers has [mean](https://artofproblemsolving.com/wiki/index.php/Mean) $12$ and [range](https://artofproblemsolving.com/wiki/index.php/Range) $18$. The [mode](https://artofproblemsolving.com/wiki/index.php/Mode) and [median](https://artofproblemsolving.com/wiki/index.php/Median) are both $8$. How many different values are possible for the second largest element of the list? \n$\\mathrm{(A) \\ 4 } \\qquad \\mathrm{(B) \\ 6 } \\qquad \\mathrm{(C) \\ 8 } \\qquad \\mathrm{(D) \\ 10 } \\qquad \\mathrm{(E) \\ 12 }$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given points $P_1, P_2,\\cdots,P_7$ on a straight line, in the order stated (not necessarily evenly spaced). \nLet $P$ be an arbitrarily selected point on the line and let $s$ be the sum of the undirected lengths\n$PP_1, PP_2, \\cdots , PP_7$. Then $s$ is smallest if and only if the point $P$ is:\n$\\textbf{(A)}\\ \\text{midway between }P_1\\text{ and }P_7\\qquad \\\\ \\textbf{(B)}\\ \\text{midway between }P_2\\text{ and }P_6\\qquad \\\\ \\textbf{(C)}\\ \\text{midway between }P_3\\text{ and }P_5\\qquad \\\\ \\textbf{(D)}\\ \\text{at }P_4 \\qquad \\textbf{(E)}\\ \\text{at }P_1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1221", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given points $P_1, P_2,\\cdots,P_7$ on a straight line, in the order stated (not necessarily evenly spaced). \nLet $P$ be an arbitrarily selected point on the line and let $s$ be the sum of the undirected lengths\n$PP_1, PP_2, \\cdots , PP_7$. Then $s$ is smallest if and only if the point $P$ is:\n$\\textbf{(A)}\\ \\text{midway between }P_1\\text{ and }P_7\\qquad \\\\ \\textbf{(B)}\\ \\text{midway between }P_2\\text{ and }P_6\\qquad \\\\ \\textbf{(C)}\\ \\text{midway between }P_3\\text{ and }P_5\\qquad \\\\ \\textbf{(D)}\\ \\text{at }P_4 \\qquad \\textbf{(E)}\\ \\text{at }P_1$" + } + }, + { + "question": "Return your final response within \\boxed{}. The first four terms of an arithmetic sequence are $p$, $9$, $3p-q$, and $3p+q$. What is the $2010^\\text{th}$ term of this sequence?\n$\\textbf{(A)}\\ 8041 \\qquad \\textbf{(B)}\\ 8043 \\qquad \\textbf{(C)}\\ 8045 \\qquad \\textbf{(D)}\\ 8047 \\qquad \\textbf{(E)}\\ 8049$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1222", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The first four terms of an arithmetic sequence are $p$, $9$, $3p-q$, and $3p+q$. What is the $2010^\\text{th}$ term of this sequence?\n$\\textbf{(A)}\\ 8041 \\qquad \\textbf{(B)}\\ 8043 \\qquad \\textbf{(C)}\\ 8045 \\qquad \\textbf{(D)}\\ 8047 \\qquad \\textbf{(E)}\\ 8049$" + } + }, + { + "question": "Return your final response within \\boxed{}. The value of $[2 - 3(2 - 3)^{-1}]^{-1}$ is:\n$\\textbf{(A)}\\ 5\\qquad \\textbf{(B)}\\ -5\\qquad \\textbf{(C)}\\ \\frac{1}{5}\\qquad \\textbf{(D)}\\ -\\frac{1}{5}\\qquad \\textbf{(E)}\\ \\frac{5}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1223", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The value of $[2 - 3(2 - 3)^{-1}]^{-1}$ is:\n$\\textbf{(A)}\\ 5\\qquad \\textbf{(B)}\\ -5\\qquad \\textbf{(C)}\\ \\frac{1}{5}\\qquad \\textbf{(D)}\\ -\\frac{1}{5}\\qquad \\textbf{(E)}\\ \\frac{5}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1224", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $\\triangle{ABC}$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP=1$, $BP=\\sqrt{3}$, and $CP=2$. What is $s$?\n$\\textbf{(A) } 1+\\sqrt{2} \\qquad \\textbf{(B) } \\sqrt{7} \\qquad \\textbf{(C) } \\frac{8}{3} \\qquad \\textbf{(D) } \\sqrt{5+\\sqrt{5}} \\qquad \\textbf{(E) } 2\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1225", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $\\triangle{ABC}$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP=1$, $BP=\\sqrt{3}$, and $CP=2$. What is $s$?\n$\\textbf{(A) } 1+\\sqrt{2} \\qquad \\textbf{(B) } \\sqrt{7} \\qquad \\textbf{(C) } \\frac{8}{3} \\qquad \\textbf{(D) } \\sqrt{5+\\sqrt{5}} \\qquad \\textbf{(E) } 2\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\\theta$ be the acute angle formed by that side and the median. What is $\\sin{\\theta}$?\n$\\textbf{(A)}\\ \\frac{3}{10}\\qquad\\textbf{(B)}\\ \\frac{1}{3}\\qquad\\textbf{(C)}\\ \\frac{9}{20}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{9}{10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1226", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\\theta$ be the acute angle formed by that side and the median. What is $\\sin{\\theta}$?\n$\\textbf{(A)}\\ \\frac{3}{10}\\qquad\\textbf{(B)}\\ \\frac{1}{3}\\qquad\\textbf{(C)}\\ \\frac{9}{20}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{9}{10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Circles of diameter $1$ inch and $3$ inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? \n[2006amc10b04.gif](https://artofproblemsolving.com/wiki/index.php/File:2006amc10b04.gif)\n$\\textbf{(A) } 2\\qquad \\textbf{(B) } 3\\qquad \\textbf{(C) } 6\\qquad \\textbf{(D) } 8\\qquad \\textbf{(E) } 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1227", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Circles of diameter $1$ inch and $3$ inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? \n[2006amc10b04.gif](https://artofproblemsolving.com/wiki/index.php/File:2006amc10b04.gif)\n$\\textbf{(A) } 2\\qquad \\textbf{(B) } 3\\qquad \\textbf{(C) } 6\\qquad \\textbf{(D) } 8\\qquad \\textbf{(E) } 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Emerson? \n$\\textbf{(A)}\\ 6 \\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 12\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1228", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Emerson? \n$\\textbf{(A)}\\ 6 \\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 12\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tom's Hat Shoppe increased all original prices by $25\\%$. Now the shoppe is having a sale where all prices are $20\\%$ off these increased prices. Which statement best describes the sale price of an item?\n$\\text{(A)}\\ \\text{The sale price is }5\\% \\text{ higher than the original price.}$\n$\\text{(B)}\\ \\text{The sale price is higher than the original price, but by less than }5\\% .$\n$\\text{(C)}\\ \\text{The sale price is higher than the original price, but by more than }5\\% .$\n$\\text{(D)}\\ \\text{The sale price is lower than the original price.}$\n$\\text{(E)}\\ \\text{The sale price is the same as the original price.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1229", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tom's Hat Shoppe increased all original prices by $25\\%$. Now the shoppe is having a sale where all prices are $20\\%$ off these increased prices. Which statement best describes the sale price of an item?\n$\\text{(A)}\\ \\text{The sale price is }5\\% \\text{ higher than the original price.}$\n$\\text{(B)}\\ \\text{The sale price is higher than the original price, but by less than }5\\% .$\n$\\text{(C)}\\ \\text{The sale price is higher than the original price, but by more than }5\\% .$\n$\\text{(D)}\\ \\text{The sale price is lower than the original price.}$\n$\\text{(E)}\\ \\text{The sale price is the same as the original price.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is:\n$\\textbf{(A)}\\ \\frac{1}{4}\\qquad \\textbf{(B)}\\ \\frac{81}{256}\\qquad \\textbf{(C)}\\ \\frac{27}{64}\\qquad \\textbf{(D)}\\ \\frac{37}{64}\\qquad \\textbf{(E)}\\ \\frac{175}{256}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1230", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is:\n$\\textbf{(A)}\\ \\frac{1}{4}\\qquad \\textbf{(B)}\\ \\frac{81}{256}\\qquad \\textbf{(C)}\\ \\frac{27}{64}\\qquad \\textbf{(D)}\\ \\frac{37}{64}\\qquad \\textbf{(E)}\\ \\frac{175}{256}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of positive integers $n$ less than $2017$ such that \\[1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}+\\frac{n^6}{6!}\\] is an integer.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "134", + "index": "Sky-T1_10k_1231", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of positive integers $n$ less than $2017$ such that \\[1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}+\\frac{n^6}{6!}\\] is an integer." + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that the set $\\{1,2,\\cdots, 1998\\}$ has been partitioned into disjoint pairs $\\{a_i,b_i\\}$ ($1\\leq i\\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \\[|a_1-b_1|+|a_2-b_2|+\\cdots +|a_{999}-b_{999}|\\] ends in the digit $9$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "9", + "index": "Sky-T1_10k_1232", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that the set $\\{1,2,\\cdots, 1998\\}$ has been partitioned into disjoint pairs $\\{a_i,b_i\\}$ ($1\\leq i\\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \\[|a_1-b_1|+|a_2-b_2|+\\cdots +|a_{999}-b_{999}|\\] ends in the digit $9$." + } + }, + { + "question": "Return your final response within \\boxed{}. The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\\angle DBA = \\angle DCB$ and $\\angle ADB = \\angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?\n$\\textbf{(A)}\\ 210 \\qquad \\textbf{(B)}\\ 220 \\qquad \\textbf{(C)}\\ 230 \\qquad \\textbf{(D)}\\ 240 \\qquad \\textbf{(E)}\\ 250$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1233", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\\angle DBA = \\angle DCB$ and $\\angle ADB = \\angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?\n$\\textbf{(A)}\\ 210 \\qquad \\textbf{(B)}\\ 220 \\qquad \\textbf{(C)}\\ 230 \\qquad \\textbf{(D)}\\ 240 \\qquad \\textbf{(E)}\\ 250$" + } + }, + { + "question": "Return your final response within \\boxed{}. The value of $\\left(256\\right)^{.16}\\left(256\\right)^{.09}$ is:\n$\\textbf{(A)}\\ 4 \\qquad \\\\ \\textbf{(B)}\\ 16\\qquad \\\\ \\textbf{(C)}\\ 64\\qquad \\\\ \\textbf{(D)}\\ 256.25\\qquad \\\\ \\textbf{(E)}\\ -16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1234", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The value of $\\left(256\\right)^{.16}\\left(256\\right)^{.09}$ is:\n$\\textbf{(A)}\\ 4 \\qquad \\\\ \\textbf{(B)}\\ 16\\qquad \\\\ \\textbf{(C)}\\ 64\\qquad \\\\ \\textbf{(D)}\\ 256.25\\qquad \\\\ \\textbf{(E)}\\ -16$" + } + }, + { + "question": "Return your final response within \\boxed{}. A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?\n$\\textbf{(A)}\\ 17\\qquad\\textbf{(B)}\\ 19\\qquad\\textbf{(C)}\\ 21\\qquad\\textbf{(D)}\\ 22\\qquad\\textbf{(E)}\\ 23$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1235", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?\n$\\textbf{(A)}\\ 17\\qquad\\textbf{(B)}\\ 19\\qquad\\textbf{(C)}\\ 21\\qquad\\textbf{(D)}\\ 22\\qquad\\textbf{(E)}\\ 23$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\frac{2+3+4}{3}=\\frac{1990+1991+1992}{N}$, then $N=$\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 6 \\qquad \\text{(C)}\\ 1990 \\qquad \\text{(D)}\\ 1991 \\qquad \\text{(E)}\\ 1992$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1236", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\frac{2+3+4}{3}=\\frac{1990+1991+1992}{N}$, then $N=$\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 6 \\qquad \\text{(C)}\\ 1990 \\qquad \\text{(D)}\\ 1991 \\qquad \\text{(E)}\\ 1992$" + } + }, + { + "question": "Return your final response within \\boxed{}. A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$?\n$\\textbf{(A)}\\ \\frac{1}{4}\\qquad\\textbf{(B)}\\ \\frac{1}{3}\\qquad\\textbf{(C)}\\ \\frac{1}{2}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1237", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$?\n$\\textbf{(A)}\\ \\frac{1}{4}\\qquad\\textbf{(B)}\\ \\frac{1}{3}\\qquad\\textbf{(C)}\\ \\frac{1}{2}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. From a group of boys and girls, $15$ girls leave. There are then left two boys for each girl. After this $45$ boys leave. There are then $5$ girls for each boy. The number of girls in the beginning was:\n$\\textbf{(A)}\\ 40 \\qquad \\textbf{(B)}\\ 43 \\qquad \\textbf{(C)}\\ 29 \\qquad \\textbf{(D)}\\ 50 \\qquad \\textbf{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1238", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. From a group of boys and girls, $15$ girls leave. There are then left two boys for each girl. After this $45$ boys leave. There are then $5$ girls for each boy. The number of girls in the beginning was:\n$\\textbf{(A)}\\ 40 \\qquad \\textbf{(B)}\\ 43 \\qquad \\textbf{(C)}\\ 29 \\qquad \\textbf{(D)}\\ 50 \\qquad \\textbf{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is: \n$\\textbf{(A)}\\ - 16 \\qquad \\textbf{(B) }\\ - 4 \\qquad \\textbf{(C) }\\ 4 \\qquad \\textbf{(D) }\\ 8 \\qquad \\textbf{(E) }\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1239", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is: \n$\\textbf{(A)}\\ - 16 \\qquad \\textbf{(B) }\\ - 4 \\qquad \\textbf{(C) }\\ 4 \\qquad \\textbf{(D) }\\ 8 \\qquad \\textbf{(E) }\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "80", + "index": "Sky-T1_10k_1240", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$." + } + }, + { + "question": "Return your final response within \\boxed{}. Successive discounts of $10\\%$ and $20\\%$ are equivalent to a single discount of:\n$\\textbf{(A)}\\ 30\\%\\qquad\\textbf{(B)}\\ 15\\%\\qquad\\textbf{(C)}\\ 72\\%\\qquad\\textbf{(D)}\\ 28\\%\\qquad\\textbf{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1241", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Successive discounts of $10\\%$ and $20\\%$ are equivalent to a single discount of:\n$\\textbf{(A)}\\ 30\\%\\qquad\\textbf{(B)}\\ 15\\%\\qquad\\textbf{(C)}\\ 72\\%\\qquad\\textbf{(D)}\\ 28\\%\\qquad\\textbf{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$, and that $65\\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?\n$\\textbf{(A) } 200 \\qquad \\textbf{(B) } 300 \\qquad \\textbf{(C) } 400 \\qquad \\textbf{(D) } 500 \\qquad \\textbf{(E) } 600$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1242", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$, and that $65\\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?\n$\\textbf{(A) } 200 \\qquad \\textbf{(B) } 300 \\qquad \\textbf{(C) } 400 \\qquad \\textbf{(D) } 500 \\qquad \\textbf{(E) } 600$" + } + }, + { + "question": "Return your final response within \\boxed{}. Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. Length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches. How high is the table?\n\n$\\textbf{(A) }28\\text{ inches}\\qquad\\textbf{(B) }29\\text{ inches}\\qquad\\textbf{(C) }30\\text{ inches}\\qquad\\textbf{(D) }31\\text{ inches}\\qquad\\textbf{(E) }32\\text{ inches}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1243", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. Length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches. How high is the table?\n\n$\\textbf{(A) }28\\text{ inches}\\qquad\\textbf{(B) }29\\text{ inches}\\qquad\\textbf{(C) }30\\text{ inches}\\qquad\\textbf{(D) }31\\text{ inches}\\qquad\\textbf{(E) }32\\text{ inches}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let the set $S = \\{P_1, P_2, \\dots, P_{12}\\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called \"communal\" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "134", + "index": "Sky-T1_10k_1244", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let the set $S = \\{P_1, P_2, \\dots, P_{12}\\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called \"communal\" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)" + } + }, + { + "question": "Return your final response within \\boxed{}. The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\n$\\textbf{(A) } 20 \\qquad\\textbf{(B) } \\dfrac{360}{17} \\qquad\\textbf{(C) } \\dfrac{107}{5} \\qquad\\textbf{(D) } \\dfrac{43}{2} \\qquad\\textbf{(E) } \\dfrac{281}{13}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1245", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\n$\\textbf{(A) } 20 \\qquad\\textbf{(B) } \\dfrac{360}{17} \\qquad\\textbf{(C) } \\dfrac{107}{5} \\qquad\\textbf{(D) } \\dfrac{43}{2} \\qquad\\textbf{(E) } \\dfrac{281}{13}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Twenty percent less than 60 is one-third more than what number?\n$\\mathrm{(A)}\\ 16\\qquad \\mathrm{(B)}\\ 30\\qquad \\mathrm{(C)}\\ 32\\qquad \\mathrm{(D)}\\ 36\\qquad \\mathrm{(E)}\\ 48$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1246", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Twenty percent less than 60 is one-third more than what number?\n$\\mathrm{(A)}\\ 16\\qquad \\mathrm{(B)}\\ 30\\qquad \\mathrm{(C)}\\ 32\\qquad \\mathrm{(D)}\\ 36\\qquad \\mathrm{(E)}\\ 48$" + } + }, + { + "question": "Return your final response within \\boxed{}. Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?\n\n\n$\\textbf{(A)}\\ 10401 \\qquad\\textbf{(B)}\\ 19801 \\qquad\\textbf{(C)}\\ 20201 \\qquad\\textbf{(D)}\\ 39801 \\qquad\\textbf{(E)}\\ 40801$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1247", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?\n\n\n$\\textbf{(A)}\\ 10401 \\qquad\\textbf{(B)}\\ 19801 \\qquad\\textbf{(C)}\\ 20201 \\qquad\\textbf{(D)}\\ 39801 \\qquad\\textbf{(E)}\\ 40801$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\\underline{32d}$ in base $n$ equals $263$, and the value of the numeral $\\underline{324}$ in base $n$ equals the value of the numeral $\\underline{11d1}$ in base six. What is $n + d ?$\n$\\textbf{(A)} ~10 \\qquad\\textbf{(B)} ~11 \\qquad\\textbf{(C)} ~13 \\qquad\\textbf{(D)} ~15 \\qquad\\textbf{(E)} ~16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1248", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\\underline{32d}$ in base $n$ equals $263$, and the value of the numeral $\\underline{324}$ in base $n$ equals the value of the numeral $\\underline{11d1}$ in base six. What is $n + d ?$\n$\\textbf{(A)} ~10 \\qquad\\textbf{(B)} ~11 \\qquad\\textbf{(C)} ~13 \\qquad\\textbf{(D)} ~15 \\qquad\\textbf{(E)} ~16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$?\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1249", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$?\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, \\[f(14)=(1+2+7+14)\\div 14=\\frac{12}{7}\\]\nWhat is $f(768)-f(384)?$\n$\\textbf{(A)}\\ \\frac{1}{768} \\qquad\\textbf{(B)}\\ \\frac{1}{192} \\qquad\\textbf{(C)}\\ 1 \\qquad\\textbf{(D)}\\ \\frac{4}{3} \\qquad\\textbf{(E)}\\ \\frac{8}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1250", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, \\[f(14)=(1+2+7+14)\\div 14=\\frac{12}{7}\\]\nWhat is $f(768)-f(384)?$\n$\\textbf{(A)}\\ \\frac{1}{768} \\qquad\\textbf{(B)}\\ \\frac{1}{192} \\qquad\\textbf{(C)}\\ 1 \\qquad\\textbf{(D)}\\ \\frac{4}{3} \\qquad\\textbf{(E)}\\ \\frac{8}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$\n$\\text{(A) } 3\\quad \\text{(B) } 3a^2\\quad \\text{(C) } 27a^2\\quad \\text{(D) } 2a^{3b}\\quad \\text{(E) } 3a^{2b}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1251", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$\n$\\text{(A) } 3\\quad \\text{(B) } 3a^2\\quad \\text{(C) } 27a^2\\quad \\text{(D) } 2a^{3b}\\quad \\text{(E) } 3a^{2b}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Square corners, 5 units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is\n[asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy]\n$\\text{(A)}\\ 300 \\qquad \\text{(B)}\\ 500 \\qquad \\text{(C)}\\ 550 \\qquad \\text{(D)}\\ 600 \\qquad \\text{(E)}\\ 1000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1252", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Square corners, 5 units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is\n[asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy]\n$\\text{(A)}\\ 300 \\qquad \\text{(B)}\\ 500 \\qquad \\text{(C)}\\ 550 \\qquad \\text{(D)}\\ 600 \\qquad \\text{(E)}\\ 1000$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation\n\\[x(x-y)+y(y-z)+z(z-x) = 1?\\]\n$\\textbf{(A)} \\: x>y$ and $y=z$\n$\\textbf{(B)} \\: x=y-1$ and $y=z-1$\n$\\textbf{(C)} \\: x=z+1$ and $y=x+1$\n$\\textbf{(D)} \\: x=z$ and $y-1=x$\n$\\textbf{(E)} \\: x+y+z=1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1253", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation\n\\[x(x-y)+y(y-z)+z(z-x) = 1?\\]\n$\\textbf{(A)} \\: x>y$ and $y=z$\n$\\textbf{(B)} \\: x=y-1$ and $y=z-1$\n$\\textbf{(C)} \\: x=z+1$ and $y=x+1$\n$\\textbf{(D)} \\: x=z$ and $y-1=x$\n$\\textbf{(E)} \\: x+y+z=1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?\n$\\textbf{(A)}\\ 22 \\qquad \\textbf{(B)}\\ 23 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 25 \\qquad \\textbf{(E)}\\ 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1254", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?\n$\\textbf{(A)}\\ 22 \\qquad \\textbf{(B)}\\ 23 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 25 \\qquad \\textbf{(E)}\\ 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $C$ be the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $xy = 1$, and denote by $C^*$ the [reflection](https://artofproblemsolving.com/wiki/index.php/Reflection) of $C$ in the line $y = 2x$. Let the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) of $C^*$ be written in the form\n\\[12x^2 + bxy + cy^2 + d = 0.\\]\nFind the product $bc$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "84", + "index": "Sky-T1_10k_1255", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $C$ be the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $xy = 1$, and denote by $C^*$ the [reflection](https://artofproblemsolving.com/wiki/index.php/Reflection) of $C$ in the line $y = 2x$. Let the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) of $C^*$ be written in the form\n\\[12x^2 + bxy + cy^2 + d = 0.\\]\nFind the product $bc$." + } + }, + { + "question": "Return your final response within \\boxed{}. There is a unique positive integer $n$ such that\\[\\log_2{(\\log_{16}{n})} = \\log_4{(\\log_4{n})}.\\]What is the sum of the digits of $n?$\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 7 \\qquad \\textbf{(C) } 8 \\qquad \\textbf{(D) } 11 \\qquad \\textbf{(E) } 13$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1256", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There is a unique positive integer $n$ such that\\[\\log_2{(\\log_{16}{n})} = \\log_4{(\\log_4{n})}.\\]What is the sum of the digits of $n?$\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 7 \\qquad \\textbf{(C) } 8 \\qquad \\textbf{(D) } 11 \\qquad \\textbf{(E) } 13$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the assertion that for each positive integer $n \\ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of 4. Either prove the assertion or find (with proof) a counter-example.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "25", + "index": "Sky-T1_10k_1257", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the assertion that for each positive integer $n \\ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of 4. Either prove the assertion or find (with proof) a counter-example." + } + }, + { + "question": "Return your final response within \\boxed{}. An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?\n$\\textbf{(A)}\\ 190 \\qquad\\textbf{(B)}\\ 191 \\qquad\\textbf{(C)}\\ 192 \\qquad\\textbf{(D)}\\ 195 \\qquad\\textbf{(E)}\\ 196$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1258", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?\n$\\textbf{(A)}\\ 190 \\qquad\\textbf{(B)}\\ 191 \\qquad\\textbf{(C)}\\ 192 \\qquad\\textbf{(D)}\\ 195 \\qquad\\textbf{(E)}\\ 196$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x, y$, and $y-\\frac{1}{x}$ are not $0$, then\n$\\frac{x-\\frac{1}{y}}{y-\\frac{1}{x}}$ equals\n$\\mathrm{(A) \\ }1 \\qquad \\mathrm{(B) \\ }\\frac{x}{y} \\qquad \\mathrm{(C) \\ } \\frac{y}{x}\\qquad \\mathrm{(D) \\ }\\frac{x}{y}-\\frac{y}{x} \\qquad \\mathrm{(E) \\ } xy-\\frac{1}{xy}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1259", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x, y$, and $y-\\frac{1}{x}$ are not $0$, then\n$\\frac{x-\\frac{1}{y}}{y-\\frac{1}{x}}$ equals\n$\\mathrm{(A) \\ }1 \\qquad \\mathrm{(B) \\ }\\frac{x}{y} \\qquad \\mathrm{(C) \\ } \\frac{y}{x}\\qquad \\mathrm{(D) \\ }\\frac{x}{y}-\\frac{y}{x} \\qquad \\mathrm{(E) \\ } xy-\\frac{1}{xy}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,a\\neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\\{a,b,c\\}=\\{r,s\\}$.)\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } 6 \\qquad\\textbf{(E) } \\text{infinitely many}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1260", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,a\\neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\\{a,b,c\\}=\\{r,s\\}$.)\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } 6 \\qquad\\textbf{(E) } \\text{infinitely many}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$\n\n$\\textbf{(A) }1 \\qquad \\textbf{(B) }2 \\qquad \\textbf{(C) }3 \\qquad \\textbf{(D) }4 \\qquad \\textbf{(E) }5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1261", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$\n\n$\\textbf{(A) }1 \\qquad \\textbf{(B) }2 \\qquad \\textbf{(C) }3 \\qquad \\textbf{(D) }4 \\qquad \\textbf{(E) }5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The set of all real numbers $x$ for which\n\\[\\log_{2004}(\\log_{2003}(\\log_{2002}(\\log_{2001}{x})))\\]\nis defined is $\\{x\\mid x > c\\}$. What is the value of $c$?\n$\\textbf {(A) } 0\\qquad \\textbf {(B) }2001^{2002} \\qquad \\textbf {(C) }2002^{2003} \\qquad \\textbf {(D) }2003^{2004} \\qquad \\textbf {(E) }2001^{2002^{2003}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1262", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The set of all real numbers $x$ for which\n\\[\\log_{2004}(\\log_{2003}(\\log_{2002}(\\log_{2001}{x})))\\]\nis defined is $\\{x\\mid x > c\\}$. What is the value of $c$?\n$\\textbf {(A) } 0\\qquad \\textbf {(B) }2001^{2002} \\qquad \\textbf {(C) }2002^{2003} \\qquad \\textbf {(D) }2003^{2004} \\qquad \\textbf {(E) }2001^{2002^{2003}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $S=1!+2!+3!+\\cdots +99!$, then the units' digit in the value of S is:\n$\\text{(A) } 9\\quad \\text{(B) } 8\\quad \\text{(C) } 5\\quad \\text{(D) } 3\\quad \\text{(E) } 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1263", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $S=1!+2!+3!+\\cdots +99!$, then the units' digit in the value of S is:\n$\\text{(A) } 9\\quad \\text{(B) } 8\\quad \\text{(C) } 5\\quad \\text{(D) } 3\\quad \\text{(E) } 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. To the nearest thousandth, $\\log_{10}2$ is $.301$ and $\\log_{10}3$ is $.477$. \nWhich of the following is the best approximation of $\\log_5 10$?\n$\\textbf{(A) }\\frac{8}{7}\\qquad \\textbf{(B) }\\frac{9}{7}\\qquad \\textbf{(C) }\\frac{10}{7}\\qquad \\textbf{(D) }\\frac{11}{7}\\qquad \\textbf{(E) }\\frac{12}{7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1264", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. To the nearest thousandth, $\\log_{10}2$ is $.301$ and $\\log_{10}3$ is $.477$. \nWhich of the following is the best approximation of $\\log_5 10$?\n$\\textbf{(A) }\\frac{8}{7}\\qquad \\textbf{(B) }\\frac{9}{7}\\qquad \\textbf{(C) }\\frac{10}{7}\\qquad \\textbf{(D) }\\frac{11}{7}\\qquad \\textbf{(E) }\\frac{12}{7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\\{ 1, 2, 3, …, 30\\}$? (For example the set $\\{1, 17, 18, 19, 30\\}$ has $2$ pairs of consecutive integers.)\n$\\textbf{(A)}\\ \\frac{2}{3} \\qquad\\textbf{(B)}\\ \\frac{29}{36} \\qquad\\textbf{(C)}\\ \\frac{5}{6} \\qquad\\textbf{(D)}\\ \\frac{29}{30} \\qquad\\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1265", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\\{ 1, 2, 3, …, 30\\}$? (For example the set $\\{1, 17, 18, 19, 30\\}$ has $2$ pairs of consecutive integers.)\n$\\textbf{(A)}\\ \\frac{2}{3} \\qquad\\textbf{(B)}\\ \\frac{29}{36} \\qquad\\textbf{(C)}\\ \\frac{5}{6} \\qquad\\textbf{(D)}\\ \\frac{29}{30} \\qquad\\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. If it is known that $\\log_2(a)+\\log_2(b) \\ge 6$, then the least value that can be taken on by $a+b$ is:\n$\\text{(A) } 2\\sqrt{6}\\quad \\text{(B) } 6\\quad \\text{(C) } 8\\sqrt{2}\\quad \\text{(D) } 16\\quad \\text{(E) none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1266", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If it is known that $\\log_2(a)+\\log_2(b) \\ge 6$, then the least value that can be taken on by $a+b$ is:\n$\\text{(A) } 2\\sqrt{6}\\quad \\text{(B) } 6\\quad \\text{(C) } 8\\sqrt{2}\\quad \\text{(D) } 16\\quad \\text{(E) none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the sum of the mean, median, and mode of the numbers $2,3,0,3,1,4,0,3$? \n$\\textbf{(A)}\\ 6.5 \\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 7.5\\qquad\\textbf{(D)}\\ 8.5\\qquad\\textbf{(E)}\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1267", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the sum of the mean, median, and mode of the numbers $2,3,0,3,1,4,0,3$? \n$\\textbf{(A)}\\ 6.5 \\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 7.5\\qquad\\textbf{(D)}\\ 8.5\\qquad\\textbf{(E)}\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. When $(a-b)^n,n\\ge2,ab\\ne0$, is expanded by the binomial theorem, it is found that when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals:\n$\\text{(A) } \\tfrac{1}{2}k(k-1)\\quad \\text{(B) } \\tfrac{1}{2}k(k+1)\\quad \\text{(C) } 2k-1\\quad \\text{(D) } 2k\\quad \\text{(E) } 2k+1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1268", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When $(a-b)^n,n\\ge2,ab\\ne0$, is expanded by the binomial theorem, it is found that when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals:\n$\\text{(A) } \\tfrac{1}{2}k(k-1)\\quad \\text{(B) } \\tfrac{1}{2}k(k+1)\\quad \\text{(C) } 2k-1\\quad \\text{(D) } 2k\\quad \\text{(E) } 2k+1$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is:\n$\\text{(A) } 3 \\qquad \\text{(B) } 2 \\qquad \\text{(C) } \\frac{27}{19} \\qquad \\text{(D) } \\frac{13}{9} \\qquad \\text{(E) } \\frac{23}{38}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1269", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is:\n$\\text{(A) } 3 \\qquad \\text{(B) } 2 \\qquad \\text{(C) } \\frac{27}{19} \\qquad \\text{(D) } \\frac{13}{9} \\qquad \\text{(E) } \\frac{23}{38}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?\n$\\textbf{(A)}\\ \\frac{5}{4}\\qquad\\textbf{(B)}\\ \\frac{3}{2}\\qquad\\textbf{(C)}\\ \\frac{9}{5}\\qquad\\textbf{(D)}\\ 2 \\qquad\\textbf{(E)}\\ \\frac{5}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1270", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?\n$\\textbf{(A)}\\ \\frac{5}{4}\\qquad\\textbf{(B)}\\ \\frac{3}{2}\\qquad\\textbf{(C)}\\ \\frac{9}{5}\\qquad\\textbf{(D)}\\ 2 \\qquad\\textbf{(E)}\\ \\frac{5}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes?\n[asy] import graph; Label f; f.p=fontsize(6); xaxis(0,3.5,Ticks(f, 1.0)); yaxis(0,4.5,Ticks(f, 1.0)); draw((0,0)--(1,0)--(1,1)--(0,1)--(0,2)--(2,2)--(2,0)--(3,0)--(3,3)--(0,3)--(0,4)--(1.5,4),blue+linewidth(2)); arrow((2,4),dir(180),blue); [/asy]\n$\\text{(A)}\\ (35,44)\\qquad\\text{(B)}\\ (36,45)\\qquad\\text{(C)}\\ (37,45)\\qquad\\text{(D)}\\ (44,35)\\qquad\\text{(E)}\\ (45,36)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1271", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes?\n[asy] import graph; Label f; f.p=fontsize(6); xaxis(0,3.5,Ticks(f, 1.0)); yaxis(0,4.5,Ticks(f, 1.0)); draw((0,0)--(1,0)--(1,1)--(0,1)--(0,2)--(2,2)--(2,0)--(3,0)--(3,3)--(0,3)--(0,4)--(1.5,4),blue+linewidth(2)); arrow((2,4),dir(180),blue); [/asy]\n$\\text{(A)}\\ (35,44)\\qquad\\text{(B)}\\ (36,45)\\qquad\\text{(C)}\\ (37,45)\\qquad\\text{(D)}\\ (44,35)\\qquad\\text{(E)}\\ (45,36)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Square $S_{1}$ is $1\\times 1.$ For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$\n[AIME 1995 Problem 1.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1995_Problem_1.png)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "255", + "index": "Sky-T1_10k_1272", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Square $S_{1}$ is $1\\times 1.$ For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$\n[AIME 1995 Problem 1.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1995_Problem_1.png)" + } + }, + { + "question": "Return your final response within \\boxed{}. Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?\n$\\mathrm{(A)}\\frac {1}{16}\\qquad \\mathrm{(B)}\\frac 18\\qquad \\mathrm{(C)}\\frac {3}{16}\\qquad \\mathrm{(D)}\\frac 14\\qquad \\mathrm{(E)}\\frac {5}{16}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1273", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?\n$\\mathrm{(A)}\\frac {1}{16}\\qquad \\mathrm{(B)}\\frac 18\\qquad \\mathrm{(C)}\\frac {3}{16}\\qquad \\mathrm{(D)}\\frac 14\\qquad \\mathrm{(E)}\\frac {5}{16}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,1)$ to $(2,1)$. What is the probability that circles $A$ and $B$ intersect?\n$\\textbf{(A)} \\; \\frac {2 + \\sqrt {2}}{4} \\qquad \\textbf{(B)} \\; \\frac {3\\sqrt {3} + 2}{8} \\qquad \\textbf{(C)} \\; \\frac {2 \\sqrt {2} - 1}{2} \\qquad \\textbf{(D)} \\; \\frac {2 + \\sqrt {3}}{4} \\qquad \\textbf{(E)} \\; \\frac {4 \\sqrt {3} - 3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1274", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,1)$ to $(2,1)$. What is the probability that circles $A$ and $B$ intersect?\n$\\textbf{(A)} \\; \\frac {2 + \\sqrt {2}}{4} \\qquad \\textbf{(B)} \\; \\frac {3\\sqrt {3} + 2}{8} \\qquad \\textbf{(C)} \\; \\frac {2 \\sqrt {2} - 1}{2} \\qquad \\textbf{(D)} \\; \\frac {2 + \\sqrt {3}}{4} \\qquad \\textbf{(E)} \\; \\frac {4 \\sqrt {3} - 3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$, $a$, $a$, $a$, $2a$, and $b$. What is the ratio of $b$ to $a$?\n$\\textbf{(A)}\\ \\sqrt{3}\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\sqrt{5}\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ \\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1275", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$, $a$, $a$, $a$, $2a$, and $b$. What is the ratio of $b$ to $a$?\n$\\textbf{(A)}\\ \\sqrt{3}\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\sqrt{5}\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ \\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. Eric plans to compete in a triathlon. He can average $2$ miles per hour in the $\\frac{1}{4}$-mile swim and $6$ miles per hour in the $3$-mile run. His goal is to finish the triathlon in $2$ hours. To accomplish his goal what must his average speed in miles per hour, be for the $15$-mile bicycle ride?\n$\\mathrm{(A)}\\ \\frac{120}{11} \\qquad \\mathrm{(B)}\\ 11 \\qquad \\mathrm{(C)}\\ \\frac{56}{5} \\qquad \\mathrm{(D)}\\ \\frac{45}{4} \\qquad \\mathrm{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1276", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Eric plans to compete in a triathlon. He can average $2$ miles per hour in the $\\frac{1}{4}$-mile swim and $6$ miles per hour in the $3$-mile run. His goal is to finish the triathlon in $2$ hours. To accomplish his goal what must his average speed in miles per hour, be for the $15$-mile bicycle ride?\n$\\mathrm{(A)}\\ \\frac{120}{11} \\qquad \\mathrm{(B)}\\ 11 \\qquad \\mathrm{(C)}\\ \\frac{56}{5} \\qquad \\mathrm{(D)}\\ \\frac{45}{4} \\qquad \\mathrm{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. According to the standard convention for exponentiation, \n\\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\\]\nIf the order in which the exponentiations are performed is changed, how many other values are possible?\n$\\textbf{(A) } 0\\qquad \\textbf{(B) } 1\\qquad \\textbf{(C) } 2\\qquad \\textbf{(D) } 3\\qquad \\textbf{(E) } 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1277", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. According to the standard convention for exponentiation, \n\\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\\]\nIf the order in which the exponentiations are performed is changed, how many other values are possible?\n$\\textbf{(A) } 0\\qquad \\textbf{(B) } 1\\qquad \\textbf{(C) } 2\\qquad \\textbf{(D) } 3\\qquad \\textbf{(E) } 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. A wooden [cube](https://artofproblemsolving.com/wiki/index.php/Cube) $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$? \n$(\\mathrm {A}) \\ 3 \\qquad (\\mathrm {B}) \\ 4 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 6 \\qquad (\\mathrm {E})\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1278", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A wooden [cube](https://artofproblemsolving.com/wiki/index.php/Cube) $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$? \n$(\\mathrm {A}) \\ 3 \\qquad (\\mathrm {B}) \\ 4 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 6 \\qquad (\\mathrm {E})\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?\n\n$\\textbf{(A)}\\ 204 \\qquad \\textbf{(B)}\\ 280 \\qquad \\textbf{(C)}\\ 320 \\qquad \\textbf{(D)}\\ 340 \\qquad \\textbf{(E)}\\ 600$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1279", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?\n\n$\\textbf{(A)}\\ 204 \\qquad \\textbf{(B)}\\ 280 \\qquad \\textbf{(C)}\\ 320 \\qquad \\textbf{(D)}\\ 340 \\qquad \\textbf{(E)}\\ 600$" + } + }, + { + "question": "Return your final response within \\boxed{}. Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, \"At least two of us have the same score.\" Marty thinks, \"I didn't get the lowest score.\" Shana thinks, \"I didn't get the highest score.\" List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S).\n$\\text{(A)}\\ \\text{S,Q,M} \\qquad \\text{(B)}\\ \\text{Q,M,S} \\qquad \\text{(C)}\\ \\text{Q,S,M} \\qquad \\text{(D)}\\ \\text{M,S,Q} \\qquad \\text{(E)}\\ \\text{S,M,Q}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1280", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, \"At least two of us have the same score.\" Marty thinks, \"I didn't get the lowest score.\" Shana thinks, \"I didn't get the highest score.\" List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S).\n$\\text{(A)}\\ \\text{S,Q,M} \\qquad \\text{(B)}\\ \\text{Q,M,S} \\qquad \\text{(C)}\\ \\text{Q,S,M} \\qquad \\text{(D)}\\ \\text{M,S,Q} \\qquad \\text{(E)}\\ \\text{S,M,Q}$" + } + }, + { + "question": "Return your final response within \\boxed{}. John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?\n$\\text {(A) } 30 \\qquad \\text {(B) } 50 \\qquad \\text {(C) } 60 \\qquad \\text {(D) } 90 \\qquad \\text {(E) } 120$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1281", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?\n$\\text {(A) } 30 \\qquad \\text {(B) } 50 \\qquad \\text {(C) } 60 \\qquad \\text {(D) } 90 \\qquad \\text {(E) } 120$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions \n(a) $0\\le Ay>0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?\n$\\textbf{(A) } \\text{Her estimate is larger than } x - y \\qquad \\textbf{(B) } \\text{Her estimate is smaller than } x - y \\qquad \\textbf{(C) } \\text{Her estimate equals } x - y$\n$\\textbf{(D) } \\text{Her estimate equals } y - x \\qquad \\textbf{(E) } \\text{Her estimate is } 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1302", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x>y>0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?\n$\\textbf{(A) } \\text{Her estimate is larger than } x - y \\qquad \\textbf{(B) } \\text{Her estimate is smaller than } x - y \\qquad \\textbf{(C) } \\text{Her estimate equals } x - y$\n$\\textbf{(D) } \\text{Her estimate equals } y - x \\qquad \\textbf{(E) } \\text{Her estimate is } 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?\n$\\textbf{(A) }2 \\qquad\\textbf{(B) } 5\\qquad\\textbf{(C) } 10\\qquad\\textbf{(D) } 25\\qquad\\textbf{(E) } 50$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1303", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?\n$\\textbf{(A) }2 \\qquad\\textbf{(B) } 5\\qquad\\textbf{(C) } 10\\qquad\\textbf{(D) } 25\\qquad\\textbf{(E) } 50$" + } + }, + { + "question": "Return your final response within \\boxed{}. A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required?\n$\\textbf{(A)} \\frac{1}{15} \\qquad \\textbf{(B)} \\frac{1}{10} \\qquad \\textbf{(C)} \\frac{1}{6} \\qquad \\textbf{(D)} \\frac{1}{5} \\qquad \\textbf{(E)} \\frac{1}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1304", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required?\n$\\textbf{(A)} \\frac{1}{15} \\qquad \\textbf{(B)} \\frac{1}{10} \\qquad \\textbf{(C)} \\frac{1}{6} \\qquad \\textbf{(D)} \\frac{1}{5} \\qquad \\textbf{(E)} \\frac{1}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube?\n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 27 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 125 \\qquad \\textbf{(E)}\\ 216$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1305", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube?\n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 27 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 125 \\qquad \\textbf{(E)}\\ 216$" + } + }, + { + "question": "Return your final response within \\boxed{}. A shopper buys a $100$ dollar coat on sale for $20\\%$ off. An additional $5$ dollars are taken off the sale price by using a discount coupon. A sales tax of $8\\%$ is paid on the final selling price. The total amount the shopper pays for the coat is\n$\\text{(A)}\\ \\text{81.00 dollars} \\qquad \\text{(B)}\\ \\text{81.40 dollars} \\qquad \\text{(C)}\\ \\text{82.00 dollars} \\qquad \\text{(D)}\\ \\text{82.08 dollars} \\qquad \\text{(E)}\\ \\text{82.40 dollars}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1306", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A shopper buys a $100$ dollar coat on sale for $20\\%$ off. An additional $5$ dollars are taken off the sale price by using a discount coupon. A sales tax of $8\\%$ is paid on the final selling price. The total amount the shopper pays for the coat is\n$\\text{(A)}\\ \\text{81.00 dollars} \\qquad \\text{(B)}\\ \\text{81.40 dollars} \\qquad \\text{(C)}\\ \\text{82.00 dollars} \\qquad \\text{(D)}\\ \\text{82.08 dollars} \\qquad \\text{(E)}\\ \\text{82.40 dollars}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For any positive integer $a, \\sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "125", + "index": "Sky-T1_10k_1307", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For any positive integer $a, \\sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$." + } + }, + { + "question": "Return your final response within \\boxed{}. Let N = $69^{5} + 5\\cdot69^{4} + 10\\cdot69^{3} + 10\\cdot69^{2} + 5\\cdot69 + 1$. How many positive integers are factors of $N$?\n$\\textbf{(A)}\\ 3\\qquad \\textbf{(B)}\\ 5\\qquad \\textbf{(C)}\\ 69\\qquad \\textbf{(D)}\\ 125\\qquad \\textbf{(E)}\\ 216$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1308", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let N = $69^{5} + 5\\cdot69^{4} + 10\\cdot69^{3} + 10\\cdot69^{2} + 5\\cdot69 + 1$. How many positive integers are factors of $N$?\n$\\textbf{(A)}\\ 3\\qquad \\textbf{(B)}\\ 5\\qquad \\textbf{(C)}\\ 69\\qquad \\textbf{(D)}\\ 125\\qquad \\textbf{(E)}\\ 216$" + } + }, + { + "question": "Return your final response within \\boxed{}. Positive integers $a$ and $b$ satisfy the condition\n\\[\\log_2(\\log_{2^a}(\\log_{2^b}(2^{1000}))) = 0.\\]\nFind the sum of all possible values of $a+b$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "881", + "index": "Sky-T1_10k_1309", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Positive integers $a$ and $b$ satisfy the condition\n\\[\\log_2(\\log_{2^a}(\\log_{2^b}(2^{1000}))) = 0.\\]\nFind the sum of all possible values of $a+b$." + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?\n$\\mathrm{(A)}\\ \\text{the empty set}\\qquad\\mathrm{(B)}\\ \\text{one point}\\qquad\\mathrm{(C)}\\ \\text{two lines}\\qquad\\mathrm{(D)}\\ \\text{a circle}\\qquad\\mathrm{(E)}\\ \\text{the entire plane}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1310", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?\n$\\mathrm{(A)}\\ \\text{the empty set}\\qquad\\mathrm{(B)}\\ \\text{one point}\\qquad\\mathrm{(C)}\\ \\text{two lines}\\qquad\\mathrm{(D)}\\ \\text{a circle}\\qquad\\mathrm{(E)}\\ \\text{the entire plane}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number halfway between $1/8$ and $1/10$ is \n$\\mathrm{(A) \\ } \\frac 1{80} \\qquad \\mathrm{(B) \\ } \\frac 1{40} \\qquad \\mathrm{(C) \\ } \\frac 1{18} \\qquad \\mathrm{(D) \\ } \\frac 1{9} \\qquad \\mathrm{(E) \\ } \\frac 9{80}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1311", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number halfway between $1/8$ and $1/10$ is \n$\\mathrm{(A) \\ } \\frac 1{80} \\qquad \\mathrm{(B) \\ } \\frac 1{40} \\qquad \\mathrm{(C) \\ } \\frac 1{18} \\qquad \\mathrm{(D) \\ } \\frac 1{9} \\qquad \\mathrm{(E) \\ } \\frac 9{80}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the ratio of $2x-y$ to $x+y$ is $\\frac{2}{3}$, what is the ratio of $x$ to $y$?\n$\\text{(A)} \\ \\frac{1}{5} \\qquad \\text{(B)} \\ \\frac{4}{5} \\qquad \\text{(C)} \\ 1 \\qquad \\text{(D)} \\ \\frac{6}{5} \\qquad \\text{(E)} \\ \\frac{5}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1312", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the ratio of $2x-y$ to $x+y$ is $\\frac{2}{3}$, what is the ratio of $x$ to $y$?\n$\\text{(A)} \\ \\frac{1}{5} \\qquad \\text{(B)} \\ \\frac{4}{5} \\qquad \\text{(C)} \\ 1 \\qquad \\text{(D)} \\ \\frac{6}{5} \\qquad \\text{(E)} \\ \\frac{5}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The numeral $47$ in base a represents the same number as $74$ in base $b$. Assuming that both bases are positive \nintegers, the least possible value of $a+b$ written as a Roman numeral, is\n$\\textbf{(A) }\\mathrm{XIII}\\qquad \\textbf{(B) }\\mathrm{XV}\\qquad \\textbf{(C) }\\mathrm{XXI}\\qquad \\textbf{(D) }\\mathrm{XXIV}\\qquad \\textbf{(E) }\\mathrm{XVI}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1313", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The numeral $47$ in base a represents the same number as $74$ in base $b$. Assuming that both bases are positive \nintegers, the least possible value of $a+b$ written as a Roman numeral, is\n$\\textbf{(A) }\\mathrm{XIII}\\qquad \\textbf{(B) }\\mathrm{XV}\\qquad \\textbf{(C) }\\mathrm{XXI}\\qquad \\textbf{(D) }\\mathrm{XXIV}\\qquad \\textbf{(E) }\\mathrm{XVI}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $(a_1, b_1), (a_2, b_2), \\ldots , (a_{100}, b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i, j)$ satisfying $1 \\le i < j \\le 100$ and $|a_ib_j - a_j b_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "197", + "index": "Sky-T1_10k_1314", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $(a_1, b_1), (a_2, b_2), \\ldots , (a_{100}, b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i, j)$ satisfying $1 \\le i < j \\le 100$ and $|a_ib_j - a_j b_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs." + } + }, + { + "question": "Return your final response within \\boxed{}. Let a binary operation $\\star$ on ordered pairs of integers be defined by $(a,b)\\star (c,d)=(a-c,b+d)$. Then, if $(3,3)\\star (0,0)$ and $(x,y)\\star (3,2)$ represent identical pairs, $x$ equals:\n$\\text{(A) } -3\\quad \\text{(B) } 0\\quad \\text{(C) } 2\\quad \\text{(D) } 3\\quad \\text{(E) } 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1315", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let a binary operation $\\star$ on ordered pairs of integers be defined by $(a,b)\\star (c,d)=(a-c,b+d)$. Then, if $(3,3)\\star (0,0)$ and $(x,y)\\star (3,2)$ represent identical pairs, $x$ equals:\n$\\text{(A) } -3\\quad \\text{(B) } 0\\quad \\text{(C) } 2\\quad \\text{(D) } 3\\quad \\text{(E) } 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\\tilde{p}(1)$?\n$\\textbf{(A) } \\dfrac{5}{16} \\qquad\\textbf{(B) } \\dfrac{1}{2} \\qquad\\textbf{(C) } \\dfrac{5}{8} \\qquad\\textbf{(D) } 1 \\qquad\\textbf{(E) } \\dfrac{9}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1316", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\\tilde{p}(1)$?\n$\\textbf{(A) } \\dfrac{5}{16} \\qquad\\textbf{(B) } \\dfrac{1}{2} \\qquad\\textbf{(C) } \\dfrac{5}{8} \\qquad\\textbf{(D) } 1 \\qquad\\textbf{(E) } \\dfrac{9}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?\n$\\text{(A)}\\ 48 \\qquad \\text{(B)}\\ 51 \\qquad \\text{(C)}\\ 52 \\qquad \\text{(D)}\\ 54 \\qquad \\text{(E)}\\ 55$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1317", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?\n$\\text{(A)}\\ 48 \\qquad \\text{(B)}\\ 51 \\qquad \\text{(C)}\\ 52 \\qquad \\text{(D)}\\ 54 \\qquad \\text{(E)}\\ 55$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $10^{\\log_{10}9} = 8x + 5$ then $x$ equals: \n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B) }\\ \\frac {1}{2} \\qquad \\textbf{(C) }\\ \\frac {5}{8} \\qquad \\textbf{(D) }\\ \\frac{9}{8}\\qquad \\textbf{(E) }\\ \\frac{2\\log_{10}3-5}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1318", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $10^{\\log_{10}9} = 8x + 5$ then $x$ equals: \n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B) }\\ \\frac {1}{2} \\qquad \\textbf{(C) }\\ \\frac {5}{8} \\qquad \\textbf{(D) }\\ \\frac{9}{8}\\qquad \\textbf{(E) }\\ \\frac{2\\log_{10}3-5}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the adjoining figure the five circles are tangent to one another consecutively and to the lines \n$L_1$ and $L_2$. \nIf the radius of the largest circle is $18$ and that of the smallest one is $8$, then the radius of the middle circle is\n\n$\\textbf{(A)} \\ 12 \\qquad \\textbf{(B)} \\ 12.5 \\qquad \\textbf{(C)} \\ 13 \\qquad \\textbf{(D)} \\ 13.5 \\qquad \\textbf{(E)} \\ 14$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1319", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the adjoining figure the five circles are tangent to one another consecutively and to the lines \n$L_1$ and $L_2$. \nIf the radius of the largest circle is $18$ and that of the smallest one is $8$, then the radius of the middle circle is\n\n$\\textbf{(A)} \\ 12 \\qquad \\textbf{(B)} \\ 12.5 \\qquad \\textbf{(C)} \\ 13 \\qquad \\textbf{(D)} \\ 13.5 \\qquad \\textbf{(E)} \\ 14$" + } + }, + { + "question": "Return your final response within \\boxed{}. When simplified and expressed with negative exponents, the expression $(x + y)^{ - 1}(x^{ - 1} + y^{ - 1})$ is equal to:\n$\\textbf{(A)}\\ x^{ - 2} + 2x^{ - 1}y^{ - 1} + y^{ - 2} \\qquad\\textbf{(B)}\\ x^{ - 2} + 2^{ - 1}x^{ - 1}y^{ - 1} + y^{ - 2} \\qquad\\textbf{(C)}\\ x^{ - 1}y^{ - 1}$\n$\\textbf{(D)}\\ x^{ - 2} + y^{ - 2} \\qquad\\textbf{(E)}\\ \\frac {1}{x^{ - 1}y^{ - 1}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1320", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When simplified and expressed with negative exponents, the expression $(x + y)^{ - 1}(x^{ - 1} + y^{ - 1})$ is equal to:\n$\\textbf{(A)}\\ x^{ - 2} + 2x^{ - 1}y^{ - 1} + y^{ - 2} \\qquad\\textbf{(B)}\\ x^{ - 2} + 2^{ - 1}x^{ - 1}y^{ - 1} + y^{ - 2} \\qquad\\textbf{(C)}\\ x^{ - 1}y^{ - 1}$\n$\\textbf{(D)}\\ x^{ - 2} + y^{ - 2} \\qquad\\textbf{(E)}\\ \\frac {1}{x^{ - 1}y^{ - 1}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A poll shows that $70\\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?\n$\\mathrm{(A)}\\ {{{0.063}}} \\qquad \\mathrm{(B)}\\ {{{0.189}}} \\qquad \\mathrm{(C)}\\ {{{0.233}}} \\qquad \\mathrm{(D)}\\ {{{0.333}}} \\qquad \\mathrm{(E)}\\ {{{0.441}}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1321", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A poll shows that $70\\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?\n$\\mathrm{(A)}\\ {{{0.063}}} \\qquad \\mathrm{(B)}\\ {{{0.189}}} \\qquad \\mathrm{(C)}\\ {{{0.233}}} \\qquad \\mathrm{(D)}\\ {{{0.333}}} \\qquad \\mathrm{(E)}\\ {{{0.441}}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the line $3x+5y=15$ and a point on this line equidistant from the coordinate axes. Such a point exists in:\n$\\textbf{(A)}\\ \\text{none of the quadrants}\\qquad\\textbf{(B)}\\ \\text{quadrant I only}\\qquad\\textbf{(C)}\\ \\text{quadrants I, II only}\\qquad$\n$\\textbf{(D)}\\ \\text{quadrants I, II, III only}\\qquad\\textbf{(E)}\\ \\text{each of the quadrants}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1322", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the line $3x+5y=15$ and a point on this line equidistant from the coordinate axes. Such a point exists in:\n$\\textbf{(A)}\\ \\text{none of the quadrants}\\qquad\\textbf{(B)}\\ \\text{quadrant I only}\\qquad\\textbf{(C)}\\ \\text{quadrants I, II only}\\qquad$\n$\\textbf{(D)}\\ \\text{quadrants I, II, III only}\\qquad\\textbf{(E)}\\ \\text{each of the quadrants}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?\n$\\textbf{(A) }1\\qquad\\textbf{(B) }2\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }4\\qquad \\textbf{(E) }5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1323", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?\n$\\textbf{(A) }1\\qquad\\textbf{(B) }2\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }4\\qquad \\textbf{(E) }5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 1 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 533$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1324", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 1 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 533$" + } + }, + { + "question": "Return your final response within \\boxed{}. Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees?\n$\\text{(A)}\\ 90 \\qquad \\text{(B)}\\ 100 \\qquad \\text{(C)}\\ 105 \\qquad \\text{(D)}\\ 120 \\qquad \\text{(E)}\\ 140$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1325", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees?\n$\\text{(A)}\\ 90 \\qquad \\text{(B)}\\ 100 \\qquad \\text{(C)}\\ 105 \\qquad \\text{(D)}\\ 120 \\qquad \\text{(E)}\\ 140$" + } + }, + { + "question": "Return your final response within \\boxed{}. Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?\n$\\textbf{(A) } 15\\qquad \\textbf{(B) } 34\\qquad \\textbf{(C) } 43\\qquad \\textbf{(D) } 51\\qquad \\textbf{(E) } 138$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1326", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?\n$\\textbf{(A) } 15\\qquad \\textbf{(B) } 34\\qquad \\textbf{(C) } 43\\qquad \\textbf{(D) } 51\\qquad \\textbf{(E) } 138$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S_n=1-2+3-4+\\cdots +(-1)^{n-1}n$, where $n=1,2,\\cdots$. Then $S_{17}+S_{33}+S_{50}$ equals:\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } -1\\quad \\text{(E) } -2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1327", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S_n=1-2+3-4+\\cdots +(-1)^{n-1}n$, where $n=1,2,\\cdots$. Then $S_{17}+S_{33}+S_{50}$ equals:\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } -1\\quad \\text{(E) } -2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\\overline{BD} \\perp \\overline{BC}$. The line $\\ell$ through $D$ parallel to line $BC$ intersects sides $\\overline{AB}$ and $\\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\\ell$ such that $F$ is between $E$ and $G$, $\\triangle AFG$ is isosceles, and the ratio of the area of $\\triangle AFG$ to the area of $\\triangle BED$ is $8:9$. Find $AF$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "336", + "index": "Sky-T1_10k_1328", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\\overline{BD} \\perp \\overline{BC}$. The line $\\ell$ through $D$ parallel to line $BC$ intersects sides $\\overline{AB}$ and $\\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\\ell$ such that $F$ is between $E$ and $G$, $\\triangle AFG$ is isosceles, and the ratio of the area of $\\triangle AFG$ to the area of $\\triangle BED$ is $8:9$. Find $AF$." + } + }, + { + "question": "Return your final response within \\boxed{}. What is the remainder when $3^0 + 3^1 + 3^2 + \\cdots + 3^{2009}$ is divided by 8?\n$\\mathrm{(A)}\\ 0\\qquad \\mathrm{(B)}\\ 1\\qquad \\mathrm{(C)}\\ 2\\qquad \\mathrm{(D)}\\ 4\\qquad \\mathrm{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1329", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the remainder when $3^0 + 3^1 + 3^2 + \\cdots + 3^{2009}$ is divided by 8?\n$\\mathrm{(A)}\\ 0\\qquad \\mathrm{(B)}\\ 1\\qquad \\mathrm{(C)}\\ 2\\qquad \\mathrm{(D)}\\ 4\\qquad \\mathrm{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle PAT,$ $\\angle P=36^{\\circ},$ $\\angle A=56^{\\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\\overline{TP}$ and $\\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\\overline{PA}$ and $\\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$\n$\\textbf{(A) } 76 \\qquad \\textbf{(B) } 77 \\qquad \\textbf{(C) } 78 \\qquad \\textbf{(D) } 79 \\qquad \\textbf{(E) } 80$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1330", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle PAT,$ $\\angle P=36^{\\circ},$ $\\angle A=56^{\\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\\overline{TP}$ and $\\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\\overline{PA}$ and $\\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$\n$\\textbf{(A) } 76 \\qquad \\textbf{(B) } 77 \\qquad \\textbf{(C) } 78 \\qquad \\textbf{(D) } 79 \\qquad \\textbf{(E) } 80$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$ with integer side lengths, $\\cos A = \\frac{11}{16}$, $\\cos B = \\frac{7}{8}$, and $\\cos C = -\\frac{1}{4}$. What is the least possible perimeter for $\\triangle ABC$?\n$\\textbf{(A) } 9 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 23 \\qquad \\textbf{(D) } 27 \\qquad \\textbf{(E) } 44$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1331", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$ with integer side lengths, $\\cos A = \\frac{11}{16}$, $\\cos B = \\frac{7}{8}$, and $\\cos C = -\\frac{1}{4}$. What is the least possible perimeter for $\\triangle ABC$?\n$\\textbf{(A) } 9 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 23 \\qquad \\textbf{(D) } 27 \\qquad \\textbf{(E) } 44$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 6 \\qquad \\textbf{(C) } 8 \\qquad \\textbf{(D) } 12 \\qquad \\textbf{(E) } 16 \\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1332", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 6 \\qquad \\textbf{(C) } 8 \\qquad \\textbf{(D) } 12 \\qquad \\textbf{(E) } 16 \\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $p$, $q$, and $M$ are positive numbers and $q<100$, then the number obtained by increasing $M$ by $p\\%$ and decreasing the result by $q\\%$ exceeds $M$ if and only if\n$\\textbf{(A)}\\ p>q \\qquad\\textbf{(B)}\\ p>\\dfrac{q}{100-q}\\qquad\\textbf{(C)}\\ p>\\dfrac{q}{1-q}\\qquad \\textbf{(D)}\\ p>\\dfrac{100q}{100+q}\\qquad\\textbf{(E)}\\ p>\\dfrac{100q}{100-q}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1333", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $p$, $q$, and $M$ are positive numbers and $q<100$, then the number obtained by increasing $M$ by $p\\%$ and decreasing the result by $q\\%$ exceeds $M$ if and only if\n$\\textbf{(A)}\\ p>q \\qquad\\textbf{(B)}\\ p>\\dfrac{q}{100-q}\\qquad\\textbf{(C)}\\ p>\\dfrac{q}{1-q}\\qquad \\textbf{(D)}\\ p>\\dfrac{100q}{100+q}\\qquad\\textbf{(E)}\\ p>\\dfrac{100q}{100-q}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? \n$\\mathrm{(A) \\ } 77\\qquad \\mathrm{(B) \\ } 91\\qquad \\mathrm{(C) \\ } 143\\qquad \\mathrm{(D) \\ } 182\\qquad \\mathrm{(E) \\ } 286$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1334", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? \n$\\mathrm{(A) \\ } 77\\qquad \\mathrm{(B) \\ } 91\\qquad \\mathrm{(C) \\ } 143\\qquad \\mathrm{(D) \\ } 182\\qquad \\mathrm{(E) \\ } 286$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the smallest prime number dividing the sum $3^{11}+5^{13}$?\n$\\mathrm{(A)\\ } 2 \\qquad \\mathrm{(B) \\ }3 \\qquad \\mathrm{(C) \\ } 5 \\qquad \\mathrm{(D) \\ } 3^{11}+5^{13} \\qquad \\mathrm{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1335", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the smallest prime number dividing the sum $3^{11}+5^{13}$?\n$\\mathrm{(A)\\ } 2 \\qquad \\mathrm{(B) \\ }3 \\qquad \\mathrm{(C) \\ } 5 \\qquad \\mathrm{(D) \\ } 3^{11}+5^{13} \\qquad \\mathrm{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $X_1, X_2, \\ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\\cap X_{i+1}=\\emptyset$ and $X_i\\cup X_{i+1}\\neq S$, for all $i\\in\\{1, \\ldots, 99\\}$. Find the smallest possible number of elements in $S$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "8", + "index": "Sky-T1_10k_1336", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $X_1, X_2, \\ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\\cap X_{i+1}=\\emptyset$ and $X_i\\cup X_{i+1}\\neq S$, for all $i\\in\\{1, \\ldots, 99\\}$. Find the smallest possible number of elements in $S$." + } + }, + { + "question": "Return your final response within \\boxed{}. Given the four equations:\n$\\textbf{(1)}\\ 3y-2x=12 \\qquad\\textbf{(2)}\\ -2x-3y=10 \\qquad\\textbf{(3)}\\ 3y+2x=12 \\qquad\\textbf{(4)}\\ 2y+3x=10$\nThe pair representing the perpendicular lines is:\n\n$\\textbf{(A)}\\ \\text{(1) and (4)}\\qquad \\textbf{(B)}\\ \\text{(1) and (3)}\\qquad \\textbf{(C)}\\ \\text{(1) and (2)}\\qquad \\textbf{(D)}\\ \\text{(2) and (4)}\\qquad \\textbf{(E)}\\ \\text{(2) and (3)}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1337", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the four equations:\n$\\textbf{(1)}\\ 3y-2x=12 \\qquad\\textbf{(2)}\\ -2x-3y=10 \\qquad\\textbf{(3)}\\ 3y+2x=12 \\qquad\\textbf{(4)}\\ 2y+3x=10$\nThe pair representing the perpendicular lines is:\n\n$\\textbf{(A)}\\ \\text{(1) and (4)}\\qquad \\textbf{(B)}\\ \\text{(1) and (3)}\\qquad \\textbf{(C)}\\ \\text{(1) and (2)}\\qquad \\textbf{(D)}\\ \\text{(2) and (4)}\\qquad \\textbf{(E)}\\ \\text{(2) and (3)}$" + } + }, + { + "question": "Return your final response within \\boxed{}. On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was 15 inches.\n$\\text{(A) } .33\\quad \\text{(B) } .34\\quad \\text{(C) } .35\\quad \\text{(D) } .38\\quad \\text{(E) } .66$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1338", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was 15 inches.\n$\\text{(A) } .33\\quad \\text{(B) } .34\\quad \\text{(C) } .35\\quad \\text{(D) } .38\\quad \\text{(E) } .66$" + } + }, + { + "question": "Return your final response within \\boxed{}. A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle?\n\n\n$\\textbf{(A)}\\ 2.18\\qquad\\textbf{(B)}\\ 2.24\\qquad\\textbf{(C)}\\ 2.31\\qquad\\textbf{(D)}\\ 2.37\\qquad\\textbf{(E)}\\ 2.41$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1339", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle?\n\n\n$\\textbf{(A)}\\ 2.18\\qquad\\textbf{(B)}\\ 2.24\\qquad\\textbf{(C)}\\ 2.31\\qquad\\textbf{(D)}\\ 2.37\\qquad\\textbf{(E)}\\ 2.41$" + } + }, + { + "question": "Return your final response within \\boxed{}. Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $$4$ per gallon. How many miles can Margie drive on $\\textdollar 20$ worth of gas?\n$\\textbf{(A) }64\\qquad\\textbf{(B) }128\\qquad\\textbf{(C) }160\\qquad\\textbf{(D) }320\\qquad \\textbf{(E) }640$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1340", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $$4$ per gallon. How many miles can Margie drive on $\\textdollar 20$ worth of gas?\n$\\textbf{(A) }64\\qquad\\textbf{(B) }128\\qquad\\textbf{(C) }160\\qquad\\textbf{(D) }320\\qquad \\textbf{(E) }640$" + } + }, + { + "question": "Return your final response within \\boxed{}. Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$?\n$\\textbf{(A) }-10\\qquad\\textbf{(B) }-6\\qquad\\textbf{(C) }0\\qquad\\textbf{(D) }6\\qquad \\textbf{(E) }10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1341", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$?\n$\\textbf{(A) }-10\\qquad\\textbf{(B) }-6\\qquad\\textbf{(C) }0\\qquad\\textbf{(D) }6\\qquad \\textbf{(E) }10$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is $\\frac{1}{n}$ , where n is a positive integer. Find n.\n[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100);[/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "429", + "index": "Sky-T1_10k_1342", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is $\\frac{1}{n}$ , where n is a positive integer. Find n.\n[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100);[/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. What is $100(100-3)-(100\\cdot100-3)$?\n$\\textbf{(A)}\\ -20,000 \\qquad \\textbf{(B)}\\ -10,000 \\qquad \\textbf{(C)}\\ -297 \\qquad \\textbf{(D)}\\ -6 \\qquad \\textbf{(E)}\\ 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1343", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is $100(100-3)-(100\\cdot100-3)$?\n$\\textbf{(A)}\\ -20,000 \\qquad \\textbf{(B)}\\ -10,000 \\qquad \\textbf{(C)}\\ -297 \\qquad \\textbf{(D)}\\ -6 \\qquad \\textbf{(E)}\\ 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two lines with slopes $\\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10  ?$\n$\\textbf{(A) } 4 \\qquad\\textbf{(B) } 4\\sqrt{2} \\qquad\\textbf{(C) } 6 \\qquad\\textbf{(D) } 8 \\qquad\\textbf{(E) } 6\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1344", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two lines with slopes $\\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10  ?$\n$\\textbf{(A) } 4 \\qquad\\textbf{(B) } 4\\sqrt{2} \\qquad\\textbf{(C) } 6 \\qquad\\textbf{(D) } 8 \\qquad\\textbf{(E) } 6\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?\n$\\mathrm{(A)}\\ 1/12\\qquad\\mathrm{(B)}\\ 1/10\\qquad\\mathrm{(C)}\\ 1/6\\qquad\\mathrm{(D)}\\ 1/3\\qquad\\mathrm{(E)}\\ 1/2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1345", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?\n$\\mathrm{(A)}\\ 1/12\\qquad\\mathrm{(B)}\\ 1/10\\qquad\\mathrm{(C)}\\ 1/6\\qquad\\mathrm{(D)}\\ 1/3\\qquad\\mathrm{(E)}\\ 1/2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then the total area of the shaded triangles is nearest\n\n$\\text{(A)}\\ 6 \\qquad \\text{(B)}\\ 7 \\qquad \\text{(C)}\\ 8 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1346", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then the total area of the shaded triangles is nearest\n\n$\\text{(A)}\\ 6 \\qquad \\text{(B)}\\ 7 \\qquad \\text{(C)}\\ 8 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the expression $\\begin{pmatrix}a & c\\\\ d & b\\end{pmatrix}$ has the value $ab-cd$ for all values of $a, b, c$ and $d$, then the equation $\\begin{pmatrix}2x & 1\\\\ x & x\\end{pmatrix}= 3$:\n$\\textbf{(A)}\\ \\text{Is satisfied for only 1 value of }x\\qquad\\\\ \\textbf{(B)}\\ \\text{Is satisified for only 2 values of }x\\qquad\\\\ \\textbf{(C)}\\ \\text{Is satisified for no values of }x\\qquad\\\\ \\textbf{(D)}\\ \\text{Is satisfied for an infinite number of values of }x\\qquad\\\\ \\textbf{(E)}\\ \\text{None of these.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1347", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the expression $\\begin{pmatrix}a & c\\\\ d & b\\end{pmatrix}$ has the value $ab-cd$ for all values of $a, b, c$ and $d$, then the equation $\\begin{pmatrix}2x & 1\\\\ x & x\\end{pmatrix}= 3$:\n$\\textbf{(A)}\\ \\text{Is satisfied for only 1 value of }x\\qquad\\\\ \\textbf{(B)}\\ \\text{Is satisified for only 2 values of }x\\qquad\\\\ \\textbf{(C)}\\ \\text{Is satisified for no values of }x\\qquad\\\\ \\textbf{(D)}\\ \\text{Is satisfied for an infinite number of values of }x\\qquad\\\\ \\textbf{(E)}\\ \\text{None of these.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\\textsc{am}$, and the sunset as $8:15\\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?\n$\\textbf{(A)}\\hspace{.05in}5:10\\textsc{pm}\\quad\\textbf{(B)}\\hspace{.05in}5:21\\textsc{pm}\\quad\\textbf{(C)}\\hspace{.05in}5:41\\textsc{pm}\\quad\\textbf{(D)}\\hspace{.05in}5:57\\textsc{pm}\\quad\\textbf{(E)}\\hspace{.05in}6:03\\textsc{pm}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1348", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\\textsc{am}$, and the sunset as $8:15\\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?\n$\\textbf{(A)}\\hspace{.05in}5:10\\textsc{pm}\\quad\\textbf{(B)}\\hspace{.05in}5:21\\textsc{pm}\\quad\\textbf{(C)}\\hspace{.05in}5:41\\textsc{pm}\\quad\\textbf{(D)}\\hspace{.05in}5:57\\textsc{pm}\\quad\\textbf{(E)}\\hspace{.05in}6:03\\textsc{pm}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Moe uses a mower to cut his rectangular $90$-foot by $150$-foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn?\n$\\textbf{(A) } 0.75 \\qquad\\textbf{(B) } 0.8 \\qquad\\textbf{(C) } 1.35 \\qquad\\textbf{(D) } 1.5 \\qquad\\textbf{(E) } 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1349", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Moe uses a mower to cut his rectangular $90$-foot by $150$-foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn?\n$\\textbf{(A) } 0.75 \\qquad\\textbf{(B) } 0.8 \\qquad\\textbf{(C) } 1.35 \\qquad\\textbf{(D) } 1.5 \\qquad\\textbf{(E) } 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?\n$\\textbf{(A)}\\hspace{.05in}3127\\qquad\\textbf{(B)}\\hspace{.05in}3133\\qquad\\textbf{(C)}\\hspace{.05in}3137\\qquad\\textbf{(D)}\\hspace{.05in}3139\\qquad\\textbf{(E)}\\hspace{.05in}3149$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1350", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?\n$\\textbf{(A)}\\hspace{.05in}3127\\qquad\\textbf{(B)}\\hspace{.05in}3133\\qquad\\textbf{(C)}\\hspace{.05in}3137\\qquad\\textbf{(D)}\\hspace{.05in}3139\\qquad\\textbf{(E)}\\hspace{.05in}3149$" + } + }, + { + "question": "Return your final response within \\boxed{}. The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome?\n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1351", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome?\n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. \"Look, daddy!\" she exclaims. \"That number is evenly divisible by the age of each of us kids!\" \"That's right,\" replies Mr. Jones, \"and the last two digits just happen to be my age.\" Which of the following is not the age of one of Mr. Jones's children? \n$\\mathrm{(A) \\ } 4\\qquad \\mathrm{(B) \\ } 5\\qquad \\mathrm{(C) \\ } 6\\qquad \\mathrm{(D) \\ } 7\\qquad \\mathrm{(E) \\ } 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1352", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. \"Look, daddy!\" she exclaims. \"That number is evenly divisible by the age of each of us kids!\" \"That's right,\" replies Mr. Jones, \"and the last two digits just happen to be my age.\" Which of the following is not the age of one of Mr. Jones's children? \n$\\mathrm{(A) \\ } 4\\qquad \\mathrm{(B) \\ } 5\\qquad \\mathrm{(C) \\ } 6\\qquad \\mathrm{(D) \\ } 7\\qquad \\mathrm{(E) \\ } 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Six pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?\n$\\textbf{(A)}\\ \\frac 12 \\qquad\\textbf{(B)}\\ \\frac 23 \\qquad\\textbf{(C)}\\ \\frac 34 \\qquad\\textbf{(D)}\\ \\frac 56 \\qquad\\textbf{(E)}\\ \\frac 78$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1353", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Six pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?\n$\\textbf{(A)}\\ \\frac 12 \\qquad\\textbf{(B)}\\ \\frac 23 \\qquad\\textbf{(C)}\\ \\frac 34 \\qquad\\textbf{(D)}\\ \\frac 56 \\qquad\\textbf{(E)}\\ \\frac 78$" + } + }, + { + "question": "Return your final response within \\boxed{}. A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:\n$\\textbf{(A)}\\ \\text{Least when the point is the center of gravity of the triangle}\\qquad\\\\ \\textbf{(B)}\\ \\text{Greater than the altitude of the triangle}\\qquad\\\\ \\textbf{(C)}\\ \\text{Equal to the altitude of the triangle}\\qquad\\\\ \\textbf{(D)}\\ \\text{One-half the sum of the sides of the triangle}\\qquad\\\\ \\textbf{(E)}\\ \\text{Greatest when the point is the center of gravity}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1354", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:\n$\\textbf{(A)}\\ \\text{Least when the point is the center of gravity of the triangle}\\qquad\\\\ \\textbf{(B)}\\ \\text{Greater than the altitude of the triangle}\\qquad\\\\ \\textbf{(C)}\\ \\text{Equal to the altitude of the triangle}\\qquad\\\\ \\textbf{(D)}\\ \\text{One-half the sum of the sides of the triangle}\\qquad\\\\ \\textbf{(E)}\\ \\text{Greatest when the point is the center of gravity}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $S = i^n + i^{-n}$, where $i = \\sqrt{-1}$ and $n$ is an integer, then the total number of possible distinct values for $S$ is: \n$\\textbf{(A)}\\ 1\\qquad \\textbf{(B)}\\ 2\\qquad \\textbf{(C)}\\ 3\\qquad \\textbf{(D)}\\ 4\\qquad \\textbf{(E)}\\ \\text{more than 4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1355", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $S = i^n + i^{-n}$, where $i = \\sqrt{-1}$ and $n$ is an integer, then the total number of possible distinct values for $S$ is: \n$\\textbf{(A)}\\ 1\\qquad \\textbf{(B)}\\ 2\\qquad \\textbf{(C)}\\ 3\\qquad \\textbf{(D)}\\ 4\\qquad \\textbf{(E)}\\ \\text{more than 4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ \\frac{1}{2} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{5}-1}{2} \\qquad \\textbf{(D)}\\ \\frac{\\sqrt{5}+1}{2} \\qquad \\textbf{(E)}\\ \\sqrt{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1356", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ \\frac{1}{2} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{5}-1}{2} \\qquad \\textbf{(D)}\\ \\frac{\\sqrt{5}+1}{2} \\qquad \\textbf{(E)}\\ \\sqrt{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $m$ and $n$ are the roots of $x^2+mx+n=0 ,m \\ne 0,n \\ne 0$, then the sum of the roots is:\n$\\text{(A) } -\\frac{1}{2}\\quad \\text{(B) } -1\\quad \\text{(C) } \\frac{1}{2}\\quad \\text{(D) } 1\\quad \\text{(E) } \\text{undetermined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1357", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $m$ and $n$ are the roots of $x^2+mx+n=0 ,m \\ne 0,n \\ne 0$, then the sum of the roots is:\n$\\text{(A) } -\\frac{1}{2}\\quad \\text{(B) } -1\\quad \\text{(C) } \\frac{1}{2}\\quad \\text{(D) } 1\\quad \\text{(E) } \\text{undetermined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For what real values of $K$ does $x = K^2 (x-1)(x-2)$ have real roots? \n$\\textbf{(A)}\\ \\text{none}\\qquad\\textbf{(B)}\\ -21\\text{ or }K<-2\\qquad\\textbf{(E)}\\ \\text{all}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1358", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For what real values of $K$ does $x = K^2 (x-1)(x-2)$ have real roots? \n$\\textbf{(A)}\\ \\text{none}\\qquad\\textbf{(B)}\\ -21\\text{ or }K<-2\\qquad\\textbf{(E)}\\ \\text{all}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 9\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1359", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 9\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $r$ be the distance from the origin to a point $P$ with coordinates $x$ and $y$. Designate the ratio $\\frac{y}{r}$ by $s$ and the ratio $\\frac{x}{r}$ by $c$. Then the values of $s^2 - c^2$ are limited to the numbers:\n$\\textbf{(A)}\\ \\text{less than }{-1}\\text{ are greater than }{+1}\\text{, both excluded}\\qquad\\\\ \\textbf{(B)}\\ \\text{less than }{-1}\\text{ are greater than }{+1}\\text{, both included}\\qquad \\\\ \\textbf{(C)}\\ \\text{between }{-1}\\text{ and }{+1}\\text{, both excluded}\\qquad \\\\ \\textbf{(D)}\\ \\text{between }{-1}\\text{ and }{+1}\\text{, both included}\\qquad \\\\ \\textbf{(E)}\\ {-1}\\text{ and }{+1}\\text{ only}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1360", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $r$ be the distance from the origin to a point $P$ with coordinates $x$ and $y$. Designate the ratio $\\frac{y}{r}$ by $s$ and the ratio $\\frac{x}{r}$ by $c$. Then the values of $s^2 - c^2$ are limited to the numbers:\n$\\textbf{(A)}\\ \\text{less than }{-1}\\text{ are greater than }{+1}\\text{, both excluded}\\qquad\\\\ \\textbf{(B)}\\ \\text{less than }{-1}\\text{ are greater than }{+1}\\text{, both included}\\qquad \\\\ \\textbf{(C)}\\ \\text{between }{-1}\\text{ and }{+1}\\text{, both excluded}\\qquad \\\\ \\textbf{(D)}\\ \\text{between }{-1}\\text{ and }{+1}\\text{, both included}\\qquad \\\\ \\textbf{(E)}\\ {-1}\\text{ and }{+1}\\text{ only}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number $a=\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying\n\\[\\lfloor x \\rfloor \\cdot \\{x\\} = a \\cdot x^2\\]\nis $420$, where $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$ and $\\{x\\}=x- \\lfloor x \\rfloor$ denotes the fractional part of $x$. What is $p+q$?\n$\\textbf{(A) } 245 \\qquad \\textbf{(B) } 593 \\qquad \\textbf{(C) } 929 \\qquad \\textbf{(D) } 1331 \\qquad \\textbf{(E) } 1332$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1361", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number $a=\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying\n\\[\\lfloor x \\rfloor \\cdot \\{x\\} = a \\cdot x^2\\]\nis $420$, where $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$ and $\\{x\\}=x- \\lfloor x \\rfloor$ denotes the fractional part of $x$. What is $p+q$?\n$\\textbf{(A) } 245 \\qquad \\textbf{(B) } 593 \\qquad \\textbf{(C) } 929 \\qquad \\textbf{(D) } 1331 \\qquad \\textbf{(E) } 1332$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of points with positive rational coordinates selected from the set of points in the $xy$-plane such that $x+y \\le 5$, is:\n$\\text{(A)} \\ 9 \\qquad \\text{(B)} \\ 10 \\qquad \\text{(C)} \\ 14 \\qquad \\text{(D)} \\ 15 \\qquad \\text{(E) infinite}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1362", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of points with positive rational coordinates selected from the set of points in the $xy$-plane such that $x+y \\le 5$, is:\n$\\text{(A)} \\ 9 \\qquad \\text{(B)} \\ 10 \\qquad \\text{(C)} \\ 14 \\qquad \\text{(D)} \\ 15 \\qquad \\text{(E) infinite}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A large urn contains $100$ balls, of which $36 \\%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \\%$? (No red balls are to be removed.)\n$\\textbf{(A)}\\ 28 \\qquad\\textbf{(B)}\\ 32 \\qquad\\textbf{(C)}\\ 36 \\qquad\\textbf{(D)}\\ 50 \\qquad\\textbf{(E)}\\ 64$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1363", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A large urn contains $100$ balls, of which $36 \\%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \\%$? (No red balls are to be removed.)\n$\\textbf{(A)}\\ 28 \\qquad\\textbf{(B)}\\ 32 \\qquad\\textbf{(C)}\\ 36 \\qquad\\textbf{(D)}\\ 50 \\qquad\\textbf{(E)}\\ 64$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be a positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) such that $\\frac 12 + \\frac 13 + \\frac 17 + \\frac 1n$ is an integer. Which of the following statements is not true:\n$\\mathrm{(A)}\\ 2\\ \\text{divides\\ }n \\qquad\\mathrm{(B)}\\ 3\\ \\text{divides\\ }n \\qquad\\mathrm{(C)}$ $\\ 6\\ \\text{divides\\ }n \\qquad\\mathrm{(D)}\\ 7\\ \\text{divides\\ }n \\qquad\\mathrm{(E)}\\ n > 84$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1364", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $n$ be a positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) such that $\\frac 12 + \\frac 13 + \\frac 17 + \\frac 1n$ is an integer. Which of the following statements is not true:\n$\\mathrm{(A)}\\ 2\\ \\text{divides\\ }n \\qquad\\mathrm{(B)}\\ 3\\ \\text{divides\\ }n \\qquad\\mathrm{(C)}$ $\\ 6\\ \\text{divides\\ }n \\qquad\\mathrm{(D)}\\ 7\\ \\text{divides\\ }n \\qquad\\mathrm{(E)}\\ n > 84$" + } + }, + { + "question": "Return your final response within \\boxed{}. Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there?\n$\\textbf{(A)}\\ 1024\\qquad\\textbf{(B)}\\ 1524\\qquad\\textbf{(C)}\\ 1533\\qquad\\textbf{(D)}\\ 1536\\qquad\\textbf{(E)}\\ 2048$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1365", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there?\n$\\textbf{(A)}\\ 1024\\qquad\\textbf{(B)}\\ 1524\\qquad\\textbf{(C)}\\ 1533\\qquad\\textbf{(D)}\\ 1536\\qquad\\textbf{(E)}\\ 2048$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x<0$, then which of the following must be positive?\n$\\textbf{(A)}\\ \\frac{x}{\\left|x\\right|} \\qquad \\textbf{(B)}\\ -x^2 \\qquad \\textbf{(C)}\\ -2^x \\qquad \\textbf{(D)}\\ -x^{-1} \\qquad \\textbf{(E)}\\ \\sqrt[3]{x}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1366", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x<0$, then which of the following must be positive?\n$\\textbf{(A)}\\ \\frac{x}{\\left|x\\right|} \\qquad \\textbf{(B)}\\ -x^2 \\qquad \\textbf{(C)}\\ -2^x \\qquad \\textbf{(D)}\\ -x^{-1} \\qquad \\textbf{(E)}\\ \\sqrt[3]{x}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The square of $5-\\sqrt{y^2-25}$ is: \n$\\textbf{(A)}\\ y^2-5\\sqrt{y^2-25} \\qquad \\textbf{(B)}\\ -y^2 \\qquad \\textbf{(C)}\\ y^2 \\\\ \\textbf{(D)}\\ (5-y)^2\\qquad\\textbf{(E)}\\ y^2-10\\sqrt{y^2-25}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1367", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The square of $5-\\sqrt{y^2-25}$ is: \n$\\textbf{(A)}\\ y^2-5\\sqrt{y^2-25} \\qquad \\textbf{(B)}\\ -y^2 \\qquad \\textbf{(C)}\\ y^2 \\\\ \\textbf{(D)}\\ (5-y)^2\\qquad\\textbf{(E)}\\ y^2-10\\sqrt{y^2-25}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the set of $n$ numbers; $n > 1$, of which one is $1 - \\frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is: \n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ n-\\frac{1}{n}\\qquad\\textbf{(C)}\\ n-\\frac{1}{n^2}\\qquad\\textbf{(D)}\\ 1-\\frac{1}{n^2}\\qquad\\textbf{(E)}\\ 1-\\frac{1}{n}-\\frac{1}{n^2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1368", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the set of $n$ numbers; $n > 1$, of which one is $1 - \\frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is: \n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ n-\\frac{1}{n}\\qquad\\textbf{(C)}\\ n-\\frac{1}{n^2}\\qquad\\textbf{(D)}\\ 1-\\frac{1}{n^2}\\qquad\\textbf{(E)}\\ 1-\\frac{1}{n}-\\frac{1}{n^2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|.$\n$\\text{(A) }0 \\qquad \\text{(B) }\\frac{1}{2002} \\qquad \\text{(C) }\\frac{1}{2001} \\qquad \\text{(D) }\\frac {2}{2001} \\qquad \\text{(E) }\\frac{1}{1000}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1369", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|.$\n$\\text{(A) }0 \\qquad \\text{(B) }\\frac{1}{2002} \\qquad \\text{(C) }\\frac{1}{2001} \\qquad \\text{(D) }\\frac {2}{2001} \\qquad \\text{(E) }\\frac{1}{1000}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\\[f(n)=\\frac{d(n)}{\\sqrt [3]n}.\\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\\ne N$. What is the sum of the digits of $N?$\n$\\textbf{(A) }5 \\qquad \\textbf{(B) }6 \\qquad \\textbf{(C) }7 \\qquad \\textbf{(D) }8\\qquad \\textbf{(E) }9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1370", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\\[f(n)=\\frac{d(n)}{\\sqrt [3]n}.\\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\\ne N$. What is the sum of the digits of $N?$\n$\\textbf{(A) }5 \\qquad \\textbf{(B) }6 \\qquad \\textbf{(C) }7 \\qquad \\textbf{(D) }8\\qquad \\textbf{(E) }9$" + } + }, + { + "question": "Return your final response within \\boxed{}. Angle $B$ of triangle $ABC$ is trisected by $BD$ and $BE$ which meet $AC$ at $D$ and $E$ respectively. Then:\n$\\textbf{(A) \\ }\\frac{AD}{EC}=\\frac{AE}{DC} \\qquad \\textbf{(B) \\ }\\frac{AD}{EC}=\\frac{AB}{BC} \\qquad \\textbf{(C) \\ }\\frac{AD}{EC}=\\frac{BD}{BE} \\qquad$\n$\\textbf{(D) \\ }\\frac{AD}{EC}=\\frac{(AB)(BD)}{(BE)(BC)} \\qquad \\textbf{(E) \\ }\\frac{AD}{EC}=\\frac{(AE)(BD)}{(DC)(BE)}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1371", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Angle $B$ of triangle $ABC$ is trisected by $BD$ and $BE$ which meet $AC$ at $D$ and $E$ respectively. Then:\n$\\textbf{(A) \\ }\\frac{AD}{EC}=\\frac{AE}{DC} \\qquad \\textbf{(B) \\ }\\frac{AD}{EC}=\\frac{AB}{BC} \\qquad \\textbf{(C) \\ }\\frac{AD}{EC}=\\frac{BD}{BE} \\qquad$\n$\\textbf{(D) \\ }\\frac{AD}{EC}=\\frac{(AB)(BD)}{(BE)(BC)} \\qquad \\textbf{(E) \\ }\\frac{AD}{EC}=\\frac{(AE)(BD)}{(DC)(BE)}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1372", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Lines $L_1,L_2,\\dots,L_{100}$ are distinct. All lines $L_{4n}, n$ a positive integer, are parallel to each other. \nAll lines $L_{4n-3}, n$ a positive integer, pass through a given point $A.$ The maximum number of points of intersection of pairs of lines from the complete set $\\{L_1,L_2,\\dots,L_{100}\\}$ is\n$\\textbf{(A) }4350\\qquad \\textbf{(B) }4351\\qquad \\textbf{(C) }4900\\qquad \\textbf{(D) }4901\\qquad \\textbf{(E) }9851$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1373", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Lines $L_1,L_2,\\dots,L_{100}$ are distinct. All lines $L_{4n}, n$ a positive integer, are parallel to each other. \nAll lines $L_{4n-3}, n$ a positive integer, pass through a given point $A.$ The maximum number of points of intersection of pairs of lines from the complete set $\\{L_1,L_2,\\dots,L_{100}\\}$ is\n$\\textbf{(A) }4350\\qquad \\textbf{(B) }4351\\qquad \\textbf{(C) }4900\\qquad \\textbf{(D) }4901\\qquad \\textbf{(E) }9851$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S$ be the sum of all positive real numbers $x$ for which\\[x^{2^{\\sqrt2}}=\\sqrt2^{2^x}.\\]Which of the following statements is true?\n$\\textbf{(A) }S<\\sqrt2 \\qquad \\textbf{(B) }S=\\sqrt2 \\qquad \\textbf{(C) }\\sqrt2 1$, then the sum of the real solutions of \n$\\sqrt{a - \\sqrt{a + x}} = x$\nis equal to \n$\\textbf{(A)}\\ \\sqrt{a} - 1\\qquad \\textbf{(B)}\\ \\dfrac{\\sqrt{a}- 1}{2}\\qquad \\textbf{(C)}\\ \\sqrt{a - 1}\\qquad \\textbf{(D)}\\ \\dfrac{\\sqrt{a - 1}}{2}\\qquad \\textbf{(E)}\\ \\dfrac{\\sqrt{4a- 3} - 1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1398", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a > 1$, then the sum of the real solutions of \n$\\sqrt{a - \\sqrt{a + x}} = x$\nis equal to \n$\\textbf{(A)}\\ \\sqrt{a} - 1\\qquad \\textbf{(B)}\\ \\dfrac{\\sqrt{a}- 1}{2}\\qquad \\textbf{(C)}\\ \\sqrt{a - 1}\\qquad \\textbf{(D)}\\ \\dfrac{\\sqrt{a - 1}}{2}\\qquad \\textbf{(E)}\\ \\dfrac{\\sqrt{4a- 3} - 1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following inequalities are satisfied for all real numbers $a, b, c, x, y, z$ which satisfy the conditions $x < a, y < b$, and $z < c$? \n$\\text{I}. \\ xy + yz + zx < ab + bc + ca \\\\ \\text{II}. \\ x^2 + y^2 + z^2 < a^2 + b^2 + c^2 \\\\ \\text{III}. \\ xyz < abc$\n$\\textbf{(A)}\\ \\text{None are satisfied.} \\qquad \\textbf{(B)}\\ \\text{I only} \\qquad \\textbf{(C)}\\ \\text{II only} \\qquad \\textbf{(D)}\\ \\text{III only} \\qquad \\textbf{(E)}\\ \\text{All are satisfied.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1399", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following inequalities are satisfied for all real numbers $a, b, c, x, y, z$ which satisfy the conditions $x < a, y < b$, and $z < c$? \n$\\text{I}. \\ xy + yz + zx < ab + bc + ca \\\\ \\text{II}. \\ x^2 + y^2 + z^2 < a^2 + b^2 + c^2 \\\\ \\text{III}. \\ xyz < abc$\n$\\textbf{(A)}\\ \\text{None are satisfied.} \\qquad \\textbf{(B)}\\ \\text{I only} \\qquad \\textbf{(C)}\\ \\text{II only} \\qquad \\textbf{(D)}\\ \\text{III only} \\qquad \\textbf{(E)}\\ \\text{All are satisfied.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?\n$\\textbf{(A)} ~0\\qquad\\textbf{(B)} ~\\frac{1}{4}\\qquad\\textbf{(C)} ~\\frac{1}{2} \\qquad\\textbf{(D)} ~1 \\qquad\\textbf{(E)} ~2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1400", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?\n$\\textbf{(A)} ~0\\qquad\\textbf{(B)} ~\\frac{1}{4}\\qquad\\textbf{(C)} ~\\frac{1}{2} \\qquad\\textbf{(D)} ~1 \\qquad\\textbf{(E)} ~2$" + } + }, + { + "question": "Return your final response within \\boxed{}. For all positive numbers $x$, $y$, $z$, the product \n\\[(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]\\] equals \n$\\textbf{(A)}\\ x^{-2}y^{-2}z^{-2}\\qquad\\textbf{(B)}\\ x^{-2}+y^{-2}+z^{-2}\\qquad\\textbf{(C)}\\ (x+y+z)^{-1}\\qquad \\textbf{(D)}\\ \\dfrac{1}{xyz}\\qquad \\\\ \\textbf{(E)}\\ \\dfrac{1}{xy+yz+xz}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1401", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For all positive numbers $x$, $y$, $z$, the product \n\\[(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]\\] equals \n$\\textbf{(A)}\\ x^{-2}y^{-2}z^{-2}\\qquad\\textbf{(B)}\\ x^{-2}+y^{-2}+z^{-2}\\qquad\\textbf{(C)}\\ (x+y+z)^{-1}\\qquad \\textbf{(D)}\\ \\dfrac{1}{xyz}\\qquad \\\\ \\textbf{(E)}\\ \\dfrac{1}{xy+yz+xz}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $X$ is the smallest set of polynomials $p(x)$ such that: \n\n1. $p(x) = x$ belongs to $X$.\n2. If $r(x)$ belongs to $X$, then $x\\cdot r(x)$ and $(x + (1 - x) \\cdot r(x) )$ both belong to $X$.\nShow that if $r(x)$ and $s(x)$ are distinct elements of $X$, then $r(x) \\neq s(x)$ for any $0 < x < 1$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "If r(x) and s(x) are distinct elements of X, then r(x) \\neq s(x) for any 0 < x < 1.", + "index": "Sky-T1_10k_1402", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $X$ is the smallest set of polynomials $p(x)$ such that: \n\n1. $p(x) = x$ belongs to $X$.\n2. If $r(x)$ belongs to $X$, then $x\\cdot r(x)$ and $(x + (1 - x) \\cdot r(x) )$ both belong to $X$.\nShow that if $r(x)$ and $s(x)$ are distinct elements of $X$, then $r(x) \\neq s(x)$ for any $0 < x < 1$." + } + }, + { + "question": "Return your final response within \\boxed{}. The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?\n$\\textbf{(A) }\\frac{1}{21}\\qquad\\textbf{(B) }\\frac{1}{14}\\qquad\\textbf{(C) }\\frac{5}{63}\\qquad\\textbf{(D) }\\frac{2}{21}\\qquad\\textbf{(E) }\\frac{1}{7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1403", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?\n$\\textbf{(A) }\\frac{1}{21}\\qquad\\textbf{(B) }\\frac{1}{14}\\qquad\\textbf{(C) }\\frac{5}{63}\\qquad\\textbf{(D) }\\frac{2}{21}\\qquad\\textbf{(E) }\\frac{1}{7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four [complex numbers](https://artofproblemsolving.com/wiki/index.php/Complex_numbers) lie at the [vertices](https://artofproblemsolving.com/wiki/index.php/Vertices) of a [square](https://artofproblemsolving.com/wiki/index.php/Square) in the [complex plane](https://artofproblemsolving.com/wiki/index.php/Complex_plane). Three of the numbers are $1+2i, -2+i$, and $-1-2i$. The fourth number is \n$\\mathrm{(A) \\ }2+i \\qquad \\mathrm{(B) \\ }2-i \\qquad \\mathrm{(C) \\ } 1-2i \\qquad \\mathrm{(D) \\ }-1+2i \\qquad \\mathrm{(E) \\ } -2-i$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1404", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four [complex numbers](https://artofproblemsolving.com/wiki/index.php/Complex_numbers) lie at the [vertices](https://artofproblemsolving.com/wiki/index.php/Vertices) of a [square](https://artofproblemsolving.com/wiki/index.php/Square) in the [complex plane](https://artofproblemsolving.com/wiki/index.php/Complex_plane). Three of the numbers are $1+2i, -2+i$, and $-1-2i$. The fourth number is \n$\\mathrm{(A) \\ }2+i \\qquad \\mathrm{(B) \\ }2-i \\qquad \\mathrm{(C) \\ } 1-2i \\qquad \\mathrm{(D) \\ }-1+2i \\qquad \\mathrm{(E) \\ } -2-i$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?\n\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 20\\qquad\\textbf{(C)}\\ 25\\qquad\\textbf{(D)}\\ 30\\qquad\\textbf{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1405", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?\n\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 20\\qquad\\textbf{(C)}\\ 25\\qquad\\textbf{(D)}\\ 30\\qquad\\textbf{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. A computer can do $10,000$ additions per second. How many additions can it do in one hour?\n$\\text{(A)}\\ 6\\text{ million} \\qquad \\text{(B)}\\ 36\\text{ million} \\qquad \\text{(C)}\\ 60\\text{ million} \\qquad \\text{(D)}\\ 216\\text{ million} \\qquad \\text{(E)}\\ 360\\text{ million}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1406", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A computer can do $10,000$ additions per second. How many additions can it do in one hour?\n$\\text{(A)}\\ 6\\text{ million} \\qquad \\text{(B)}\\ 36\\text{ million} \\qquad \\text{(C)}\\ 60\\text{ million} \\qquad \\text{(D)}\\ 216\\text{ million} \\qquad \\text{(E)}\\ 360\\text{ million}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?\n$\\mathrm{(A)}\\frac {1}{16}\\qquad \\mathrm{(B)}\\frac 18\\qquad \\mathrm{(C)}\\frac {3}{16}\\qquad \\mathrm{(D)}\\frac 14\\qquad \\mathrm{(E)}\\frac {5}{16}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1407", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?\n$\\mathrm{(A)}\\frac {1}{16}\\qquad \\mathrm{(B)}\\frac 18\\qquad \\mathrm{(C)}\\frac {3}{16}\\qquad \\mathrm{(D)}\\frac 14\\qquad \\mathrm{(E)}\\frac {5}{16}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Thirty-one books are arranged from left to right in order of increasing prices. \nThe price of each book differs by $\\textdollar{2}$ from that of each adjacent book. \nFor the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then:\n$\\textbf{(A)}\\ \\text{The adjacent book referred to is at the left of the middle book}\\qquad \\\\ \\textbf{(B)}\\ \\text{The middle book sells for \\textdollar 36} \\qquad \\\\ \\textbf{(C)}\\ \\text{The cheapest book sells for \\textdollar4} \\qquad \\\\ \\textbf{(D)}\\ \\text{The most expensive book sells for \\textdollar64 } \\qquad \\\\ \\textbf{(E)}\\ \\text{None of these is correct }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1408", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Thirty-one books are arranged from left to right in order of increasing prices. \nThe price of each book differs by $\\textdollar{2}$ from that of each adjacent book. \nFor the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then:\n$\\textbf{(A)}\\ \\text{The adjacent book referred to is at the left of the middle book}\\qquad \\\\ \\textbf{(B)}\\ \\text{The middle book sells for \\textdollar 36} \\qquad \\\\ \\textbf{(C)}\\ \\text{The cheapest book sells for \\textdollar4} \\qquad \\\\ \\textbf{(D)}\\ \\text{The most expensive book sells for \\textdollar64 } \\qquad \\\\ \\textbf{(E)}\\ \\text{None of these is correct }$" + } + }, + { + "question": "Return your final response within \\boxed{}. An inverted cone with base radius $12 \\mathrm{cm}$ and height $18 \\mathrm{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \\mathrm{cm}$. What is the height in centimeters of the water in the cylinder?\n$\\textbf{(A)} ~1.5 \\qquad\\textbf{(B)} ~3 \\qquad\\textbf{(C)} ~4 \\qquad\\textbf{(D)} ~4.5 \\qquad\\textbf{(E)} ~6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1409", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An inverted cone with base radius $12 \\mathrm{cm}$ and height $18 \\mathrm{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \\mathrm{cm}$. What is the height in centimeters of the water in the cylinder?\n$\\textbf{(A)} ~1.5 \\qquad\\textbf{(B)} ~3 \\qquad\\textbf{(C)} ~4 \\qquad\\textbf{(D)} ~4.5 \\qquad\\textbf{(E)} ~6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Lilypads $1,2,3,\\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly with probability $\\tfrac{1}{2}$ and independently of other jumps. The probability that the frog visits pad $7$ is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "107", + "index": "Sky-T1_10k_1410", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Lilypads $1,2,3,\\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly with probability $\\tfrac{1}{2}$ and independently of other jumps. The probability that the frog visits pad $7$ is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$." + } + }, + { + "question": "Return your final response within \\boxed{}. A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2$ feet in t seconds and that the velocity of sound is $1120$ feet per second. The depth of the well is:\n$\\textbf{(A)}\\ 784\\text{ ft.}\\qquad\\textbf{(B)}\\ 342\\text{ ft.}\\qquad\\textbf{(C)}\\ 1568\\text{ ft.}\\qquad\\textbf{(D)}\\ 156.8\\text{ ft.}\\qquad\\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1411", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2$ feet in t seconds and that the velocity of sound is $1120$ feet per second. The depth of the well is:\n$\\textbf{(A)}\\ 784\\text{ ft.}\\qquad\\textbf{(B)}\\ 342\\text{ ft.}\\qquad\\textbf{(C)}\\ 1568\\text{ ft.}\\qquad\\textbf{(D)}\\ 156.8\\text{ ft.}\\qquad\\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Josanna's test scores to date are $90, 80, 70, 60,$ and $85$. Her goal is to raise here test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal?\n$\\textbf{(A)}\\ 80 \\qquad\\textbf{(B)}\\ 82 \\qquad\\textbf{(C)}\\ 85 \\qquad\\textbf{(D)}\\ 90 \\qquad\\textbf{(E)}\\ 95$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1412", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Josanna's test scores to date are $90, 80, 70, 60,$ and $85$. Her goal is to raise here test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal?\n$\\textbf{(A)}\\ 80 \\qquad\\textbf{(B)}\\ 82 \\qquad\\textbf{(C)}\\ 85 \\qquad\\textbf{(D)}\\ 90 \\qquad\\textbf{(E)}\\ 95$" + } + }, + { + "question": "Return your final response within \\boxed{}. A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following 2 properties:\n(a) It is a palindrome\n(b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome. \nHow many years in the millenium between 1000 and 2000 have properties (a) and (b)?\n$\\text{(A) } 1\\quad \\text{(B) } 2\\quad \\text{(C) } 3\\quad \\text{(D) } 4\\quad \\text{(E) } 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1413", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following 2 properties:\n(a) It is a palindrome\n(b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome. \nHow many years in the millenium between 1000 and 2000 have properties (a) and (b)?\n$\\text{(A) } 1\\quad \\text{(B) } 2\\quad \\text{(C) } 3\\quad \\text{(D) } 4\\quad \\text{(E) } 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many solutions does the equation $\\tan(2x)=\\cos(\\tfrac{x}{2})$ have on the interval $[0,2\\pi]?$\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1414", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many solutions does the equation $\\tan(2x)=\\cos(\\tfrac{x}{2})$ have on the interval $[0,2\\pi]?$\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. [asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16); draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N); label(\"S\", (18,8)); label(\"S\", (50,8)); label(\"Figure 1\", (A+B)/2, S); label(\"Figure 2\", (E+F)/2, S); label(\"10'\", (I+J)/2, S); label(\"8'\", (12,12)); label(\"8'\", (L+M)/2, S); label(\"10'\", (42,11)); label(\"table\", (5,12)); label(\"table\", (36,11)); [/asy]\nAn $8'\\times 10'$ table sits in the corner of a square room, as in Figure $1$ below. \nThe owners desire to move the table to the position shown in Figure $2$. \nThe side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart?\n$\\textbf{(A)}\\ 11\\qquad \\textbf{(B)}\\ 12\\qquad \\textbf{(C)}\\ 13\\qquad \\textbf{(D)}\\ 14\\qquad \\textbf{(E)}\\ 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1415", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16); draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N); label(\"S\", (18,8)); label(\"S\", (50,8)); label(\"Figure 1\", (A+B)/2, S); label(\"Figure 2\", (E+F)/2, S); label(\"10'\", (I+J)/2, S); label(\"8'\", (12,12)); label(\"8'\", (L+M)/2, S); label(\"10'\", (42,11)); label(\"table\", (5,12)); label(\"table\", (36,11)); [/asy]\nAn $8'\\times 10'$ table sits in the corner of a square room, as in Figure $1$ below. \nThe owners desire to move the table to the position shown in Figure $2$. \nThe side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart?\n$\\textbf{(A)}\\ 11\\qquad \\textbf{(B)}\\ 12\\qquad \\textbf{(C)}\\ 13\\qquad \\textbf{(D)}\\ 14\\qquad \\textbf{(E)}\\ 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. The Y2K Game is played on a $1 \\times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The second player has a winning strategy.", + "index": "Sky-T1_10k_1416", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The Y2K Game is played on a $1 \\times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "750", + "index": "Sky-T1_10k_1417", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$." + } + }, + { + "question": "Return your final response within \\boxed{}. Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point \n$\\mathrm{(A) \\ 1 } \\qquad \\mathrm{(B) \\ 2 } \\qquad \\mathrm{(C) \\ 3 } \\qquad \\mathrm{(D) \\ 4 } \\qquad \\mathrm{(E) \\ 5 }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1418", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point \n$\\mathrm{(A) \\ 1 } \\qquad \\mathrm{(B) \\ 2 } \\qquad \\mathrm{(C) \\ 3 } \\qquad \\mathrm{(D) \\ 4 } \\qquad \\mathrm{(E) \\ 5 }$" + } + }, + { + "question": "Return your final response within \\boxed{}. A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\\triangle ABE$ and $\\triangle CDE$ are $105$ and $91$, respectively. What is the volume of the pyramid?\n$\\textbf{(A)}\\ 392 \\qquad \\textbf{(B)}\\ 196\\sqrt {6} \\qquad \\textbf{(C)}\\ 392\\sqrt {2} \\qquad \\textbf{(D)}\\ 392\\sqrt {3} \\qquad \\textbf{(E)}\\ 784$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1419", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\\triangle ABE$ and $\\triangle CDE$ are $105$ and $91$, respectively. What is the volume of the pyramid?\n$\\textbf{(A)}\\ 392 \\qquad \\textbf{(B)}\\ 196\\sqrt {6} \\qquad \\textbf{(C)}\\ 392\\sqrt {2} \\qquad \\textbf{(D)}\\ 392\\sqrt {3} \\qquad \\textbf{(E)}\\ 784$" + } + }, + { + "question": "Return your final response within \\boxed{}. The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002$ in$^3$. Find the minimum possible sum of the three dimensions.\n$\\text{(A) }36 \\qquad \\text{(B) }38 \\qquad \\text{(C) }42 \\qquad \\text{(D) }44 \\qquad \\text{(E) }92$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1420", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002$ in$^3$. Find the minimum possible sum of the three dimensions.\n$\\text{(A) }36 \\qquad \\text{(B) }38 \\qquad \\text{(C) }42 \\qquad \\text{(D) }44 \\qquad \\text{(E) }92$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $z$ be a complex number satisfying $12|z|^2=2|z+2|^2+|z^2+1|^2+31.$ What is the value of $z+\\frac 6z?$\n$\\textbf{(A) }-2 \\qquad \\textbf{(B) }-1 \\qquad \\textbf{(C) }\\frac12\\qquad \\textbf{(D) }1 \\qquad \\textbf{(E) }4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1421", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $z$ be a complex number satisfying $12|z|^2=2|z+2|^2+|z^2+1|^2+31.$ What is the value of $z+\\frac 6z?$\n$\\textbf{(A) }-2 \\qquad \\textbf{(B) }-1 \\qquad \\textbf{(C) }\\frac12\\qquad \\textbf{(D) }1 \\qquad \\textbf{(E) }4$" + } + }, + { + "question": "Return your final response within \\boxed{}. A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days?\n$\\text{(A)}\\ 140 \\qquad \\text{(B)}\\ 170 \\qquad \\text{(C)}\\ 185 \\qquad \\text{(D)}\\ 198.5 \\qquad \\text{(E)}\\ 199.85$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1422", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days?\n$\\text{(A)}\\ 140 \\qquad \\text{(B)}\\ 170 \\qquad \\text{(C)}\\ 185 \\qquad \\text{(D)}\\ 198.5 \\qquad \\text{(E)}\\ 199.85$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $A$ be a set with $|A| = 225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1$, $\\dots$, $A_{11}$ of $A$ such that $|A_i | = 45$ for $1 \\le i \\le 11$ and $|A_i \\cap A_j| = 9$ for $1 \\le i < j \\le 11$. Prove that $|A_1 \\cup A_2 \\cup \\dots \\cup A_{11}| \\ge 165$, and give an example for which equality holds.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "165", + "index": "Sky-T1_10k_1423", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $A$ be a set with $|A| = 225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1$, $\\dots$, $A_{11}$ of $A$ such that $|A_i | = 45$ for $1 \\le i \\le 11$ and $|A_i \\cap A_j| = 9$ for $1 \\le i < j \\le 11$. Prove that $|A_1 \\cup A_2 \\cup \\dots \\cup A_{11}| \\ge 165$, and give an example for which equality holds." + } + }, + { + "question": "Return your final response within \\boxed{}. How many 4-digit numbers greater than 1000 are there that use the four digits of 2012?\n$\\textbf{(A)}\\hspace{.05in}6\\qquad\\textbf{(B)}\\hspace{.05in}7\\qquad\\textbf{(C)}\\hspace{.05in}8\\qquad\\textbf{(D)}\\hspace{.05in}9\\qquad\\textbf{(E)}\\hspace{.05in}12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1424", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many 4-digit numbers greater than 1000 are there that use the four digits of 2012?\n$\\textbf{(A)}\\hspace{.05in}6\\qquad\\textbf{(B)}\\hspace{.05in}7\\qquad\\textbf{(C)}\\hspace{.05in}8\\qquad\\textbf{(D)}\\hspace{.05in}9\\qquad\\textbf{(E)}\\hspace{.05in}12$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\\frac{1}{3}$ of all the ninth graders are paired with $\\frac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?\n$\\textbf{(A) } \\frac{2}{15} \\qquad\\textbf{(B) } \\frac{4}{11} \\qquad\\textbf{(C) } \\frac{11}{30} \\qquad\\textbf{(D) } \\frac{3}{8} \\qquad\\textbf{(E) } \\frac{11}{15}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1425", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\\frac{1}{3}$ of all the ninth graders are paired with $\\frac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?\n$\\textbf{(A) } \\frac{2}{15} \\qquad\\textbf{(B) } \\frac{4}{11} \\qquad\\textbf{(C) } \\frac{11}{30} \\qquad\\textbf{(D) } \\frac{3}{8} \\qquad\\textbf{(E) } \\frac{11}{15}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?\n[asy] size(7cm); pathpen = linewidth(0.7); D(CR((0,0),10)); D(CR((0,0),9.5)); D(CR((0,-18.5),9.5)); D(CR((0,-18.5),9)); MP(\"$\\vdots$\",(0,-31),(0,0)); D(CR((0,-39),3)); D(CR((0,-39),2.5)); D(CR((0,-43.5),2.5)); D(CR((0,-43.5),2)); D(CR((0,-47),2)); D(CR((0,-47),1.5)); D(CR((0,-49.5),1.5)); D(CR((0,-49.5),1.0)); D((12,-10)--(12,10)); MP('20',(12,0),E); D((12,-51)--(12,-48)); MP('3',(12,-49.5),E);[/asy]\n$\\textbf{(A) } 171\\qquad\\textbf{(B) } 173\\qquad\\textbf{(C) } 182\\qquad\\textbf{(D) } 188\\qquad\\textbf{(E) } 210\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1426", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?\n[asy] size(7cm); pathpen = linewidth(0.7); D(CR((0,0),10)); D(CR((0,0),9.5)); D(CR((0,-18.5),9.5)); D(CR((0,-18.5),9)); MP(\"$\\vdots$\",(0,-31),(0,0)); D(CR((0,-39),3)); D(CR((0,-39),2.5)); D(CR((0,-43.5),2.5)); D(CR((0,-43.5),2)); D(CR((0,-47),2)); D(CR((0,-47),1.5)); D(CR((0,-49.5),1.5)); D(CR((0,-49.5),1.0)); D((12,-10)--(12,10)); MP('20',(12,0),E); D((12,-51)--(12,-48)); MP('3',(12,-49.5),E);[/asy]\n$\\textbf{(A) } 171\\qquad\\textbf{(B) } 173\\qquad\\textbf{(C) } 182\\qquad\\textbf{(D) } 188\\qquad\\textbf{(E) } 210\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?\n$\\textbf{(A)}\\hspace{.05in}4\\qquad\\textbf{(B)}\\hspace{.05in}5\\qquad\\textbf{(C)}\\hspace{.05in}6\\qquad\\textbf{(D)}\\hspace{.05in}7\\qquad\\textbf{(E)}\\hspace{.05in}8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1427", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?\n$\\textbf{(A)}\\hspace{.05in}4\\qquad\\textbf{(B)}\\hspace{.05in}5\\qquad\\textbf{(C)}\\hspace{.05in}6\\qquad\\textbf{(D)}\\hspace{.05in}7\\qquad\\textbf{(E)}\\hspace{.05in}8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\\{A, B, C\\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$\n$\\textbf{(A)}\\ 385 \\qquad \\textbf{(B)}\\ 665 \\qquad \\textbf{(C)}\\ 945 \\qquad \\textbf{(D)}\\ 1140 \\qquad \\textbf{(E)}\\ 1330$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1428", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\\{A, B, C\\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$\n$\\textbf{(A)}\\ 385 \\qquad \\textbf{(B)}\\ 665 \\qquad \\textbf{(C)}\\ 945 \\qquad \\textbf{(D)}\\ 1140 \\qquad \\textbf{(E)}\\ 1330$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?\n$\\textbf{(A)}\\ 30\\%\\qquad\\textbf{(B)}\\ 40\\%\\qquad\\textbf{(C)}\\ 50\\%\\qquad\\textbf{(D)}\\ 60\\%\\qquad\\textbf{(E)}\\ 70\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1429", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?\n$\\textbf{(A)}\\ 30\\%\\qquad\\textbf{(B)}\\ 40\\%\\qquad\\textbf{(C)}\\ 50\\%\\qquad\\textbf{(D)}\\ 60\\%\\qquad\\textbf{(E)}\\ 70\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that in a certain society, each pair of persons can be classified as either amicable or hostile. We shall say that each member of an amicable pair is a friend of the other, and each member of a hostile pair is a foe of the other. Suppose that the society has $\\, n \\,$ persons and $\\, q \\,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\\, q(1 - 4q/n^2) \\,$ or fewer amicable pairs.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "q\\left(1 - \\frac{4q}{n^2}\\right)", + "index": "Sky-T1_10k_1430", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that in a certain society, each pair of persons can be classified as either amicable or hostile. We shall say that each member of an amicable pair is a friend of the other, and each member of a hostile pair is a foe of the other. Suppose that the society has $\\, n \\,$ persons and $\\, q \\,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\\, q(1 - 4q/n^2) \\,$ or fewer amicable pairs." + } + }, + { + "question": "Return your final response within \\boxed{}. Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?\n\n$\\textbf{(A)}\\ 60\\qquad\\textbf{(B)}\\ 70\\qquad\\textbf{(C)}\\ 75\\qquad\\textbf{(D)}\\ 80\\qquad\\textbf{(E)}\\ 85$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1431", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?\n\n$\\textbf{(A)}\\ 60\\qquad\\textbf{(B)}\\ 70\\qquad\\textbf{(C)}\\ 75\\qquad\\textbf{(D)}\\ 80\\qquad\\textbf{(E)}\\ 85$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given a [nonnegative](https://artofproblemsolving.com/wiki/index.php/Nonnegative) real number $x$, let $\\langle x\\rangle$ denote the fractional part of $x$; that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor$, where $\\lfloor x\\rfloor$ denotes the [greatest integer](https://artofproblemsolving.com/wiki/index.php/Greatest_integer) less than or equal to $x$. Suppose that $a$ is positive, $\\langle a^{-1}\\rangle=\\langle a^2\\rangle$, and $2y$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1442", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is $y$ miles per hour. Then we always have \n$\\textbf{(A)}\\ x \\leq y\\qquad \\textbf{(B)}\\ x \\geq y \\qquad \\textbf{(C)}\\ x=y \\qquad \\textbf{(D)}\\ xy$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each day Walter gets $3$ dollars for doing his chores or $5$ dollars for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $36$ dollars. On how many days did Walter do them exceptionally well?\n$\\text{(A)}\\ 3\\qquad\\text{(B)}\\ 4\\qquad\\text{(C)}\\ 5\\qquad\\text{(D)}\\ 6\\qquad\\text{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1443", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each day Walter gets $3$ dollars for doing his chores or $5$ dollars for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $36$ dollars. On how many days did Walter do them exceptionally well?\n$\\text{(A)}\\ 3\\qquad\\text{(B)}\\ 4\\qquad\\text{(C)}\\ 5\\qquad\\text{(D)}\\ 6\\qquad\\text{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. All of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have?\n$\\textbf{(A) } 1 \\qquad \\textbf{(B) } 2 \\qquad \\textbf{(C) } 7 \\qquad \\textbf{(D) } 8 \\qquad \\textbf{(E) } 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1444", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. All of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have?\n$\\textbf{(A) } 1 \\qquad \\textbf{(B) } 2 \\qquad \\textbf{(C) } 7 \\qquad \\textbf{(D) } 8 \\qquad \\textbf{(E) } 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. There is a unique positive real number $x$ such that the three numbers $\\log_8{2x}$, $\\log_4{x}$, and $\\log_2{x}$, in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "17", + "index": "Sky-T1_10k_1445", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There is a unique positive real number $x$ such that the three numbers $\\log_8{2x}$, $\\log_4{x}$, and $\\log_2{x}$, in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a_1, a_2, \\ldots$ and $b_1, b_2, \\ldots$ be arithmetic progressions such that $a_1 = 25, b_1 = 75$, and $a_{100} + b_{100} = 100$. \nFind the sum of the first hundred terms of the progression $a_1 + b_1, a_2 + b_2, \\ldots$\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 100 \\qquad \\textbf{(C)}\\ 10,000 \\qquad \\textbf{(D)}\\ 505,000 \\qquad \\\\ \\textbf{(E)}\\ \\text{not enough information given to solve the problem}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1446", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a_1, a_2, \\ldots$ and $b_1, b_2, \\ldots$ be arithmetic progressions such that $a_1 = 25, b_1 = 75$, and $a_{100} + b_{100} = 100$. \nFind the sum of the first hundred terms of the progression $a_1 + b_1, a_2 + b_2, \\ldots$\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 100 \\qquad \\textbf{(C)}\\ 10,000 \\qquad \\textbf{(D)}\\ 505,000 \\qquad \\\\ \\textbf{(E)}\\ \\text{not enough information given to solve the problem}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In an All-Area track meet, $216$ sprinters enter a $100-$meter dash competition. The track has $6$ lanes, so only $6$ sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?\n$\\textbf{(A)}\\mbox{ }36\\qquad\\textbf{(B)}\\mbox{ }42\\qquad\\textbf{(C)}\\mbox{ }43\\qquad\\textbf{(D)}\\mbox{ }60\\qquad\\textbf{(E)}\\mbox{ }72$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1447", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In an All-Area track meet, $216$ sprinters enter a $100-$meter dash competition. The track has $6$ lanes, so only $6$ sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?\n$\\textbf{(A)}\\mbox{ }36\\qquad\\textbf{(B)}\\mbox{ }42\\qquad\\textbf{(C)}\\mbox{ }43\\qquad\\textbf{(D)}\\mbox{ }60\\qquad\\textbf{(E)}\\mbox{ }72$" + } + }, + { + "question": "Return your final response within \\boxed{}. For any integer $k\\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2.$ Let $\\{x_n\\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\\geq 0.$ Find the smallest positive integer $t$ such that $x_t=2090.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "149", + "index": "Sky-T1_10k_1448", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For any integer $k\\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2.$ Let $\\{x_n\\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\\geq 0.$ Find the smallest positive integer $t$ such that $x_t=2090.$" + } + }, + { + "question": "Return your final response within \\boxed{}. What number should be removed from the list\n\\[1,2,3,4,5,6,7,8,9,10,11\\]\nso that the average of the remaining numbers is $6.1$?\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1449", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What number should be removed from the list\n\\[1,2,3,4,5,6,7,8,9,10,11\\]\nso that the average of the remaining numbers is $6.1$?\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Estimate the population of Nisos in the year 2050.\n$\\text{(A)}\\ 600 \\qquad \\text{(B)}\\ 800 \\qquad \\text{(C)}\\ 1000 \\qquad \\text{(D)}\\ 2000 \\qquad \\text{(E)}\\ 3000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1450", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Estimate the population of Nisos in the year 2050.\n$\\text{(A)}\\ 600 \\qquad \\text{(B)}\\ 800 \\qquad \\text{(C)}\\ 1000 \\qquad \\text{(D)}\\ 2000 \\qquad \\text{(E)}\\ 3000$" + } + }, + { + "question": "Return your final response within \\boxed{}. Annie and Bonnie are running laps around a $400$-meter oval track. They started together, but Annie has pulled ahead, because she runs $25\\%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?\n$\\textbf{(A) }1\\dfrac{1}{4}\\qquad\\textbf{(B) }3\\dfrac{1}{3}\\qquad\\textbf{(C) }4\\qquad\\textbf{(D) }5\\qquad \\textbf{(E) }25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1451", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Annie and Bonnie are running laps around a $400$-meter oval track. They started together, but Annie has pulled ahead, because she runs $25\\%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?\n$\\textbf{(A) }1\\dfrac{1}{4}\\qquad\\textbf{(B) }3\\dfrac{1}{3}\\qquad\\textbf{(C) }4\\qquad\\textbf{(D) }5\\qquad \\textbf{(E) }25$" + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?\n$\\textbf{(A)}\\ 52 \\qquad\\textbf{(B)}\\ 58 \\qquad\\textbf{(C)}\\ 62 \\qquad\\textbf{(D)}\\ 68 \\qquad\\textbf{(E)}\\ 70$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1452", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?\n$\\textbf{(A)}\\ 52 \\qquad\\textbf{(B)}\\ 58 \\qquad\\textbf{(C)}\\ 62 \\qquad\\textbf{(D)}\\ 68 \\qquad\\textbf{(E)}\\ 70$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the number $74982.1035$ the value of the place occupied by the digit 9 is how many times as great as the value of the place occupied by the digit 3?\n$\\text{(A)}\\ 1,000 \\qquad \\text{(B)}\\ 10,000 \\qquad \\text{(C)}\\ 100,000 \\qquad \\text{(D)}\\ 1,000,000 \\qquad \\text{(E)}\\ 10,000,000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1453", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the number $74982.1035$ the value of the place occupied by the digit 9 is how many times as great as the value of the place occupied by the digit 3?\n$\\text{(A)}\\ 1,000 \\qquad \\text{(B)}\\ 10,000 \\qquad \\text{(C)}\\ 100,000 \\qquad \\text{(D)}\\ 1,000,000 \\qquad \\text{(E)}\\ 10,000,000$" + } + }, + { + "question": "Return your final response within \\boxed{}. One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $\\frac{2}{3}$ of the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that a girl is chosen. The [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) of the number of boys to the total number of boys and girls is\n$\\mathrm{(A)\\ } \\frac{1}{3} \\qquad \\mathrm{(B) \\ }\\frac{2}{5} \\qquad \\mathrm{(C) \\ } \\frac{1}{2} \\qquad \\mathrm{(D) \\ } \\frac{3}{5} \\qquad \\mathrm{(E) \\ }\\frac{2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1454", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $\\frac{2}{3}$ of the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that a girl is chosen. The [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) of the number of boys to the total number of boys and girls is\n$\\mathrm{(A)\\ } \\frac{1}{3} \\qquad \\mathrm{(B) \\ }\\frac{2}{5} \\qquad \\mathrm{(C) \\ } \\frac{1}{2} \\qquad \\mathrm{(D) \\ } \\frac{3}{5} \\qquad \\mathrm{(E) \\ }\\frac{2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "222", + "index": "Sky-T1_10k_1455", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$." + } + }, + { + "question": "Return your final response within \\boxed{}. Find the sum of all [positive](https://artofproblemsolving.com/wiki/index.php/Positive_number) [rational numbers](https://artofproblemsolving.com/wiki/index.php/Rational_number) that are less than 10 and that have [denominator](https://artofproblemsolving.com/wiki/index.php/Denominator) 30 when written in [ lowest terms](https://artofproblemsolving.com/wiki/index.php/Reduced_fraction).", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "400", + "index": "Sky-T1_10k_1456", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the sum of all [positive](https://artofproblemsolving.com/wiki/index.php/Positive_number) [rational numbers](https://artofproblemsolving.com/wiki/index.php/Rational_number) that are less than 10 and that have [denominator](https://artofproblemsolving.com/wiki/index.php/Denominator) 30 when written in [ lowest terms](https://artofproblemsolving.com/wiki/index.php/Reduced_fraction)." + } + }, + { + "question": "Return your final response within \\boxed{}. An open box is constructed by starting with a rectangular sheet of metal $10$ in. by $14$ in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:\n$\\textbf{(A)}\\ 140x - 48x^2 + 4x^3 \\qquad \\textbf{(B)}\\ 140x + 48x^2 + 4x^3\\qquad \\\\ \\textbf{(C)}\\ 140x+24x^2+x^3\\qquad \\textbf{(D)}\\ 140x-24x^2+x^3\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1457", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An open box is constructed by starting with a rectangular sheet of metal $10$ in. by $14$ in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:\n$\\textbf{(A)}\\ 140x - 48x^2 + 4x^3 \\qquad \\textbf{(B)}\\ 140x + 48x^2 + 4x^3\\qquad \\\\ \\textbf{(C)}\\ 140x+24x^2+x^3\\qquad \\textbf{(D)}\\ 140x-24x^2+x^3\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "147", + "index": "Sky-T1_10k_1458", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$." + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{9}{7\\times 53} =$\n$\\text{(A)}\\ \\frac{.9}{.7\\times 53} \\qquad \\text{(B)}\\ \\frac{.9}{.7\\times .53} \\qquad \\text{(C)}\\ \\frac{.9}{.7\\times 5.3} \\qquad \\text{(D)}\\ \\frac{.9}{7\\times .53} \\qquad \\text{(E)}\\ \\frac{.09}{.07\\times .53}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1459", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{9}{7\\times 53} =$\n$\\text{(A)}\\ \\frac{.9}{.7\\times 53} \\qquad \\text{(B)}\\ \\frac{.9}{.7\\times .53} \\qquad \\text{(C)}\\ \\frac{.9}{.7\\times 5.3} \\qquad \\text{(D)}\\ \\frac{.9}{7\\times .53} \\qquad \\text{(E)}\\ \\frac{.09}{.07\\times .53}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $K$ takes $30$ minutes less time than $M$ to travel a distance of $30$ miles. $K$ travels $\\frac {1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hours, then $K$'s time for the distance is: \n$\\textbf{(A)}\\ \\dfrac{x + \\frac {1}{3}}{30} \\qquad \\textbf{(B)}\\ \\dfrac{x - \\frac {1}{3}}{30} \\qquad \\textbf{(C)}\\ \\frac{30}{x+\\frac{1}{3}}\\qquad \\textbf{(D)}\\ \\frac{30}{x}\\qquad \\textbf{(E)}\\ \\frac{x}{30}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1460", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $K$ takes $30$ minutes less time than $M$ to travel a distance of $30$ miles. $K$ travels $\\frac {1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hours, then $K$'s time for the distance is: \n$\\textbf{(A)}\\ \\dfrac{x + \\frac {1}{3}}{30} \\qquad \\textbf{(B)}\\ \\dfrac{x - \\frac {1}{3}}{30} \\qquad \\textbf{(C)}\\ \\frac{30}{x+\\frac{1}{3}}\\qquad \\textbf{(D)}\\ \\frac{30}{x}\\qquad \\textbf{(E)}\\ \\frac{x}{30}$" + } + }, + { + "question": "Return your final response within \\boxed{}. An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\\in\\overline{AB},$ $Q\\in\\overline{AC},$ and $N, P\\in\\overline{BC}.$ Let $S$ be the intersection of lines $MN$ and $PQ.$ Denote by $\\ell$ the angle bisector of $\\angle MSQ.$\nProve that $\\overline{OI}$ is parallel to $\\ell,$ where $O$ is the circumcenter of triangle $ABC,$ and $I$ is the incenter of triangle $ABC.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\overline{OI} \\parallel \\ell", + "index": "Sky-T1_10k_1461", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\\in\\overline{AB},$ $Q\\in\\overline{AC},$ and $N, P\\in\\overline{BC}.$ Let $S$ be the intersection of lines $MN$ and $PQ.$ Denote by $\\ell$ the angle bisector of $\\angle MSQ.$\nProve that $\\overline{OI}$ is parallel to $\\ell,$ where $O$ is the circumcenter of triangle $ABC,$ and $I$ is the incenter of triangle $ABC.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Five cards are lying on a table as shown.\n\\[\\begin{matrix} & \\qquad & \\boxed{\\tt{P}} & \\qquad & \\boxed{\\tt{Q}} \\\\ \\\\ \\boxed{\\tt{3}} & \\qquad & \\boxed{\\tt{4}} & \\qquad & \\boxed{\\tt{6}} \\end{matrix}\\]\nEach card has a letter on one side and a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) on the other side. Jane said, \"If a vowel is on one side of any card, then an [even number](https://artofproblemsolving.com/wiki/index.php/Even_number) is on the other side.\" Mary showed Jane was wrong by turning over one card. Which card did Mary turn over? (Each card number is the one with the number on it. For example card 4 is the one with 4 on it, not the fourth card from the left/right)\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ \\text{P} \\qquad \\text{(E)}\\ \\text{Q}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1462", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five cards are lying on a table as shown.\n\\[\\begin{matrix} & \\qquad & \\boxed{\\tt{P}} & \\qquad & \\boxed{\\tt{Q}} \\\\ \\\\ \\boxed{\\tt{3}} & \\qquad & \\boxed{\\tt{4}} & \\qquad & \\boxed{\\tt{6}} \\end{matrix}\\]\nEach card has a letter on one side and a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) on the other side. Jane said, \"If a vowel is on one side of any card, then an [even number](https://artofproblemsolving.com/wiki/index.php/Even_number) is on the other side.\" Mary showed Jane was wrong by turning over one card. Which card did Mary turn over? (Each card number is the one with the number on it. For example card 4 is the one with 4 on it, not the fourth card from the left/right)\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ \\text{P} \\qquad \\text{(E)}\\ \\text{Q}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the graphs $y=Ax^2$ and $y^2+3=x^2+4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect?\n$\\mathrm{(A) \\ }\\text{exactly }4 \\qquad \\mathrm{(B) \\ }\\text{exactly }2 \\qquad$\n$\\mathrm{(C) \\ }\\text{at least }1,\\text{ but the number varies for different positive values of }A \\qquad$\n$\\mathrm{(D) \\ }0\\text{ for at least one positive value of }A \\qquad \\mathrm{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1463", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the graphs $y=Ax^2$ and $y^2+3=x^2+4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect?\n$\\mathrm{(A) \\ }\\text{exactly }4 \\qquad \\mathrm{(B) \\ }\\text{exactly }2 \\qquad$\n$\\mathrm{(C) \\ }\\text{at least }1,\\text{ but the number varies for different positive values of }A \\qquad$\n$\\mathrm{(D) \\ }0\\text{ for at least one positive value of }A \\qquad \\mathrm{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In this diagram $AB$ and $AC$ are the equal sides of an isosceles $\\triangle ABC$, in which is inscribed equilateral $\\triangle DEF$. \nDesignate $\\angle BFD$ by $a$, $\\angle ADE$ by $b$, and $\\angle FEC$ by $c$. Then:\n\n$\\textbf{(A)}\\ b=\\frac{a+c}{2}\\qquad \\textbf{(B)}\\ b=\\frac{a-c}{2}\\qquad \\textbf{(C)}\\ a=\\frac{b-c}{2} \\qquad \\textbf{(D)}\\ a=\\frac{b+c}{2}\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1464", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In this diagram $AB$ and $AC$ are the equal sides of an isosceles $\\triangle ABC$, in which is inscribed equilateral $\\triangle DEF$. \nDesignate $\\angle BFD$ by $a$, $\\angle ADE$ by $b$, and $\\angle FEC$ by $c$. Then:\n\n$\\textbf{(A)}\\ b=\\frac{a+c}{2}\\qquad \\textbf{(B)}\\ b=\\frac{a-c}{2}\\qquad \\textbf{(C)}\\ a=\\frac{b-c}{2} \\qquad \\textbf{(D)}\\ a=\\frac{b+c}{2}\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?\n$\\textbf{(A)}\\ \\frac{5}{4}\\qquad\\textbf{(B)}\\ \\frac{3}{2}\\qquad\\textbf{(C)}\\ \\frac{9}{5}\\qquad\\textbf{(D)}\\ 2 \\qquad\\textbf{(E)}\\ \\frac{5}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1465", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?\n$\\textbf{(A)}\\ \\frac{5}{4}\\qquad\\textbf{(B)}\\ \\frac{3}{2}\\qquad\\textbf{(C)}\\ \\frac{9}{5}\\qquad\\textbf{(D)}\\ 2 \\qquad\\textbf{(E)}\\ \\frac{5}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$?\n$\\textbf{(A)} \\ 55 \\qquad \\textbf{(B)} \\ 56 \\qquad \\textbf{(C)} \\ 57 \\qquad \\textbf{(D)} \\ 58 \\qquad \\textbf{(E)} \\ 59$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1466", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$?\n$\\textbf{(A)} \\ 55 \\qquad \\textbf{(B)} \\ 56 \\qquad \\textbf{(C)} \\ 57 \\qquad \\textbf{(D)} \\ 58 \\qquad \\textbf{(E)} \\ 59$" + } + }, + { + "question": "Return your final response within \\boxed{}. The expression $2 + \\sqrt{2} + \\frac{1}{2 + \\sqrt{2}} + \\frac{1}{\\sqrt{2} - 2}$ equals:\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 2 - \\sqrt{2}\\qquad \\textbf{(C)}\\ 2 + \\sqrt{2}\\qquad \\textbf{(D)}\\ 2\\sqrt{2}\\qquad \\textbf{(E)}\\ \\frac{\\sqrt{2}}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1467", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The expression $2 + \\sqrt{2} + \\frac{1}{2 + \\sqrt{2}} + \\frac{1}{\\sqrt{2} - 2}$ equals:\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 2 - \\sqrt{2}\\qquad \\textbf{(C)}\\ 2 + \\sqrt{2}\\qquad \\textbf{(D)}\\ 2\\sqrt{2}\\qquad \\textbf{(E)}\\ \\frac{\\sqrt{2}}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The difference between the larger root and the smaller root of $x^2 - px + (p^2 - 1)/4 = 0$ is: \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ p\\qquad\\textbf{(E)}\\ p+1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1468", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The difference between the larger root and the smaller root of $x^2 - px + (p^2 - 1)/4 = 0$ is: \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ p\\qquad\\textbf{(E)}\\ p+1$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\log_2(\\log_3(\\log_4 x))=\\log_3(\\log_4(\\log_2 y))=\\log_4(\\log_2(\\log_3 z))=0$, then the sum $x+y+z$ is equal to\n$\\textbf{(A) }50\\qquad \\textbf{(B) }58\\qquad \\textbf{(C) }89\\qquad \\textbf{(D) }111\\qquad \\textbf{(E) }1296$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1469", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\log_2(\\log_3(\\log_4 x))=\\log_3(\\log_4(\\log_2 y))=\\log_4(\\log_2(\\log_3 z))=0$, then the sum $x+y+z$ is equal to\n$\\textbf{(A) }50\\qquad \\textbf{(B) }58\\qquad \\textbf{(C) }89\\qquad \\textbf{(D) }111\\qquad \\textbf{(E) }1296$" + } + }, + { + "question": "Return your final response within \\boxed{}. A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?\n[asy] draw((0,0)--(100,0)--(100,50)--(0,50)--cycle); draw((50,0)--(50,50)); draw((0,25)--(100,25)); [/asy]\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 64 \\qquad \\textbf{(C)}\\ 84 \\qquad \\textbf{(D)}\\ 90 \\qquad \\textbf{(E)}\\ 144$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1470", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?\n[asy] draw((0,0)--(100,0)--(100,50)--(0,50)--cycle); draw((50,0)--(50,50)); draw((0,25)--(100,25)); [/asy]\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 64 \\qquad \\textbf{(C)}\\ 84 \\qquad \\textbf{(D)}\\ 90 \\qquad \\textbf{(E)}\\ 144$" + } + }, + { + "question": "Return your final response within \\boxed{}. A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes alone is:\n$\\textbf{(A)}\\ 8-x=2 \\qquad \\textbf{(B)}\\ \\dfrac{1}{8}+\\dfrac{1}{x}=\\dfrac{1}{2} \\qquad \\textbf{(C)}\\ \\dfrac{500}{8}+\\dfrac{500}{x}=500 \\qquad \\textbf{(D)}\\ \\dfrac{x}{2}+\\dfrac{x}{8}=1 \\qquad\\\\ \\textbf{(E)}\\ \\text{None of these answers}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1471", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes alone is:\n$\\textbf{(A)}\\ 8-x=2 \\qquad \\textbf{(B)}\\ \\dfrac{1}{8}+\\dfrac{1}{x}=\\dfrac{1}{2} \\qquad \\textbf{(C)}\\ \\dfrac{500}{8}+\\dfrac{500}{x}=500 \\qquad \\textbf{(D)}\\ \\dfrac{x}{2}+\\dfrac{x}{8}=1 \\qquad\\\\ \\textbf{(E)}\\ \\text{None of these answers}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?\n$\\textbf{(A)}\\ 585 \\qquad \\textbf{(B)}\\ 594 \\qquad \\textbf{(C)}\\ 672 \\qquad \\textbf{(D)}\\ 679 \\qquad \\textbf{(E)}\\ 694$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1472", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?\n$\\textbf{(A)}\\ 585 \\qquad \\textbf{(B)}\\ 594 \\qquad \\textbf{(C)}\\ 672 \\qquad \\textbf{(D)}\\ 679 \\qquad \\textbf{(E)}\\ 694$" + } + }, + { + "question": "Return your final response within \\boxed{}. For all integers $n \\geq 9,$ the value of\n\\[\\frac{(n+2)!-(n+1)!}{n!}\\]is always which of the following?\n$\\textbf{(A) } \\text{a multiple of 4} \\qquad \\textbf{(B) } \\text{a multiple of 10} \\qquad \\textbf{(C) } \\text{a prime number} \\qquad \\textbf{(D) } \\text{a perfect square} \\qquad \\textbf{(E) } \\text{a perfect cube}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1473", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For all integers $n \\geq 9,$ the value of\n\\[\\frac{(n+2)!-(n+1)!}{n!}\\]is always which of the following?\n$\\textbf{(A) } \\text{a multiple of 4} \\qquad \\textbf{(B) } \\text{a multiple of 10} \\qquad \\textbf{(C) } \\text{a prime number} \\qquad \\textbf{(D) } \\text{a perfect square} \\qquad \\textbf{(E) } \\text{a perfect cube}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose\\[2+\\frac{1}{1+\\frac{1}{2+\\frac{2}{3+x}}}=\\frac{144}{53}.\\]What is the value of $x?$\n$\\textbf{(A) }\\frac34 \\qquad \\textbf{(B) }\\frac78 \\qquad \\textbf{(C) }\\frac{14}{15} \\qquad \\textbf{(D) }\\frac{37}{38} \\qquad \\textbf{(E) }\\frac{52}{53}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1474", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose\\[2+\\frac{1}{1+\\frac{1}{2+\\frac{2}{3+x}}}=\\frac{144}{53}.\\]What is the value of $x?$\n$\\textbf{(A) }\\frac34 \\qquad \\textbf{(B) }\\frac78 \\qquad \\textbf{(C) }\\frac{14}{15} \\qquad \\textbf{(D) }\\frac{37}{38} \\qquad \\textbf{(E) }\\frac{52}{53}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?\n\n$\\mathrm{(A) \\ } 15\\qquad \\mathrm{(B) \\ } 17\\qquad \\mathrm{(C) \\ } \\frac{35}{2}\\qquad \\mathrm{(D) \\ } 18\\qquad \\mathrm{(E) \\ } \\frac{55}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1475", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?\n\n$\\mathrm{(A) \\ } 15\\qquad \\mathrm{(B) \\ } 17\\qquad \\mathrm{(C) \\ } \\frac{35}{2}\\qquad \\mathrm{(D) \\ } 18\\qquad \\mathrm{(E) \\ } \\frac{55}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x<-2$, then $|1-|1+x||$ equals \n$\\mathrm{(A)\\ } 2+x \\qquad \\mathrm{(B) \\ }-2-x \\qquad \\mathrm{(C) \\ } x \\qquad \\mathrm{(D) \\ } -x \\qquad \\mathrm{(E) \\ }-2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1476", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x<-2$, then $|1-|1+x||$ equals \n$\\mathrm{(A)\\ } 2+x \\qquad \\mathrm{(B) \\ }-2-x \\qquad \\mathrm{(C) \\ } x \\qquad \\mathrm{(D) \\ } -x \\qquad \\mathrm{(E) \\ }-2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the two triangles $\\triangle ABC$ and $\\triangle PQR$ shown in Figure 1. In $\\triangle ABC$, $\\angle ADB = \\angle BDC = \\angle CDA = 120^\\circ$. Prove that $x=u+v+w$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "x = u + v + w", + "index": "Sky-T1_10k_1477", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the two triangles $\\triangle ABC$ and $\\triangle PQR$ shown in Figure 1. In $\\triangle ABC$, $\\angle ADB = \\angle BDC = \\angle CDA = 120^\\circ$. Prove that $x=u+v+w$." + } + }, + { + "question": "Return your final response within \\boxed{}. Positive real numbers $a$ and $b$ have the property that\n\\[\\sqrt{\\log{a}} + \\sqrt{\\log{b}} + \\log \\sqrt{a} + \\log \\sqrt{b} = 100\\]\nand all four terms on the left are positive integers, where log denotes the base 10 logarithm. What is $ab$?\n$\\textbf{(A) } 10^{52} \\qquad \\textbf{(B) } 10^{100} \\qquad \\textbf{(C) } 10^{144} \\qquad \\textbf{(D) } 10^{164} \\qquad \\textbf{(E) } 10^{200}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1478", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Positive real numbers $a$ and $b$ have the property that\n\\[\\sqrt{\\log{a}} + \\sqrt{\\log{b}} + \\log \\sqrt{a} + \\log \\sqrt{b} = 100\\]\nand all four terms on the left are positive integers, where log denotes the base 10 logarithm. What is $ab$?\n$\\textbf{(A) } 10^{52} \\qquad \\textbf{(B) } 10^{100} \\qquad \\textbf{(C) } 10^{144} \\qquad \\textbf{(D) } 10^{164} \\qquad \\textbf{(E) } 10^{200}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$\n$\\textbf{(A) }98\\qquad\\textbf{(B) }100\\qquad\\textbf{(C) }117\\qquad\\textbf{(D) }119\\qquad\\textbf{(E) }121$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1479", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$\n$\\textbf{(A) }98\\qquad\\textbf{(B) }100\\qquad\\textbf{(C) }117\\qquad\\textbf{(D) }119\\qquad\\textbf{(E) }121$" + } + }, + { + "question": "Return your final response within \\boxed{}. A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\\%$ of the original price, by what percent was the price increased and decreased$?$\n$\\textbf{(A) }16\\qquad\\textbf{(B) }20\\qquad\\textbf{(C) }28\\qquad\\textbf{(D) }36\\qquad\\textbf{(E) }40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1480", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\\%$ of the original price, by what percent was the price increased and decreased$?$\n$\\textbf{(A) }16\\qquad\\textbf{(B) }20\\qquad\\textbf{(C) }28\\qquad\\textbf{(D) }36\\qquad\\textbf{(E) }40$" + } + }, + { + "question": "Return your final response within \\boxed{}. In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to \n\n$\\mathrm{(A)\\ } k \\qquad \\mathrm{(B) \\ }\\frac{6k}{5} \\qquad \\mathrm{(C) \\ } \\frac{9k}{8} \\qquad \\mathrm{(D) \\ } \\frac{5k}{4} \\qquad \\mathrm{(E) \\ }2k$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1481", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to \n\n$\\mathrm{(A)\\ } k \\qquad \\mathrm{(B) \\ }\\frac{6k}{5} \\qquad \\mathrm{(C) \\ } \\frac{9k}{8} \\qquad \\mathrm{(D) \\ } \\frac{5k}{4} \\qquad \\mathrm{(E) \\ }2k$" + } + }, + { + "question": "Return your final response within \\boxed{}. Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\\%$ of the group are girls. How many girls were initially in the group?\n$\\textbf{(A) } 4 \\qquad\\textbf{(B) } 6 \\qquad\\textbf{(C) } 8 \\qquad\\textbf{(D) } 10 \\qquad\\textbf{(E) } 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1482", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\\%$ of the group are girls. How many girls were initially in the group?\n$\\textbf{(A) } 4 \\qquad\\textbf{(B) } 6 \\qquad\\textbf{(C) } 8 \\qquad\\textbf{(D) } 10 \\qquad\\textbf{(E) } 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many (not necessarily positive) integer values of $n$ is the value of $4000\\cdot \\left(\\tfrac{2}{5}\\right)^n$ an integer?\n$\\textbf{(A) }3 \\qquad \\textbf{(B) }4 \\qquad \\textbf{(C) }6 \\qquad \\textbf{(D) }8 \\qquad \\textbf{(E) }9 \\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1483", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many (not necessarily positive) integer values of $n$ is the value of $4000\\cdot \\left(\\tfrac{2}{5}\\right)^n$ an integer?\n$\\textbf{(A) }3 \\qquad \\textbf{(B) }4 \\qquad \\textbf{(C) }6 \\qquad \\textbf{(D) }8 \\qquad \\textbf{(E) }9 \\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Prove that for any integer $n$, there exists a unique polynomial $Q(x)$ with coefficients in $\\{0,1,...,9\\}$ such that $Q(-2)=Q(-5)=n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "Such a polynomial Q(x) exists and is unique for any integer n.", + "index": "Sky-T1_10k_1484", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Prove that for any integer $n$, there exists a unique polynomial $Q(x)$ with coefficients in $\\{0,1,...,9\\}$ such that $Q(-2)=Q(-5)=n$." + } + }, + { + "question": "Return your final response within \\boxed{}. If $a-1=b+2=c-3=d+4$, which of the four quantities $a,b,c,d$ is the largest?\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ b \\qquad \\textbf{(C)}\\ c \\qquad \\textbf{(D)}\\ d \\qquad \\textbf{(E)}\\ \\text{no one is always largest}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1485", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a-1=b+2=c-3=d+4$, which of the four quantities $a,b,c,d$ is the largest?\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ b \\qquad \\textbf{(C)}\\ c \\qquad \\textbf{(D)}\\ d \\qquad \\textbf{(E)}\\ \\text{no one is always largest}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $f(x)=4^x$ then $f(x+1)-f(x)$ equals:\n$\\text{(A)}\\ 4\\qquad\\text{(B)}\\ f(x)\\qquad\\text{(C)}\\ 2f(x)\\qquad\\text{(D)}\\ 3f(x)\\qquad\\text{(E)}\\ 4f(x)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1486", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $f(x)=4^x$ then $f(x+1)-f(x)$ equals:\n$\\text{(A)}\\ 4\\qquad\\text{(B)}\\ f(x)\\qquad\\text{(C)}\\ 2f(x)\\qquad\\text{(D)}\\ 3f(x)\\qquad\\text{(E)}\\ 4f(x)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with $2$ goes into cup $E$ and a slip with $3$ goes into cup $B$, then the slip with $3.5$ must go into what cup?\n$\\textbf{(A) } A \\qquad \\textbf{(B) } B \\qquad \\textbf{(C) } C \\qquad \\textbf{(D) } D \\qquad \\textbf{(E) } E$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1487", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with $2$ goes into cup $E$ and a slip with $3$ goes into cup $B$, then the slip with $3.5$ must go into what cup?\n$\\textbf{(A) } A \\qquad \\textbf{(B) } B \\qquad \\textbf{(C) } C \\qquad \\textbf{(D) } D \\qquad \\textbf{(E) } E$" + } + }, + { + "question": "Return your final response within \\boxed{}. Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?\n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 20\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 40\\qquad\\textbf{(E)}\\ 50$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1488", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?\n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 20\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 40\\qquad\\textbf{(E)}\\ 50$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many positive integers less than $1000$ are $6$ times the sum of their digits?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1489", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many positive integers less than $1000$ are $6$ times the sum of their digits?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Roy's cat eats $\\frac{1}{3}$ of a can of cat food every morning and $\\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?\n$\\textbf{(A)}\\ \\text{Tuesday}\\qquad\\textbf{(B)}\\ \\text{Wednesday}\\qquad\\textbf{(C)}\\ \\text{Thursday}\\qquad\\textbf{(D)}\\ \\text{Friday}\\qquad\\textbf{(E)}\\ \\text{Saturday}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1490", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Roy's cat eats $\\frac{1}{3}$ of a can of cat food every morning and $\\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?\n$\\textbf{(A)}\\ \\text{Tuesday}\\qquad\\textbf{(B)}\\ \\text{Wednesday}\\qquad\\textbf{(C)}\\ \\text{Thursday}\\qquad\\textbf{(D)}\\ \\text{Friday}\\qquad\\textbf{(E)}\\ \\text{Saturday}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many positive integers $n$ is $\\frac{n}{30-n}$ also a positive integer?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1491", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many positive integers $n$ is $\\frac{n}{30-n}$ also a positive integer?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $N$ be the positive integer $7777\\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is\\[f(2) + f(3) + f(4) + f(5)+ f(6)?\\]$(\\textbf{A})\\: 8\\qquad(\\textbf{B}) \\: 9\\qquad(\\textbf{C}) \\: 11\\qquad(\\textbf{D}) \\: 22\\qquad(\\textbf{E}) \\: 29$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1492", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $N$ be the positive integer $7777\\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is\\[f(2) + f(3) + f(4) + f(5)+ f(6)?\\]$(\\textbf{A})\\: 8\\qquad(\\textbf{B}) \\: 9\\qquad(\\textbf{C}) \\: 11\\qquad(\\textbf{D}) \\: 22\\qquad(\\textbf{E}) \\: 29$" + } + }, + { + "question": "Return your final response within \\boxed{}. Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book?\n\n$\\textbf{(A) }240\\qquad\\textbf{(B) }250\\qquad\\textbf{(C) }260\\qquad\\textbf{(D) }270\\qquad \\textbf{(E) }280$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1493", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book?\n\n$\\textbf{(A) }240\\qquad\\textbf{(B) }250\\qquad\\textbf{(C) }260\\qquad\\textbf{(D) }270\\qquad \\textbf{(E) }280$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?\n$\\textbf{(A)}\\ 52\\qquad\\textbf{(B)}\\ 57\\qquad\\textbf{(C)}\\ 62\\qquad\\textbf{(D)}\\ 67\\qquad\\textbf{(E)}\\ 72$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1494", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?\n$\\textbf{(A)}\\ 52\\qquad\\textbf{(B)}\\ 57\\qquad\\textbf{(C)}\\ 62\\qquad\\textbf{(D)}\\ 67\\qquad\\textbf{(E)}\\ 72$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\text{o}$ be an odd whole number and let $\\text{n}$ be any whole number. Which of the following statements about the whole number $(\\text{o}^2+\\text{no})$ is always true?\n$\\text{(A)}\\ \\text{it is always odd} \\qquad \\text{(B)}\\ \\text{it is always even}$\n$\\text{(C)}\\ \\text{it is even only if n is even} \\qquad \\text{(D)}\\ \\text{it is odd only if n is odd}$\n$\\text{(E)}\\ \\text{it is odd only if n is even}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1495", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\text{o}$ be an odd whole number and let $\\text{n}$ be any whole number. Which of the following statements about the whole number $(\\text{o}^2+\\text{no})$ is always true?\n$\\text{(A)}\\ \\text{it is always odd} \\qquad \\text{(B)}\\ \\text{it is always even}$\n$\\text{(C)}\\ \\text{it is even only if n is even} \\qquad \\text{(D)}\\ \\text{it is odd only if n is odd}$\n$\\text{(E)}\\ \\text{it is odd only if n is even}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?\n$\\textbf{(A)}\\ 480 \\qquad\\textbf{(B)}\\ 481 \\qquad\\textbf{(C)}\\ 482 \\qquad\\textbf{(D)}\\ 483 \\qquad\\textbf{(E)}\\ 484$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1496", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?\n$\\textbf{(A)}\\ 480 \\qquad\\textbf{(B)}\\ 481 \\qquad\\textbf{(C)}\\ 482 \\qquad\\textbf{(D)}\\ 483 \\qquad\\textbf{(E)}\\ 484$" + } + }, + { + "question": "Return your final response within \\boxed{}. The letters $A$, $B$, $C$ and $D$ represent digits. If $\\begin{tabular}{ccc}&A&B\\\\ +&C&A\\\\ \\hline &D&A\\end{tabular}$and $\\begin{tabular}{ccc}&A&B\\\\ -&C&A\\\\ \\hline &&A\\end{tabular}$,what digit does $D$ represent?\n$\\textbf{(A)}\\ 5\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 7\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1497", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The letters $A$, $B$, $C$ and $D$ represent digits. If $\\begin{tabular}{ccc}&A&B\\\\ +&C&A\\\\ \\hline &D&A\\end{tabular}$and $\\begin{tabular}{ccc}&A&B\\\\ -&C&A\\\\ \\hline &&A\\end{tabular}$,what digit does $D$ represent?\n$\\textbf{(A)}\\ 5\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 7\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. If you walk for $45$ minutes at a [rate](https://artofproblemsolving.com/wiki/index.php/Rate) of $4 \\text{ mph}$ and then run for $30$ minutes at a rate of $10\\text{ mph}$, how many miles will you have gone at the end of one hour and $15$ minutes?\n$\\text{(A)}\\ 3.5\\text{ miles} \\qquad \\text{(B)}\\ 8\\text{ miles} \\qquad \\text{(C)}\\ 9\\text{ miles} \\qquad \\text{(D)}\\ 25\\frac{1}{3}\\text{ miles} \\qquad \\text{(E)}\\ 480\\text{ miles}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1498", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If you walk for $45$ minutes at a [rate](https://artofproblemsolving.com/wiki/index.php/Rate) of $4 \\text{ mph}$ and then run for $30$ minutes at a rate of $10\\text{ mph}$, how many miles will you have gone at the end of one hour and $15$ minutes?\n$\\text{(A)}\\ 3.5\\text{ miles} \\qquad \\text{(B)}\\ 8\\text{ miles} \\qquad \\text{(C)}\\ 9\\text{ miles} \\qquad \\text{(D)}\\ 25\\frac{1}{3}\\text{ miles} \\qquad \\text{(E)}\\ 480\\text{ miles}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the product shown, $\\text{B}$ is a digit. The value of $\\text{B}$ is\n\\[\\begin{array}{rr} &\\text{B}2 \\\\ \\times& 7\\text{B} \\\\ \\hline &6396 \\\\ \\end{array}\\]\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1499", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the product shown, $\\text{B}$ is a digit. The value of $\\text{B}$ is\n\\[\\begin{array}{rr} &\\text{B}2 \\\\ \\times& 7\\text{B} \\\\ \\hline &6396 \\\\ \\end{array}\\]\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box?\n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 7\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1500", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box?\n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 7\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the binary operation $\\star$ defined by $a\\star b=a^b$ for all positive numbers $a$ and $b$. Then for all positive $a,b,c,n$, we have\n$\\text{(A) } a\\star b=b\\star a\\quad\\qquad\\qquad\\ \\text{(B) } a\\star (b\\star c)=(a\\star b) \\star c\\quad\\\\ \\text{(C) } (a\\star b^n)=(a \\star n) \\star b\\quad \\text{(D) } (a\\star b)^n =a\\star (bn)\\quad\\\\ \\text{(E) None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1501", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the binary operation $\\star$ defined by $a\\star b=a^b$ for all positive numbers $a$ and $b$. Then for all positive $a,b,c,n$, we have\n$\\text{(A) } a\\star b=b\\star a\\quad\\qquad\\qquad\\ \\text{(B) } a\\star (b\\star c)=(a\\star b) \\star c\\quad\\\\ \\text{(C) } (a\\star b^n)=(a \\star n) \\star b\\quad \\text{(D) } (a\\star b)^n =a\\star (bn)\\quad\\\\ \\text{(E) None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money?\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 6.5\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 13$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1502", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money?\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 6.5\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 13$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $D=a^2+b^2+c^2$, where $a$, $b$, are consecutive integers and $c=ab$. Then $\\sqrt{D}$ is:\n$\\textbf{(A)}\\ \\text{always an even integer}\\qquad \\textbf{(B)}\\ \\text{sometimes an odd integer, sometimes not}\\\\ \\textbf{(C)}\\ \\text{always an odd integer}\\qquad \\textbf{(D)}\\ \\text{sometimes rational, sometimes not}\\\\ \\textbf{(E)}\\ \\text{always irrational}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1503", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $D=a^2+b^2+c^2$, where $a$, $b$, are consecutive integers and $c=ab$. Then $\\sqrt{D}$ is:\n$\\textbf{(A)}\\ \\text{always an even integer}\\qquad \\textbf{(B)}\\ \\text{sometimes an odd integer, sometimes not}\\\\ \\textbf{(C)}\\ \\text{always an odd integer}\\qquad \\textbf{(D)}\\ \\text{sometimes rational, sometimes not}\\\\ \\textbf{(E)}\\ \\text{always irrational}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Sandwiches at Joe's Fast Food cost $$3$ each and sodas cost $$2$ each. How many dollars will it cost to purchase $5$ sandwiches and $8$ sodas?\n$\\textbf{(A)}\\ 31\\qquad\\textbf{(B)}\\ 32\\qquad\\textbf{(C)}\\ 33\\qquad\\textbf{(D)}\\ 34\\qquad\\textbf{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1504", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Sandwiches at Joe's Fast Food cost $$3$ each and sodas cost $$2$ each. How many dollars will it cost to purchase $5$ sandwiches and $8$ sodas?\n$\\textbf{(A)}\\ 31\\qquad\\textbf{(B)}\\ 32\\qquad\\textbf{(C)}\\ 33\\qquad\\textbf{(D)}\\ 34\\qquad\\textbf{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\mathit{B}$ is a point on circle $\\mathit{C}$ with center $\\mathit{P}$, then the set of all points $\\mathit{A}$ in the plane of circle $\\mathit{C}$ such that the distance between $\\mathit{A}$ and $\\mathit{B}$ is less than or equal to the distance between $\\mathit{A}$ \nand any other point on circle $\\mathit{C}$ is\n$\\textbf{(A) }\\text{the line segment from }P \\text{ to }B\\qquad\\\\ \\textbf{(B) }\\text{the ray beginning at }P \\text{ and passing through }B\\qquad\\\\ \\textbf{(C) }\\text{a ray beginning at }B\\qquad\\\\ \\textbf{(D) }\\text{a circle whose center is }P\\qquad\\\\ \\textbf{(E) }\\text{a circle whose center is }B$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1505", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\mathit{B}$ is a point on circle $\\mathit{C}$ with center $\\mathit{P}$, then the set of all points $\\mathit{A}$ in the plane of circle $\\mathit{C}$ such that the distance between $\\mathit{A}$ and $\\mathit{B}$ is less than or equal to the distance between $\\mathit{A}$ \nand any other point on circle $\\mathit{C}$ is\n$\\textbf{(A) }\\text{the line segment from }P \\text{ to }B\\qquad\\\\ \\textbf{(B) }\\text{the ray beginning at }P \\text{ and passing through }B\\qquad\\\\ \\textbf{(C) }\\text{a ray beginning at }B\\qquad\\\\ \\textbf{(D) }\\text{a circle whose center is }P\\qquad\\\\ \\textbf{(E) }\\text{a circle whose center is }B$" + } + }, + { + "question": "Return your final response within \\boxed{}. A circle passes through the vertices of a triangle with side-lengths $7\\tfrac{1}{2},10,12\\tfrac{1}{2}.$ The radius of the circle is:\n$\\text{(A) } \\frac{15}{4}\\quad \\text{(B) } 5\\quad \\text{(C) } \\frac{25}{4}\\quad \\text{(D) } \\frac{35}{4}\\quad \\text{(E) } \\frac{15\\sqrt{2}}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1506", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A circle passes through the vertices of a triangle with side-lengths $7\\tfrac{1}{2},10,12\\tfrac{1}{2}.$ The radius of the circle is:\n$\\text{(A) } \\frac{15}{4}\\quad \\text{(B) } 5\\quad \\text{(C) } \\frac{25}{4}\\quad \\text{(D) } \\frac{35}{4}\\quad \\text{(E) } \\frac{15\\sqrt{2}}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The lines $x = \\frac 14y + a$ and $y = \\frac 14x + b$ intersect at the point $(1,2)$. What is $a + b$?\n$\\text {(A) } 0 \\qquad \\text {(B) } \\frac 34 \\qquad \\text {(C) } 1 \\qquad \\text {(D) } 2 \\qquad \\text {(E) } \\frac 94$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1507", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The lines $x = \\frac 14y + a$ and $y = \\frac 14x + b$ intersect at the point $(1,2)$. What is $a + b$?\n$\\text {(A) } 0 \\qquad \\text {(B) } \\frac 34 \\qquad \\text {(C) } 1 \\qquad \\text {(D) } 2 \\qquad \\text {(E) } \\frac 94$" + } + }, + { + "question": "Return your final response within \\boxed{}. A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1508", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$?\n$\\textbf{(A)} \\text{ 13} \\qquad \\textbf{(B)} \\text{ 26} \\qquad \\textbf{(C)} \\text{ 32} \\qquad \\textbf{(D)} \\text{ 39} \\qquad \\textbf{(E)} \\text{ 42}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1509", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$?\n$\\textbf{(A)} \\text{ 13} \\qquad \\textbf{(B)} \\text{ 26} \\qquad \\textbf{(C)} \\text{ 32} \\qquad \\textbf{(D)} \\text{ 39} \\qquad \\textbf{(E)} \\text{ 42}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of $2010$ boxes in a line contains a single red marble, and for $1 \\le k \\le 2010$, the box in the $k\\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \\frac{1}{2010}$?\n$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 63 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 201 \\qquad \\textbf{(E)}\\ 1005$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1510", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of $2010$ boxes in a line contains a single red marble, and for $1 \\le k \\le 2010$, the box in the $k\\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \\frac{1}{2010}$?\n$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 63 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 201 \\qquad \\textbf{(E)}\\ 1005$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?\n\n$\\textbf{(A)}\\ 72\\qquad\\textbf{(B)}\\ 78\\qquad\\textbf{(C)}\\ 90\\qquad\\textbf{(D)}\\ 120\\qquad\\textbf{(E)}\\ 150$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1511", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?\n\n$\\textbf{(A)}\\ 72\\qquad\\textbf{(B)}\\ 78\\qquad\\textbf{(C)}\\ 90\\qquad\\textbf{(D)}\\ 120\\qquad\\textbf{(E)}\\ 150$" + } + }, + { + "question": "Return your final response within \\boxed{}. Handy Aaron helped a neighbor $1 \\frac14$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\\textdollar 3$ per hour. How much did he earn for the week?\n$\\textbf{(A)}\\ \\textdollar 8 \\qquad \\textbf{(B)}\\ \\textdollar 9 \\qquad \\textbf{(C)}\\ \\textdollar 10 \\qquad \\textbf{(D)}\\ \\textdollar 12 \\qquad \\textbf{(E)}\\ \\textdollar 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1512", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Handy Aaron helped a neighbor $1 \\frac14$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\\textdollar 3$ per hour. How much did he earn for the week?\n$\\textbf{(A)}\\ \\textdollar 8 \\qquad \\textbf{(B)}\\ \\textdollar 9 \\qquad \\textbf{(C)}\\ \\textdollar 10 \\qquad \\textbf{(D)}\\ \\textdollar 12 \\qquad \\textbf{(E)}\\ \\textdollar 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. For real numbers $x$ and $y$, define $x \\spadesuit y = (x+y)(x-y)$. What is $3 \\spadesuit (4 \\spadesuit 5)$?\n$\\mathrm{(A) \\ } -72\\qquad \\mathrm{(B) \\ } -27\\qquad \\mathrm{(C) \\ } -24\\qquad \\mathrm{(D) \\ } 24\\qquad \\mathrm{(E) \\ } 72$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1513", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For real numbers $x$ and $y$, define $x \\spadesuit y = (x+y)(x-y)$. What is $3 \\spadesuit (4 \\spadesuit 5)$?\n$\\mathrm{(A) \\ } -72\\qquad \\mathrm{(B) \\ } -27\\qquad \\mathrm{(C) \\ } -24\\qquad \\mathrm{(D) \\ } 24\\qquad \\mathrm{(E) \\ } 72$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many distinguishable rearrangements of the letters in $CONTEST$ have both the vowels first? (For instance, $OETCNST$ is one such arrangement but $OTETSNC$ is not.)\n$\\mathrm{(A)\\ } 60 \\qquad \\mathrm{(B) \\ }120 \\qquad \\mathrm{(C) \\ } 240 \\qquad \\mathrm{(D) \\ } 720 \\qquad \\mathrm{(E) \\ }2520$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1514", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many distinguishable rearrangements of the letters in $CONTEST$ have both the vowels first? (For instance, $OETCNST$ is one such arrangement but $OTETSNC$ is not.)\n$\\mathrm{(A)\\ } 60 \\qquad \\mathrm{(B) \\ }120 \\qquad \\mathrm{(C) \\ } 240 \\qquad \\mathrm{(D) \\ } 720 \\qquad \\mathrm{(E) \\ }2520$" + } + }, + { + "question": "Return your final response within \\boxed{}. Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?\n$\\textbf{(A) } 15\\qquad \\textbf{(B) } 34\\qquad \\textbf{(C) } 43\\qquad \\textbf{(D) } 51\\qquad \\textbf{(E) } 138$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1515", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?\n$\\textbf{(A) } 15\\qquad \\textbf{(B) } 34\\qquad \\textbf{(C) } 43\\qquad \\textbf{(D) } 51\\qquad \\textbf{(E) } 138$" + } + }, + { + "question": "Return your final response within \\boxed{}. The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves\nthe room. What is the average age of the four remaining people?\n$\\mathrm{(A)}\\ 25 \\qquad\\mathrm{(B)}\\ 26 \\qquad\\mathrm{(C)}\\ 29 \\qquad\\mathrm{(D)}\\ 33 \\qquad\\mathrm{(E)}\\ 36$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1516", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves\nthe room. What is the average age of the four remaining people?\n$\\mathrm{(A)}\\ 25 \\qquad\\mathrm{(B)}\\ 26 \\qquad\\mathrm{(C)}\\ 29 \\qquad\\mathrm{(D)}\\ 33 \\qquad\\mathrm{(E)}\\ 36$" + } + }, + { + "question": "Return your final response within \\boxed{}. Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1517", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a,b,c$ are real numbers such that $a^2 + 2b =7$, $b^2 + 4c= -7,$ and $c^2 + 6a= -14$, find $a^2 + b^2 + c^2.$\n$\\text{(A) }14 \\qquad \\text{(B) }21 \\qquad \\text{(C) }28 \\qquad \\text{(D) }35 \\qquad \\text{(E) }49$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1518", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a,b,c$ are real numbers such that $a^2 + 2b =7$, $b^2 + 4c= -7,$ and $c^2 + 6a= -14$, find $a^2 + b^2 + c^2.$\n$\\text{(A) }14 \\qquad \\text{(B) }21 \\qquad \\text{(C) }28 \\qquad \\text{(D) }35 \\qquad \\text{(E) }49$" + } + }, + { + "question": "Return your final response within \\boxed{}. Of the following statements, the one that is incorrect is:\n$\\textbf{(A)}\\ \\text{Doubling the base of a given rectangle doubles the area.}$\n$\\textbf{(B)}\\ \\text{Doubling the altitude of a triangle doubles the area.}$\n$\\textbf{(C)}\\ \\text{Doubling the radius of a given circle doubles the area.}$\n$\\textbf{(D)}\\ \\text{Doubling the divisor of a fraction and dividing its numerator by 2 changes the quotient.}$\n$\\textbf{(E)}\\ \\text{Doubling a given quantity may make it less than it originally was.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1519", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Of the following statements, the one that is incorrect is:\n$\\textbf{(A)}\\ \\text{Doubling the base of a given rectangle doubles the area.}$\n$\\textbf{(B)}\\ \\text{Doubling the altitude of a triangle doubles the area.}$\n$\\textbf{(C)}\\ \\text{Doubling the radius of a given circle doubles the area.}$\n$\\textbf{(D)}\\ \\text{Doubling the divisor of a fraction and dividing its numerator by 2 changes the quotient.}$\n$\\textbf{(E)}\\ \\text{Doubling a given quantity may make it less than it originally was.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For real numbers $a$ and $b$, define $a\\textdollar b = (a - b)^2$. What is $(x - y)^2\\textdollar(y - x)^2$?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ x^2 + y^2 \\qquad \\textbf{(C)}\\ 2x^2 \\qquad \\textbf{(D)}\\ 2y^2 \\qquad \\textbf{(E)}\\ 4xy$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1520", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For real numbers $a$ and $b$, define $a\\textdollar b = (a - b)^2$. What is $(x - y)^2\\textdollar(y - x)^2$?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ x^2 + y^2 \\qquad \\textbf{(C)}\\ 2x^2 \\qquad \\textbf{(D)}\\ 2y^2 \\qquad \\textbf{(E)}\\ 4xy$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $4^x - 4^{x - 1} = 24$, then $(2x)^x$ equals:\n$\\textbf{(A)}\\ 5\\sqrt{5}\\qquad \\textbf{(B)}\\ \\sqrt{5}\\qquad \\textbf{(C)}\\ 25\\sqrt{5}\\qquad \\textbf{(D)}\\ 125\\qquad \\textbf{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1521", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $4^x - 4^{x - 1} = 24$, then $(2x)^x$ equals:\n$\\textbf{(A)}\\ 5\\sqrt{5}\\qquad \\textbf{(B)}\\ \\sqrt{5}\\qquad \\textbf{(C)}\\ 25\\sqrt{5}\\qquad \\textbf{(D)}\\ 125\\qquad \\textbf{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. Quadrilateral $ABCD$ satisfies $\\angle ABC = \\angle ACD = 90^{\\circ}, AC=20,$ and $CD=30.$ Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$\n$\\textbf{(A) } 330 \\qquad \\textbf{(B) } 340 \\qquad \\textbf{(C) } 350 \\qquad \\textbf{(D) } 360 \\qquad \\textbf{(E) } 370$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1522", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Quadrilateral $ABCD$ satisfies $\\angle ABC = \\angle ACD = 90^{\\circ}, AC=20,$ and $CD=30.$ Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$\n$\\textbf{(A) } 330 \\qquad \\textbf{(B) } 340 \\qquad \\textbf{(C) } 350 \\qquad \\textbf{(D) } 360 \\qquad \\textbf{(E) } 370$" + } + }, + { + "question": "Return your final response within \\boxed{}. The values of $y$ which will satisfy the equations $2x^{2}+6x+5y+1=0, 2x+y+3=0$ may be found by solving:\n$\\textbf{(A)}\\ y^{2}+14y-7=0\\qquad\\textbf{(B)}\\ y^{2}+8y+1=0\\qquad\\textbf{(C)}\\ y^{2}+10y-7=0\\qquad\\\\ \\textbf{(D)}\\ y^{2}+y-12=0\\qquad \\textbf{(E)}\\ \\text{None of these equations}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1523", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The values of $y$ which will satisfy the equations $2x^{2}+6x+5y+1=0, 2x+y+3=0$ may be found by solving:\n$\\textbf{(A)}\\ y^{2}+14y-7=0\\qquad\\textbf{(B)}\\ y^{2}+8y+1=0\\qquad\\textbf{(C)}\\ y^{2}+10y-7=0\\qquad\\\\ \\textbf{(D)}\\ y^{2}+y-12=0\\qquad \\textbf{(E)}\\ \\text{None of these equations}$" + } + }, + { + "question": "Return your final response within \\boxed{}. An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = 1.60, how much lire will the traveler receive in exchange for 1.00?\n$\\text{(A)}\\ 180 \\qquad \\text{(B)}\\ 480 \\qquad \\text{(C)}\\ 1800 \\qquad \\text{(D)}\\ 1875 \\qquad \\text{(E)}\\ 4875$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1524", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = 1.60, how much lire will the traveler receive in exchange for 1.00?\n$\\text{(A)}\\ 180 \\qquad \\text{(B)}\\ 480 \\qquad \\text{(C)}\\ 1800 \\qquad \\text{(D)}\\ 1875 \\qquad \\text{(E)}\\ 4875$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are $N$ [permutations](https://artofproblemsolving.com/wiki/index.php/Permutation) $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \\ldots, 30$ such that for $m \\in \\left\\{{2, 3, 5}\\right\\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \\leq n < n+m \\leq 30$. Find the remainder when $N$ is divided by $1000$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "440", + "index": "Sky-T1_10k_1525", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are $N$ [permutations](https://artofproblemsolving.com/wiki/index.php/Permutation) $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \\ldots, 30$ such that for $m \\in \\left\\{{2, 3, 5}\\right\\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \\leq n < n+m \\leq 30$. Find the remainder when $N$ is divided by $1000$." + } + }, + { + "question": "Return your final response within \\boxed{}. Isabella has $6$ coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every $10$ days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the $6$ dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?\n$\\textbf{(A) }\\text{Monday}\\qquad\\textbf{(B) }\\text{Tuesday}\\qquad\\textbf{(C) }\\text{Wednesday}\\qquad\\textbf{(D) }\\text{Thursday}\\qquad\\textbf{(E) }\\text{Friday}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1526", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Isabella has $6$ coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every $10$ days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the $6$ dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?\n$\\textbf{(A) }\\text{Monday}\\qquad\\textbf{(B) }\\text{Tuesday}\\qquad\\textbf{(C) }\\text{Wednesday}\\qquad\\textbf{(D) }\\text{Thursday}\\qquad\\textbf{(E) }\\text{Friday}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the probability that a randomly drawn positive factor of $60$ is less than $7$?\n$\\mathrm{(A) \\ } \\frac{1}{10}\\qquad \\mathrm{(B) \\ } \\frac{1}{6}\\qquad \\mathrm{(C) \\ } \\frac{1}{4}\\qquad \\mathrm{(D) \\ } \\frac{1}{3}\\qquad \\mathrm{(E) \\ } \\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1527", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the probability that a randomly drawn positive factor of $60$ is less than $7$?\n$\\mathrm{(A) \\ } \\frac{1}{10}\\qquad \\mathrm{(B) \\ } \\frac{1}{6}\\qquad \\mathrm{(C) \\ } \\frac{1}{4}\\qquad \\mathrm{(D) \\ } \\frac{1}{3}\\qquad \\mathrm{(E) \\ } \\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. All three vertices of $\\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1528", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. All three vertices of $\\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| < 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then\n$\\textbf{(A)}\\ 1 < r < \\dfrac{3}{2}\\qquad \\textbf{(B)}\\ \\dfrac{3}{2} < r < 2\\qquad \\textbf{(C)}\\ 2 < r < \\dfrac{5}{2}\\qquad \\textbf{(D)}\\ \\dfrac{5}{2} < r < 3\\qquad \\\\ \\textbf{(E)}\\ 3 < r < \\dfrac{7}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1529", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| < 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then\n$\\textbf{(A)}\\ 1 < r < \\dfrac{3}{2}\\qquad \\textbf{(B)}\\ \\dfrac{3}{2} < r < 2\\qquad \\textbf{(C)}\\ 2 < r < \\dfrac{5}{2}\\qquad \\textbf{(D)}\\ \\dfrac{5}{2} < r < 3\\qquad \\\\ \\textbf{(E)}\\ 3 < r < \\dfrac{7}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k}$ is equal to\n$\\textbf{(A) }2^{-2k}\\qquad \\textbf{(B) }2^{-(2k-1)}\\qquad \\textbf{(C) }-2^{-(2k+1)}\\qquad \\textbf{(D) }0\\qquad \\textbf{(E) }2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1530", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k}$ is equal to\n$\\textbf{(A) }2^{-2k}\\qquad \\textbf{(B) }2^{-(2k-1)}\\qquad \\textbf{(C) }-2^{-(2k+1)}\\qquad \\textbf{(D) }0\\qquad \\textbf{(E) }2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $p$ be a prime, and let $a_1, \\dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \\[a_1 + k, a_2 + 2k, \\dots, a_p + pk\\]produce at least $\\tfrac{1}{2} p$ distinct remainders upon division by $p$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "Such an integer k exists.", + "index": "Sky-T1_10k_1531", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $p$ be a prime, and let $a_1, \\dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \\[a_1 + k, a_2 + 2k, \\dots, a_p + pk\\]produce at least $\\tfrac{1}{2} p$ distinct remainders upon division by $p$." + } + }, + { + "question": "Return your final response within \\boxed{}. For positive real numbers $s$, let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "736", + "index": "Sky-T1_10k_1532", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For positive real numbers $s$, let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$." + } + }, + { + "question": "Return your final response within \\boxed{}. How many integers $x$ satisfy the equation $(x^2-x-1)^{x+2}=1?$\n$\\mathrm{(A)\\ } 2 \\qquad \\mathrm{(B) \\ }3 \\qquad \\mathrm{(C) \\ } 4 \\qquad \\mathrm{(D) \\ } 5 \\qquad \\mathrm{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1533", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many integers $x$ satisfy the equation $(x^2-x-1)^{x+2}=1?$\n$\\mathrm{(A)\\ } 2 \\qquad \\mathrm{(B) \\ }3 \\qquad \\mathrm{(C) \\ } 4 \\qquad \\mathrm{(D) \\ } 5 \\qquad \\mathrm{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $x_1$ and $x_2$ be such that $x_1\\not=x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals\n$\\mathrm{(A)\\ } -\\frac{h}{3} \\qquad \\mathrm{(B) \\ }\\frac{h}{3} \\qquad \\mathrm{(C) \\ } \\frac{b}{3} \\qquad \\mathrm{(D) \\ } 2b \\qquad \\mathrm{(E) \\ }-\\frac{b}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1534", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $x_1$ and $x_2$ be such that $x_1\\not=x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals\n$\\mathrm{(A)\\ } -\\frac{h}{3} \\qquad \\mathrm{(B) \\ }\\frac{h}{3} \\qquad \\mathrm{(C) \\ } \\frac{b}{3} \\qquad \\mathrm{(D) \\ } 2b \\qquad \\mathrm{(E) \\ }-\\frac{b}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\sqrt{\\frac{1}{9}+\\frac{1}{16}}=$\n$\\textrm{(A)}\\ \\frac{1}5\\qquad\\textrm{(B)}\\ \\frac{1}4\\qquad\\textrm{(C)}\\ \\frac{2}7\\qquad\\textrm{(D)}\\ \\frac{5}{12}\\qquad\\textrm{(E)}\\ \\frac{7}{12}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1535", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\sqrt{\\frac{1}{9}+\\frac{1}{16}}=$\n$\\textrm{(A)}\\ \\frac{1}5\\qquad\\textrm{(B)}\\ \\frac{1}4\\qquad\\textrm{(C)}\\ \\frac{2}7\\qquad\\textrm{(D)}\\ \\frac{5}{12}\\qquad\\textrm{(E)}\\ \\frac{7}{12}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The altitude drawn to the base of an isosceles triangle is $8$, and the perimeter $32$. The area of the triangle is:\n$\\textbf{(A)}\\ 56\\qquad \\textbf{(B)}\\ 48\\qquad \\textbf{(C)}\\ 40\\qquad \\textbf{(D)}\\ 32\\qquad \\textbf{(E)}\\ 24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1536", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The altitude drawn to the base of an isosceles triangle is $8$, and the perimeter $32$. The area of the triangle is:\n$\\textbf{(A)}\\ 56\\qquad \\textbf{(B)}\\ 48\\qquad \\textbf{(C)}\\ 40\\qquad \\textbf{(D)}\\ 32\\qquad \\textbf{(E)}\\ 24$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many ordered pairs of integers $(x, y)$ satisfy the equation \\[x^{2020}+y^2=2y?\\]\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } 2 \\qquad\\textbf{(C) } 3 \\qquad\\textbf{(D) } 4 \\qquad\\textbf{(E) } \\text{infinitely many}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1537", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many ordered pairs of integers $(x, y)$ satisfy the equation \\[x^{2020}+y^2=2y?\\]\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } 2 \\qquad\\textbf{(C) } 3 \\qquad\\textbf{(D) } 4 \\qquad\\textbf{(E) } \\text{infinitely many}$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are twenty-four $4$-digit numbers that use each of the four digits $2$, $4$, $5$, and $7$ exactly once. Listed in numerical order from smallest to largest, the number in the $17\\text{th}$ position in the list is\n$\\text{(A)}\\ 4527 \\qquad \\text{(B)}\\ 5724 \\qquad \\text{(C)}\\ 5742 \\qquad \\text{(D)}\\ 7245 \\qquad \\text{(E)}\\ 7524$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1538", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are twenty-four $4$-digit numbers that use each of the four digits $2$, $4$, $5$, and $7$ exactly once. Listed in numerical order from smallest to largest, the number in the $17\\text{th}$ position in the list is\n$\\text{(A)}\\ 4527 \\qquad \\text{(B)}\\ 5724 \\qquad \\text{(C)}\\ 5742 \\qquad \\text{(D)}\\ 7245 \\qquad \\text{(E)}\\ 7524$" + } + }, + { + "question": "Return your final response within \\boxed{}. Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$? \n$\\textbf{(A)}\\ -100 \\qquad \\textbf{(B)}\\ -82 \\qquad \\textbf{(C)}\\ -73 \\qquad \\textbf{(D)}\\ -64 \\qquad \\textbf{(E)}\\ 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1539", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$? \n$\\textbf{(A)}\\ -100 \\qquad \\textbf{(B)}\\ -82 \\qquad \\textbf{(C)}\\ -73 \\qquad \\textbf{(D)}\\ -64 \\qquad \\textbf{(E)}\\ 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{15^{30}}{45^{15}} =$\n$\\text{(A) } \\left(\\frac{1}{3}\\right)^{15}\\quad \\text{(B) } \\left(\\frac{1}{3}\\right)^{2}\\quad \\text{(C) } 1\\quad \\text{(D) } 3^{15}\\quad \\text{(E) } 5^{15}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1540", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{15^{30}}{45^{15}} =$\n$\\text{(A) } \\left(\\frac{1}{3}\\right)^{15}\\quad \\text{(B) } \\left(\\frac{1}{3}\\right)^{2}\\quad \\text{(C) } 1\\quad \\text{(D) } 3^{15}\\quad \\text{(E) } 5^{15}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $r$ and $s$ are the roots of $x^2-px+q=0$, then $r^2+s^2$ equals: \n$\\textbf{(A)}\\ p^2+2q\\qquad\\textbf{(B)}\\ p^2-2q\\qquad\\textbf{(C)}\\ p^2+q^2\\qquad\\textbf{(D)}\\ p^2-q^2\\qquad\\textbf{(E)}\\ p^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1541", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $r$ and $s$ are the roots of $x^2-px+q=0$, then $r^2+s^2$ equals: \n$\\textbf{(A)}\\ p^2+2q\\qquad\\textbf{(B)}\\ p^2-2q\\qquad\\textbf{(C)}\\ p^2+q^2\\qquad\\textbf{(D)}\\ p^2-q^2\\qquad\\textbf{(E)}\\ p^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is\n$\\text{(A)} \\ 5 \\qquad \\text{(B)} \\ 7 \\qquad \\text{(C)} \\ 12 \\qquad \\text{(D)} \\ 17 \\qquad \\text{(E)} \\ 72$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1542", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is\n$\\text{(A)} \\ 5 \\qquad \\text{(B)} \\ 7 \\qquad \\text{(C)} \\ 12 \\qquad \\text{(D)} \\ 17 \\qquad \\text{(E)} \\ 72$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. How many such pairs are there? \n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ \\text{infinitely many}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1543", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. How many such pairs are there? \n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ \\text{infinitely many}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled?\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 24 \\qquad \\text{(C)}\\ 32 \\qquad \\text{(D)}\\ 33 \\qquad \\text{(E)}\\ 40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1544", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled?\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 24 \\qquad \\text{(C)}\\ 32 \\qquad \\text{(D)}\\ 33 \\qquad \\text{(E)}\\ 40$" + } + }, + { + "question": "Return your final response within \\boxed{}. The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?\n$\\textbf {(A) } 1 \\qquad \\textbf {(B) } 2 \\qquad \\textbf {(C) } 3 \\qquad \\textbf {(D) } 4 \\qquad \\textbf {(E) } 5$\n.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1545", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?\n$\\textbf {(A) } 1 \\qquad \\textbf {(B) } 2 \\qquad \\textbf {(C) } 3 \\qquad \\textbf {(D) } 4 \\qquad \\textbf {(E) } 5$\n." + } + }, + { + "question": "Return your final response within \\boxed{}. Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either $12$ pieces of red candy, $14$ pieces of green candy, $15$ pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs $20$ cents. What is the smallest possible value of $n$?\n$\\textbf{(A) } 18 \\qquad \\textbf{(B) } 21 \\qquad \\textbf{(C) } 24\\qquad \\textbf{(D) } 25 \\qquad \\textbf{(E) } 28$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1546", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either $12$ pieces of red candy, $14$ pieces of green candy, $15$ pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs $20$ cents. What is the smallest possible value of $n$?\n$\\textbf{(A) } 18 \\qquad \\textbf{(B) } 21 \\qquad \\textbf{(C) } 24\\qquad \\textbf{(D) } 25 \\qquad \\textbf{(E) } 28$" + } + }, + { + "question": "Return your final response within \\boxed{}. The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is\n$\\textbf{(A) }9.7\\qquad \\textbf{(B) }10.1\\qquad \\textbf{(C) }10.5\\qquad \\textbf{(D) }10.9\\qquad \\textbf{(E) }20.5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1547", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is\n$\\textbf{(A) }9.7\\qquad \\textbf{(B) }10.1\\qquad \\textbf{(C) }10.5\\qquad \\textbf{(D) }10.9\\qquad \\textbf{(E) }20.5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Assume the adjoining chart shows the $1980$ U.S. population, in millions, for each region by ethnic group. To the nearest [percent](https://artofproblemsolving.com/wiki/index.php/Percent), what percent of the U.S. Black population lived in the South?\n\\[\\begin{tabular}[t]{c|cccc} & NE & MW & South & West \\\\ \\hline White & 42 & 52 & 57 & 35 \\\\ Black & 5 & 5 & 15 & 2 \\\\ Asian & 1 & 1 & 1 & 3 \\\\ Other & 1 & 1 & 2 & 4 \\end{tabular}\\]\n$\\text{(A)}\\ 20\\% \\qquad \\text{(B)}\\ 25\\% \\qquad \\text{(C)}\\ 40\\% \\qquad \\text{(D)}\\ 56\\% \\qquad \\text{(E)}\\ 80\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1548", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Assume the adjoining chart shows the $1980$ U.S. population, in millions, for each region by ethnic group. To the nearest [percent](https://artofproblemsolving.com/wiki/index.php/Percent), what percent of the U.S. Black population lived in the South?\n\\[\\begin{tabular}[t]{c|cccc} & NE & MW & South & West \\\\ \\hline White & 42 & 52 & 57 & 35 \\\\ Black & 5 & 5 & 15 & 2 \\\\ Asian & 1 & 1 & 1 & 3 \\\\ Other & 1 & 1 & 2 & 4 \\end{tabular}\\]\n$\\text{(A)}\\ 20\\% \\qquad \\text{(B)}\\ 25\\% \\qquad \\text{(C)}\\ 40\\% \\qquad \\text{(D)}\\ 56\\% \\qquad \\text{(E)}\\ 80\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$?\n$\\mathrm{(A) \\ } 4000\\qquad \\mathrm{(B) \\ } 4040\\qquad \\mathrm{(C) \\ } 4400\\qquad \\mathrm{(D) \\ } 40,000\\qquad \\mathrm{(E) \\ } 40,400$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1549", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$?\n$\\mathrm{(A) \\ } 4000\\qquad \\mathrm{(B) \\ } 4040\\qquad \\mathrm{(C) \\ } 4400\\qquad \\mathrm{(D) \\ } 40,000\\qquad \\mathrm{(E) \\ } 40,400$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a$ and $b$ are positive numbers such that $a^b=b^a$ and $b=9a$, then the value of $a$ is\n$\\mathrm{(A)\\ } 9 \\qquad \\mathrm{(B) \\ }\\frac{1}{9} \\qquad \\mathrm{(C) \\ } \\sqrt[9]{9} \\qquad \\mathrm{(D) \\ } \\sqrt[3]{9} \\qquad \\mathrm{(E) \\ }\\sqrt[4]{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1550", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a$ and $b$ are positive numbers such that $a^b=b^a$ and $b=9a$, then the value of $a$ is\n$\\mathrm{(A)\\ } 9 \\qquad \\mathrm{(B) \\ }\\frac{1}{9} \\qquad \\mathrm{(C) \\ } \\sqrt[9]{9} \\qquad \\mathrm{(D) \\ } \\sqrt[3]{9} \\qquad \\mathrm{(E) \\ }\\sqrt[4]{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $r>0$, then for all $p$ and $q$ such that $pq\\ne 0$ and $pr>qr$, we have\n$\\text{(A) } -p>-q\\quad \\text{(B) } -p>q\\quad \\text{(C) } 1>-q/p\\quad \\text{(D) } 10$, then for all $p$ and $q$ such that $pq\\ne 0$ and $pr>qr$, we have\n$\\text{(A) } -p>-q\\quad \\text{(B) } -p>q\\quad \\text{(C) } 1>-q/p\\quad \\text{(D) } 1 1$ have the property that there exists an infinite sequence $a_1$, $a_2$, $a_3$, $\\dots$ of nonzero integers such that the equality\n\\[a_k + 2a_{2k} + \\dots + na_{nk} = 0\\]\nholds for every positive integer $k$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "3", + "index": "Sky-T1_10k_1566", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1$, $a_2$, $a_3$, $\\dots$ of nonzero integers such that the equality\n\\[a_k + 2a_{2k} + \\dots + na_{nk} = 0\\]\nholds for every positive integer $k$." + } + }, + { + "question": "Return your final response within \\boxed{}. A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?\n[asy]draw((0,0)--(8,15)--(16,0)--(0,0)); draw(arc((8,0),7.0588,0,180));[/asy]\n$\\textbf{(A) }4 \\sqrt{3}\\qquad\\textbf{(B) } \\dfrac{120}{17}\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }\\dfrac{17\\sqrt{2}}{2}\\qquad \\textbf{(E)} \\dfrac{17\\sqrt{3}}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1567", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?\n[asy]draw((0,0)--(8,15)--(16,0)--(0,0)); draw(arc((8,0),7.0588,0,180));[/asy]\n$\\textbf{(A) }4 \\sqrt{3}\\qquad\\textbf{(B) } \\dfrac{120}{17}\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }\\dfrac{17\\sqrt{2}}{2}\\qquad \\textbf{(E)} \\dfrac{17\\sqrt{3}}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Of the following \n(1) $a(x-y)=ax-ay$\n(2) $a^{x-y}=a^x-a^y$\n(3) $\\log (x-y)=\\log x-\\log y$\n(4) $\\frac{\\log x}{\\log y}=\\log{x}-\\log{y}$\n(5) $a(xy)=ax \\cdot ay$\n$\\textbf{(A)}\\text{Only 1 and 4 are true}\\qquad\\\\\\textbf{(B)}\\ \\text{Only 1 and 5 are true}\\qquad\\\\\\textbf{(C)}\\ \\text{Only 1 and 3 are true}\\qquad\\\\\\textbf{(D)}\\ \\text{Only 1 and 2 are true}\\qquad\\\\\\textbf{(E)}\\ \\text{Only 1 is true}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1568", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Of the following \n(1) $a(x-y)=ax-ay$\n(2) $a^{x-y}=a^x-a^y$\n(3) $\\log (x-y)=\\log x-\\log y$\n(4) $\\frac{\\log x}{\\log y}=\\log{x}-\\log{y}$\n(5) $a(xy)=ax \\cdot ay$\n$\\textbf{(A)}\\text{Only 1 and 4 are true}\\qquad\\\\\\textbf{(B)}\\ \\text{Only 1 and 5 are true}\\qquad\\\\\\textbf{(C)}\\ \\text{Only 1 and 3 are true}\\qquad\\\\\\textbf{(D)}\\ \\text{Only 1 and 2 are true}\\qquad\\\\\\textbf{(E)}\\ \\text{Only 1 is true}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following numbers has the largest [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal)?\n$\\text{(A)}\\ \\frac{1}{3} \\qquad \\text{(B)}\\ \\frac{2}{5} \\qquad \\text{(C)}\\ 1 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 1986$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1569", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following numbers has the largest [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal)?\n$\\text{(A)}\\ \\frac{1}{3} \\qquad \\text{(B)}\\ \\frac{2}{5} \\qquad \\text{(C)}\\ 1 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 1986$" + } + }, + { + "question": "Return your final response within \\boxed{}. The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $61$ points. How many free throws did they make?\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 14 \\qquad \\textbf{(C)}\\ 15 \\qquad \\textbf{(D)}\\ 16 \\qquad \\textbf{(E)}\\ 17$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1570", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $61$ points. How many free throws did they make?\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 14 \\qquad \\textbf{(C)}\\ 15 \\qquad \\textbf{(D)}\\ 16 \\qquad \\textbf{(E)}\\ 17$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which one of the following statements is false? All equilateral triangles are\n$\\textbf{(A)}\\ \\text{ equiangular}\\qquad \\textbf{(B)}\\ \\text{isosceles}\\qquad \\textbf{(C)}\\ \\text{regular polygons }\\qquad\\\\ \\textbf{(D)}\\ \\text{congruent to each other}\\qquad \\textbf{(E)}\\ \\text{similar to each other}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1571", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which one of the following statements is false? All equilateral triangles are\n$\\textbf{(A)}\\ \\text{ equiangular}\\qquad \\textbf{(B)}\\ \\text{isosceles}\\qquad \\textbf{(C)}\\ \\text{regular polygons }\\qquad\\\\ \\textbf{(D)}\\ \\text{congruent to each other}\\qquad \\textbf{(E)}\\ \\text{similar to each other}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $2005$ Tycoon Tammy invested $100$ dollars for two years. During the first year\nher investment suffered a $15\\%$ loss, but during the second year the remaining\ninvestment showed a $20\\%$ gain. Over the two-year period, what was the change\nin Tammy's investment?\n$\\textbf{(A)}\\ 5\\%\\text{ loss}\\qquad \\textbf{(B)}\\ 2\\%\\text{ loss}\\qquad \\textbf{(C)}\\ 1\\%\\text{ gain}\\qquad \\textbf{(D)}\\ 2\\% \\text{ gain} \\qquad \\textbf{(E)}\\ 5\\%\\text{ gain}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1572", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $2005$ Tycoon Tammy invested $100$ dollars for two years. During the first year\nher investment suffered a $15\\%$ loss, but during the second year the remaining\ninvestment showed a $20\\%$ gain. Over the two-year period, what was the change\nin Tammy's investment?\n$\\textbf{(A)}\\ 5\\%\\text{ loss}\\qquad \\textbf{(B)}\\ 2\\%\\text{ loss}\\qquad \\textbf{(C)}\\ 1\\%\\text{ gain}\\qquad \\textbf{(D)}\\ 2\\% \\text{ gain} \\qquad \\textbf{(E)}\\ 5\\%\\text{ gain}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob's sequence is an [integer](https://artofproblemsolving.com/wiki/index.php/Integer)?\n$\\mathrm{(A)}\\ \\frac{1}{6}\\qquad\\mathrm{(B)}\\ \\frac{1}{3}\\qquad\\mathrm{(C)}\\ \\frac{1}{2}\\qquad\\mathrm{(D)}\\ \\frac{5}{8}\\qquad\\mathrm{(E)}\\ \\frac{3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1573", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob's sequence is an [integer](https://artofproblemsolving.com/wiki/index.php/Integer)?\n$\\mathrm{(A)}\\ \\frac{1}{6}\\qquad\\mathrm{(B)}\\ \\frac{1}{3}\\qquad\\mathrm{(C)}\\ \\frac{1}{2}\\qquad\\mathrm{(D)}\\ \\frac{5}{8}\\qquad\\mathrm{(E)}\\ \\frac{3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a @ b$ represent the operation on two numbers, $a$ and $b$, which selects the larger of the two numbers, with $a@a = a$. Let $a ! b$ represent the operator which selects the smaller of the two numbers, with $a ! a = a$. Which of the following three rules is (are) correct? \n$\\textbf{(1)}\\ a@b = b@a\\qquad\\textbf{(2)}\\ a@(b@c) = (a@b)@c\\qquad\\textbf{(3)}\\ a ! (b@c) = (a ! b) @ (a ! c)$\n\n$\\textbf{(A)}\\ (1)\\text{ only}\\qquad\\textbf{(B)}\\ (2)\\text{ only}\\qquad\\textbf{(C)}\\ \\text{(1) and (2) only}$\n$\\textbf{(D)}\\ \\text{(1) and (3) only}\\qquad\\textbf{(E)}\\ \\text{all three}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1574", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a @ b$ represent the operation on two numbers, $a$ and $b$, which selects the larger of the two numbers, with $a@a = a$. Let $a ! b$ represent the operator which selects the smaller of the two numbers, with $a ! a = a$. Which of the following three rules is (are) correct? \n$\\textbf{(1)}\\ a@b = b@a\\qquad\\textbf{(2)}\\ a@(b@c) = (a@b)@c\\qquad\\textbf{(3)}\\ a ! (b@c) = (a ! b) @ (a ! c)$\n\n$\\textbf{(A)}\\ (1)\\text{ only}\\qquad\\textbf{(B)}\\ (2)\\text{ only}\\qquad\\textbf{(C)}\\ \\text{(1) and (2) only}$\n$\\textbf{(D)}\\ \\text{(1) and (3) only}\\qquad\\textbf{(E)}\\ \\text{all three}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $[t]$ denote the greatest integer $\\leq t$ where $t \\geq 0$ and $S = \\{(x,y): (x-T)^2 + y^2 \\leq T^2 \\text{ where } T = t - [t]\\}$. Then we have \n$\\textbf{(A)}\\ \\text{the point } (0,0) \\text{ does not belong to } S \\text{ for any } t \\qquad$\n$\\textbf{(B)}\\ 0 \\leq \\text{Area } S \\leq \\pi \\text{ for all } t \\qquad$\n$\\textbf{(C)}\\ S \\text{ is contained in the first quadrant for all } t \\geq 5 \\qquad$\n$\\textbf{(D)}\\ \\text{the center of } S \\text{ for any } t \\text{ is on the line } y=x \\qquad$\n$\\textbf{(E)}\\ \\text{none of the other statements is true}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1575", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $[t]$ denote the greatest integer $\\leq t$ where $t \\geq 0$ and $S = \\{(x,y): (x-T)^2 + y^2 \\leq T^2 \\text{ where } T = t - [t]\\}$. Then we have \n$\\textbf{(A)}\\ \\text{the point } (0,0) \\text{ does not belong to } S \\text{ for any } t \\qquad$\n$\\textbf{(B)}\\ 0 \\leq \\text{Area } S \\leq \\pi \\text{ for all } t \\qquad$\n$\\textbf{(C)}\\ S \\text{ is contained in the first quadrant for all } t \\geq 5 \\qquad$\n$\\textbf{(D)}\\ \\text{the center of } S \\text{ for any } t \\text{ is on the line } y=x \\qquad$\n$\\textbf{(E)}\\ \\text{none of the other statements is true}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is the correct order of the fractions $\\frac{15}{11},\\frac{19}{15},$ and $\\frac{17}{13},$ from least to greatest? \n$\\textbf{(A) }\\frac{15}{11}< \\frac{17}{13}< \\frac{19}{15} \\qquad\\textbf{(B) }\\frac{15}{11}< \\frac{19}{15}<\\frac{17}{13} \\qquad\\textbf{(C) }\\frac{17}{13}<\\frac{19}{15}<\\frac{15}{11} \\qquad\\textbf{(D) } \\frac{19}{15}<\\frac{15}{11}<\\frac{17}{13} \\qquad\\textbf{(E) } \\frac{19}{15}<\\frac{17}{13}<\\frac{15}{11}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1576", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is the correct order of the fractions $\\frac{15}{11},\\frac{19}{15},$ and $\\frac{17}{13},$ from least to greatest? \n$\\textbf{(A) }\\frac{15}{11}< \\frac{17}{13}< \\frac{19}{15} \\qquad\\textbf{(B) }\\frac{15}{11}< \\frac{19}{15}<\\frac{17}{13} \\qquad\\textbf{(C) }\\frac{17}{13}<\\frac{19}{15}<\\frac{15}{11} \\qquad\\textbf{(D) } \\frac{19}{15}<\\frac{15}{11}<\\frac{17}{13} \\qquad\\textbf{(E) } \\frac{19}{15}<\\frac{17}{13}<\\frac{15}{11}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and \nobtains $8$ and $2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and \nfind $-9$ and $-1$ for the roots. The correct equation was: \n$\\textbf{(A)}\\ x^2-10x+9=0 \\qquad \\textbf{(B)}\\ x^2+10x+9=0 \\qquad \\textbf{(C)}\\ x^2-10x+16=0\\\\ \\textbf{(D)}\\ x^2-8x-9=0\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1577", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and \nobtains $8$ and $2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and \nfind $-9$ and $-1$ for the roots. The correct equation was: \n$\\textbf{(A)}\\ x^2-10x+9=0 \\qquad \\textbf{(B)}\\ x^2+10x+9=0 \\qquad \\textbf{(C)}\\ x^2-10x+16=0\\\\ \\textbf{(D)}\\ x^2-8x-9=0\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1578", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\\frac{1}{2}$ times the \nexterior angle at the same vertex. Then\n$\\textbf{(A)}\\ S=2660^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{may be regular}\\qquad \\\\ \\textbf{(B)}\\ S=2660^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{is not regular}\\qquad \\\\ \\textbf{(C)}\\ S=2700^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{is regular}\\qquad \\\\ \\textbf{(D)}\\ S=2700^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{is not regular}\\qquad \\\\ \\textbf{(E)}\\ S=2700^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{may or may not be regular}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1579", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\\frac{1}{2}$ times the \nexterior angle at the same vertex. Then\n$\\textbf{(A)}\\ S=2660^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{may be regular}\\qquad \\\\ \\textbf{(B)}\\ S=2660^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{is not regular}\\qquad \\\\ \\textbf{(C)}\\ S=2700^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{is regular}\\qquad \\\\ \\textbf{(D)}\\ S=2700^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{is not regular}\\qquad \\\\ \\textbf{(E)}\\ S=2700^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{may or may not be regular}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is: \n$\\textbf{(A)}\\ \\frac {\\sqrt {8d^2 - p^2}}{2} \\qquad \\textbf{(B)}\\ \\frac {\\sqrt {8d^2 + p^2}}{2} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{6d^2-p^2}}{2}\\qquad\\\\ \\textbf{(D)}\\ \\frac {\\sqrt {6d^2 + p^2}}{2} \\qquad \\textbf{(E)}\\ \\frac {8d^2 - p^2}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1580", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is: \n$\\textbf{(A)}\\ \\frac {\\sqrt {8d^2 - p^2}}{2} \\qquad \\textbf{(B)}\\ \\frac {\\sqrt {8d^2 + p^2}}{2} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{6d^2-p^2}}{2}\\qquad\\\\ \\textbf{(D)}\\ \\frac {\\sqrt {6d^2 + p^2}}{2} \\qquad \\textbf{(E)}\\ \\frac {8d^2 - p^2}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph of $x^2-4y^2=0$ is:\n$\\textbf{(A)}\\ \\text{a parabola} \\qquad \\textbf{(B)}\\ \\text{an ellipse} \\qquad \\textbf{(C)}\\ \\text{a pair of straight lines}\\qquad \\\\ \\textbf{(D)}\\ \\text{a point}\\qquad \\textbf{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1581", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph of $x^2-4y^2=0$ is:\n$\\textbf{(A)}\\ \\text{a parabola} \\qquad \\textbf{(B)}\\ \\text{an ellipse} \\qquad \\textbf{(C)}\\ \\text{a pair of straight lines}\\qquad \\\\ \\textbf{(D)}\\ \\text{a point}\\qquad \\textbf{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f(x) = \\frac{x+1}{x-1}$. Then for $x^2 \\neq 1, f(-x)$ is\n$\\textbf{(A)}\\ \\frac{1}{f(x)}\\qquad \\textbf{(B)}\\ -f(x)\\qquad \\textbf{(C)}\\ \\frac{1}{f(-x)}\\qquad \\textbf{(D)}\\ -f(-x)\\qquad \\textbf{(E)}\\ f(x)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1582", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f(x) = \\frac{x+1}{x-1}$. Then for $x^2 \\neq 1, f(-x)$ is\n$\\textbf{(A)}\\ \\frac{1}{f(x)}\\qquad \\textbf{(B)}\\ -f(x)\\qquad \\textbf{(C)}\\ \\frac{1}{f(-x)}\\qquad \\textbf{(D)}\\ -f(-x)\\qquad \\textbf{(E)}\\ f(x)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider all [triangles](https://artofproblemsolving.com/wiki/index.php/Triangle) $ABC$ satisfying in the following conditions: $AB = AC$, $D$ is a point on $\\overline{AC}$ for which $\\overline{BD} \\perp \\overline{AC}$, $AC$ and $CD$ are integers, and $BD^{2} = 57$. Among all such triangles, the smallest possible value of $AC$ is\n\n$\\textrm{(A)} \\ 9 \\qquad \\textrm{(B)} \\ 10 \\qquad \\textrm{(C)} \\ 11 \\qquad \\textrm{(D)} \\ 12 \\qquad \\textrm{(E)} \\ 13$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1583", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider all [triangles](https://artofproblemsolving.com/wiki/index.php/Triangle) $ABC$ satisfying in the following conditions: $AB = AC$, $D$ is a point on $\\overline{AC}$ for which $\\overline{BD} \\perp \\overline{AC}$, $AC$ and $CD$ are integers, and $BD^{2} = 57$. Among all such triangles, the smallest possible value of $AC$ is\n\n$\\textrm{(A)} \\ 9 \\qquad \\textrm{(B)} \\ 10 \\qquad \\textrm{(C)} \\ 11 \\qquad \\textrm{(D)} \\ 12 \\qquad \\textrm{(E)} \\ 13$" + } + }, + { + "question": "Return your final response within \\boxed{}. The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "150", + "index": "Sky-T1_10k_1584", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?" + } + }, + { + "question": "Return your final response within \\boxed{}. For every real number $x$, let $\\lfloor x\\rfloor$ denote the greatest integer not exceeding $x$, and let \\[f(x)=\\lfloor x\\rfloor(2014^{x-\\lfloor x\\rfloor}-1).\\] The set of all numbers $x$ such that $1\\leq x<2014$ and $f(x)\\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?\n$\\textbf{(A) }1\\qquad \\textbf{(B) }\\dfrac{\\log 2015}{\\log 2014}\\qquad \\textbf{(C) }\\dfrac{\\log 2014}{\\log 2013}\\qquad \\textbf{(D) }\\dfrac{2014}{2013}\\qquad \\textbf{(E) }2014^{\\frac1{2014}}\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1585", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For every real number $x$, let $\\lfloor x\\rfloor$ denote the greatest integer not exceeding $x$, and let \\[f(x)=\\lfloor x\\rfloor(2014^{x-\\lfloor x\\rfloor}-1).\\] The set of all numbers $x$ such that $1\\leq x<2014$ and $f(x)\\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?\n$\\textbf{(A) }1\\qquad \\textbf{(B) }\\dfrac{\\log 2015}{\\log 2014}\\qquad \\textbf{(C) }\\dfrac{\\log 2014}{\\log 2013}\\qquad \\textbf{(D) }\\dfrac{2014}{2013}\\qquad \\textbf{(E) }2014^{\\frac1{2014}}\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the set of $x$-values satisfying the inequality $|\\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive, \n$-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $111\\qquad \\textbf{(E)}\\ |x| < 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1586", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the set of $x$-values satisfying the inequality $|\\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive, \n$-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $111\\qquad \\textbf{(E)}\\ |x| < 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many whole numbers are between $\\sqrt{8}$ and $\\sqrt{80}$?\n$\\text{(A)}\\ 5 \\qquad \\text{(B)}\\ 6 \\qquad \\text{(C)}\\ 7 \\qquad \\text{(D)}\\ 8 \\qquad \\text{(E)}\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1587", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many whole numbers are between $\\sqrt{8}$ and $\\sqrt{80}$?\n$\\text{(A)}\\ 5 \\qquad \\text{(B)}\\ 6 \\qquad \\text{(C)}\\ 7 \\qquad \\text{(D)}\\ 8 \\qquad \\text{(E)}\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. $ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\\overline{AB}$. $\\overline{PS} \\perp \\overline{BD}$ and $\\overline{PR} \\perp \\overline{AC}$. $\\overline{AF} \\perp \\overline{BD}$ and $\\overline{PQ} \\perp \\overline{AF}$. Then $PR + PS$ is equal to:\n\n$\\textbf{(A)}\\ PQ\\qquad \\textbf{(B)}\\ AE\\qquad \\textbf{(C)}\\ PT + AT\\qquad \\textbf{(D)}\\ AF\\qquad \\textbf{(E)}\\ EF$\n\\usepackage{amssymb}", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1588", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\\overline{AB}$. $\\overline{PS} \\perp \\overline{BD}$ and $\\overline{PR} \\perp \\overline{AC}$. $\\overline{AF} \\perp \\overline{BD}$ and $\\overline{PQ} \\perp \\overline{AF}$. Then $PR + PS$ is equal to:\n\n$\\textbf{(A)}\\ PQ\\qquad \\textbf{(B)}\\ AE\\qquad \\textbf{(C)}\\ PT + AT\\qquad \\textbf{(D)}\\ AF\\qquad \\textbf{(E)}\\ EF$\n\\usepackage{amssymb}" + } + }, + { + "question": "Return your final response within \\boxed{}. In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.\nBrian: \"Mike and I are different species.\"\nChris: \"LeRoy is a frog.\"\nLeRoy: \"Chris is a frog.\"\nMike: \"Of the four of us, at least two are toads.\"\nHow many of these amphibians are frogs?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1589", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.\nBrian: \"Mike and I are different species.\"\nChris: \"LeRoy is a frog.\"\nLeRoy: \"Chris is a frog.\"\nMike: \"Of the four of us, at least two are toads.\"\nHow many of these amphibians are frogs?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions \"left\" and \"right\" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?\n$\\textbf{(A) }1 \\qquad \\textbf{(B) } 2 \\qquad \\textbf{(C) } 3 \\qquad \\textbf{(D) } 4\\qquad \\textbf{(E) } 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1590", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions \"left\" and \"right\" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?\n$\\textbf{(A) }1 \\qquad \\textbf{(B) } 2 \\qquad \\textbf{(C) } 3 \\qquad \\textbf{(D) } 4\\qquad \\textbf{(E) } 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1591", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighbourhood picnic?\n$\\textbf{(A)}\\hspace{.05in}6 \\qquad \\textbf{(B)}\\hspace{.05in}6\\dfrac23 \\qquad \\textbf{(C)}\\hspace{.05in}7\\dfrac12 \\qquad \\textbf{(D)}\\hspace{.05in}8 \\qquad \\textbf{(E)}\\hspace{.05in}9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1592", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighbourhood picnic?\n$\\textbf{(A)}\\hspace{.05in}6 \\qquad \\textbf{(B)}\\hspace{.05in}6\\dfrac23 \\qquad \\textbf{(C)}\\hspace{.05in}7\\dfrac12 \\qquad \\textbf{(D)}\\hspace{.05in}8 \\qquad \\textbf{(E)}\\hspace{.05in}9$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $m\\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\\le a_i \\le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial\n\\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\\]such that $D(m) = q(m)$ for all odd integers $m\\ge 5$. What is $c_1?$\n$\\textbf{(A)}\\ {-}6\\qquad\\textbf{(B)}\\ {-}1\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1593", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $m\\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\\le a_i \\le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial\n\\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\\]such that $D(m) = q(m)$ for all odd integers $m\\ge 5$. What is $c_1?$\n$\\textbf{(A)}\\ {-}6\\qquad\\textbf{(B)}\\ {-}1\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$?\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1594", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$?\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $f(n)=\\tfrac{1}{3} n(n+1)(n+2)$, then $f(r)-f(r-1)$ equals:\n$\\text{(A) } r(r+1)\\quad \\text{(B) } (r+1)(r+2)\\quad \\text{(C) } \\tfrac{1}{3} r(r+1)\\quad \\\\ \\text{(D) } \\tfrac{1}{3} (r+1)(r+2)\\quad \\text{(E )} \\tfrac{1}{3} r(r+1)(2r+1)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1595", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $f(n)=\\tfrac{1}{3} n(n+1)(n+2)$, then $f(r)-f(r-1)$ equals:\n$\\text{(A) } r(r+1)\\quad \\text{(B) } (r+1)(r+2)\\quad \\text{(C) } \\tfrac{1}{3} r(r+1)\\quad \\\\ \\text{(D) } \\tfrac{1}{3} (r+1)(r+2)\\quad \\text{(E )} \\tfrac{1}{3} r(r+1)(2r+1)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$\n\"chunks\" and the channel can transmit $120$ chunks per second.\n$\\textbf{(A)}\\ 0.04 \\text{ seconds}\\qquad \\textbf{(B)}\\ 0.4 \\text{ seconds}\\qquad \\textbf{(C)}\\ 4 \\text{ seconds}\\qquad \\textbf{(D)}\\ 4\\text{ minutes}\\qquad \\textbf{(E)}\\ 4\\text{ hours}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1596", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$\n\"chunks\" and the channel can transmit $120$ chunks per second.\n$\\textbf{(A)}\\ 0.04 \\text{ seconds}\\qquad \\textbf{(B)}\\ 0.4 \\text{ seconds}\\qquad \\textbf{(C)}\\ 4 \\text{ seconds}\\qquad \\textbf{(D)}\\ 4\\text{ minutes}\\qquad \\textbf{(E)}\\ 4\\text{ hours}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. The number of cows was: \n$\\textbf{(A)}\\ 5 \\qquad\\textbf{(B)}\\ 7 \\qquad\\textbf{(C)}\\ 10 \\qquad\\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 14$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1597", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. The number of cows was: \n$\\textbf{(A)}\\ 5 \\qquad\\textbf{(B)}\\ 7 \\qquad\\textbf{(C)}\\ 10 \\qquad\\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 14$" + } + }, + { + "question": "Return your final response within \\boxed{}. Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?\n$\\textbf{(A)}\\ 6 \\qquad\\textbf{(B)}\\ 10\\qquad\\textbf{(C)}\\ 15\\qquad\\textbf{(D)}\\ 25\\qquad\\textbf{(E)}\\ 99$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1598", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?\n$\\textbf{(A)}\\ 6 \\qquad\\textbf{(B)}\\ 10\\qquad\\textbf{(C)}\\ 15\\qquad\\textbf{(D)}\\ 25\\qquad\\textbf{(E)}\\ 99$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many integers between $1000$ and $9999$ have four distinct digits?\n$\\textbf{(A) }3024\\qquad\\textbf{(B) }4536\\qquad\\textbf{(C) }5040\\qquad\\textbf{(D) }6480\\qquad \\textbf{(E) }6561$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1599", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many integers between $1000$ and $9999$ have four distinct digits?\n$\\textbf{(A) }3024\\qquad\\textbf{(B) }4536\\qquad\\textbf{(C) }5040\\qquad\\textbf{(D) }6480\\qquad \\textbf{(E) }6561$" + } + }, + { + "question": "Return your final response within \\boxed{}. A merchant buys goods at $25\\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\\%$ on the marked price and still clear a profit of $25\\%$ on the selling price. What percent of the list price must he mark the goods?\n$\\textbf{(A)}\\ 125\\% \\qquad \\textbf{(B)}\\ 100\\% \\qquad \\textbf{(C)}\\ 120\\% \\qquad \\textbf{(D)}\\ 80\\% \\qquad \\textbf{(E)}\\ 75\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1600", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A merchant buys goods at $25\\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\\%$ on the marked price and still clear a profit of $25\\%$ on the selling price. What percent of the list price must he mark the goods?\n$\\textbf{(A)}\\ 125\\% \\qquad \\textbf{(B)}\\ 100\\% \\qquad \\textbf{(C)}\\ 120\\% \\qquad \\textbf{(D)}\\ 80\\% \\qquad \\textbf{(E)}\\ 75\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is:\n$\\text{(A) } y+3x-4=0\\quad \\text{(B) } y+3x+4=0\\quad \\text{(C) } y-3x-4=0\\quad \\\\ \\text{(D) } 3y+x-12=0\\quad \\text{(E) } 3y-x-12=0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1601", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is:\n$\\text{(A) } y+3x-4=0\\quad \\text{(B) } y+3x+4=0\\quad \\text{(C) } y-3x-4=0\\quad \\\\ \\text{(D) } 3y+x-12=0\\quad \\text{(E) } 3y-x-12=0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle $ABCD$?\n\n$\\textbf{(A) }45\\qquad\\textbf{(B) }75\\qquad\\textbf{(C) }100\\qquad\\textbf{(D) }125\\qquad\\textbf{(E) }150$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1602", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle $ABCD$?\n\n$\\textbf{(A) }45\\qquad\\textbf{(B) }75\\qquad\\textbf{(C) }100\\qquad\\textbf{(D) }125\\qquad\\textbf{(E) }150$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$?\n$\\textbf{(A)}\\ 70\\qquad\\textbf{(B)}\\ 97\\qquad\\textbf{(C)}\\ 127\\qquad\\textbf{(D)}\\ 159\\qquad\\textbf{(E)}\\ 729$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1603", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$?\n$\\textbf{(A)}\\ 70\\qquad\\textbf{(B)}\\ 97\\qquad\\textbf{(C)}\\ 127\\qquad\\textbf{(D)}\\ 159\\qquad\\textbf{(E)}\\ 729$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $Q(z)$ and $R(z)$ be the unique polynomials such that\\[z^{2021}+1=(z^2+z+1)Q(z)+R(z)\\]and the degree of $R$ is less than $2.$ What is $R(z)?$\n$\\textbf{(A) }{-}z \\qquad \\textbf{(B) }{-}1 \\qquad \\textbf{(C) }2021\\qquad \\textbf{(D) }z+1 \\qquad \\textbf{(E) }2z+1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1604", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $Q(z)$ and $R(z)$ be the unique polynomials such that\\[z^{2021}+1=(z^2+z+1)Q(z)+R(z)\\]and the degree of $R$ is less than $2.$ What is $R(z)?$\n$\\textbf{(A) }{-}z \\qquad \\textbf{(B) }{-}1 \\qquad \\textbf{(C) }2021\\qquad \\textbf{(D) }z+1 \\qquad \\textbf{(E) }2z+1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f(x)=\\sum_{k=2}^{10}(\\lfloor kx \\rfloor -k \\lfloor x \\rfloor)$, where $\\lfloor r \\rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \\ge 0$?\n$\\textbf{(A)}\\ 32\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 45\\qquad\\textbf{(D)}\\ 46\\qquad\\textbf{(E)}\\ \\text{infinitely many}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1605", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f(x)=\\sum_{k=2}^{10}(\\lfloor kx \\rfloor -k \\lfloor x \\rfloor)$, where $\\lfloor r \\rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \\ge 0$?\n$\\textbf{(A)}\\ 32\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 45\\qquad\\textbf{(D)}\\ 46\\qquad\\textbf{(E)}\\ \\text{infinitely many}$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Kiran Kedlaya) Suppose $a_1, \\dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties:\n(a) For $i = 1, \\dots, n$, $a_i \\in S$.\n(b) For $i,j = 1, \\dots, n$ (not necessarily distinct), $a_i - a_j \\in S$.\n(c) For any integers $x,y \\in S$, if $x + y \\in S$, then $x - y \\in S$.\nProve that $S$ must be equal to the set of all integers.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "S is the set of all integers", + "index": "Sky-T1_10k_1606", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Kiran Kedlaya) Suppose $a_1, \\dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties:\n(a) For $i = 1, \\dots, n$, $a_i \\in S$.\n(b) For $i,j = 1, \\dots, n$ (not necessarily distinct), $a_i - a_j \\in S$.\n(c) For any integers $x,y \\in S$, if $x + y \\in S$, then $x - y \\in S$.\nProve that $S$ must be equal to the set of all integers." + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of sets $\\{a,b,c\\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "728", + "index": "Sky-T1_10k_1607", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of sets $\\{a,b,c\\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$." + } + }, + { + "question": "Return your final response within \\boxed{}. The [function](https://artofproblemsolving.com/wiki/index.php/Function) $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x$. If the equation $f(x)=0$ has exactly four distinct real [roots](https://artofproblemsolving.com/wiki/index.php/Root_(polynomial)), then the sum of these roots is\n$\\mathrm{(A) \\ }0 \\qquad \\mathrm{(B) \\ }2 \\qquad \\mathrm{(C) \\ } 4 \\qquad \\mathrm{(D) \\ }6 \\qquad \\mathrm{(E) \\ } 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1608", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The [function](https://artofproblemsolving.com/wiki/index.php/Function) $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x$. If the equation $f(x)=0$ has exactly four distinct real [roots](https://artofproblemsolving.com/wiki/index.php/Root_(polynomial)), then the sum of these roots is\n$\\mathrm{(A) \\ }0 \\qquad \\mathrm{(B) \\ }2 \\qquad \\mathrm{(C) \\ } 4 \\qquad \\mathrm{(D) \\ }6 \\qquad \\mathrm{(E) \\ } 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the discriminant of $ax^2+2bx+c=0$ is zero, then another true statement about $a, b$, and $c$ is that: \n$\\textbf{(A)}\\ \\text{they form an arithmetic progression}\\\\ \\textbf{(B)}\\ \\text{they form a geometric progression}\\\\ \\textbf{(C)}\\ \\text{they are unequal}\\\\ \\textbf{(D)}\\ \\text{they are all negative numbers}\\\\ \\textbf{(E)}\\ \\text{only b is negative and a and c are positive}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1609", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the discriminant of $ax^2+2bx+c=0$ is zero, then another true statement about $a, b$, and $c$ is that: \n$\\textbf{(A)}\\ \\text{they form an arithmetic progression}\\\\ \\textbf{(B)}\\ \\text{they form a geometric progression}\\\\ \\textbf{(C)}\\ \\text{they are unequal}\\\\ \\textbf{(D)}\\ \\text{they are all negative numbers}\\\\ \\textbf{(E)}\\ \\text{only b is negative and a and c are positive}$" + } + }, + { + "question": "Return your final response within \\boxed{}. On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?\n$\\textbf{(A) } 21 \\qquad \\textbf{(B) } 30 \\qquad \\textbf{(C) } 60 \\qquad \\textbf{(D) } 90 \\qquad \\textbf{(E) } 1080$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1610", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?\n$\\textbf{(A) } 21 \\qquad \\textbf{(B) } 30 \\qquad \\textbf{(C) } 60 \\qquad \\textbf{(D) } 90 \\qquad \\textbf{(E) } 1080$" + } + }, + { + "question": "Return your final response within \\boxed{}. Call a fraction $\\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?\n$\\textbf{(A)}\\ 9 \\qquad\\textbf{(B)}\\ 10 \\qquad\\textbf{(C)}\\ 11 \\qquad\\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 13$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1611", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Call a fraction $\\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?\n$\\textbf{(A)}\\ 9 \\qquad\\textbf{(B)}\\ 10 \\qquad\\textbf{(C)}\\ 11 \\qquad\\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 13$" + } + }, + { + "question": "Return your final response within \\boxed{}. Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?\n\n$\\textbf{(A) }12\\qquad\\textbf{(B) }13\\qquad\\textbf{(C) }14\\qquad\\textbf{(D) }15\\qquad\\textbf{(E) }16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1612", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?\n\n$\\textbf{(A) }12\\qquad\\textbf{(B) }13\\qquad\\textbf{(C) }14\\qquad\\textbf{(D) }15\\qquad\\textbf{(E) }16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $q = \\dfrac{3p-5}{2}$ where $p$ is an odd prime, and let\n\n\n\\[S_q = \\frac{1}{2\\cdot 3 \\cdot 4} + \\frac{1}{5\\cdot 6 \\cdot 7} + \\cdots + \\frac{1}{q\\cdot (q+1) \\cdot (q+2)}.\\]\n\n\nProve that if $\\dfrac{1}{p}-2S_q = \\dfrac{m}{n}$ for integers\n$m$ and $n$, then $m-n$ is divisible by $p$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "m - n is divisible by p", + "index": "Sky-T1_10k_1613", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $q = \\dfrac{3p-5}{2}$ where $p$ is an odd prime, and let\n\n\n\\[S_q = \\frac{1}{2\\cdot 3 \\cdot 4} + \\frac{1}{5\\cdot 6 \\cdot 7} + \\cdots + \\frac{1}{q\\cdot (q+1) \\cdot (q+2)}.\\]\n\n\nProve that if $\\dfrac{1}{p}-2S_q = \\dfrac{m}{n}$ for integers\n$m$ and $n$, then $m-n$ is divisible by $p$." + } + }, + { + "question": "Return your final response within \\boxed{}. Three $\\text{A's}$, three $\\text{B's}$, and three $\\text{C's}$ are placed in the nine spaces so that each row and column contains one of each letter. If $\\text{A}$ is placed in the upper left corner, how many arrangements are possible?\n\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1614", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three $\\text{A's}$, three $\\text{B's}$, and three $\\text{C's}$ are placed in the nine spaces so that each row and column contains one of each letter. If $\\text{A}$ is placed in the upper left corner, how many arrangements are possible?\n\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now, she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?\n$\\textbf{(A)}\\ \\frac{1}{2}\\qquad\\textbf{(B)}\\ \\frac{3}{4}\\qquad\\textbf{(C)}\\ 1\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1615", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now, she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?\n$\\textbf{(A)}\\ \\frac{1}{2}\\qquad\\textbf{(B)}\\ \\frac{3}{4}\\qquad\\textbf{(C)}\\ 1\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $(a, b)$ and $(c, d)$ are two points on the line whose equation is $y=mx+k$, then the distance between $(a, b)$ and $(c, d)$, in terms of $a, c,$ and $m$ is \n$\\mathrm{(A)\\ } |a-c|\\sqrt{1+m^2} \\qquad \\mathrm{(B) \\ }|a+c|\\sqrt{1+m^2} \\qquad \\mathrm{(C) \\ } \\frac{|a-c|}{\\sqrt{1+m^2}} \\qquad$\n$\\mathrm{(D) \\ } |a-c|(1+m^2) \\qquad \\mathrm{(E) \\ }|a-c|\\,|m|$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1616", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $(a, b)$ and $(c, d)$ are two points on the line whose equation is $y=mx+k$, then the distance between $(a, b)$ and $(c, d)$, in terms of $a, c,$ and $m$ is \n$\\mathrm{(A)\\ } |a-c|\\sqrt{1+m^2} \\qquad \\mathrm{(B) \\ }|a+c|\\sqrt{1+m^2} \\qquad \\mathrm{(C) \\ } \\frac{|a-c|}{\\sqrt{1+m^2}} \\qquad$\n$\\mathrm{(D) \\ } |a-c|(1+m^2) \\qquad \\mathrm{(E) \\ }|a-c|\\,|m|$" + } + }, + { + "question": "Return your final response within \\boxed{}. On a beach $50$ people are wearing sunglasses and $35$ people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap? \n$\\textbf{(A) }\\frac{14}{85}\\qquad\\textbf{(B) }\\frac{7}{25}\\qquad\\textbf{(C) }\\frac{2}{5}\\qquad\\textbf{(D) }\\frac{4}{7}\\qquad\\textbf{(E) }\\frac{7}{10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1617", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On a beach $50$ people are wearing sunglasses and $35$ people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap? \n$\\textbf{(A) }\\frac{14}{85}\\qquad\\textbf{(B) }\\frac{7}{25}\\qquad\\textbf{(C) }\\frac{2}{5}\\qquad\\textbf{(D) }\\frac{4}{7}\\qquad\\textbf{(E) }\\frac{7}{10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is $k\\%$ acid. From jar C, $\\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "85", + "index": "Sky-T1_10k_1618", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is $k\\%$ acid. From jar C, $\\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. Business is a little slow at Lou's Fine Shoes, so Lou decides to have a\nsale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the\nweekend, Lou advertises the sale: \"Ten percent off the listed price. Sale\nstarts Monday.\" How much does a pair of shoes cost on Monday that\ncost $40$ dollars on Thursday?\n$\\textbf{(A)}\\ 36\\qquad\\textbf{(B)}\\ 39.60\\qquad\\textbf{(C)}\\ 40\\qquad\\textbf{(D)}\\ 40.40\\qquad\\textbf{(E)}\\ 44$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1619", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Business is a little slow at Lou's Fine Shoes, so Lou decides to have a\nsale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the\nweekend, Lou advertises the sale: \"Ten percent off the listed price. Sale\nstarts Monday.\" How much does a pair of shoes cost on Monday that\ncost $40$ dollars on Thursday?\n$\\textbf{(A)}\\ 36\\qquad\\textbf{(B)}\\ 39.60\\qquad\\textbf{(C)}\\ 40\\qquad\\textbf{(D)}\\ 40.40\\qquad\\textbf{(E)}\\ 44$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?\n\n$\\textbf{(A) }\\frac{1}{2}(\\sqrt{6}-2)\\qquad\\textbf{(B) }\\frac{1}{4}\\qquad\\textbf{(C) }2-\\sqrt{3}\\qquad\\textbf{(D) }\\frac{\\sqrt{3}}{6}\\qquad\\textbf{(E) } 1-\\frac{\\sqrt{2}}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1620", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?\n\n$\\textbf{(A) }\\frac{1}{2}(\\sqrt{6}-2)\\qquad\\textbf{(B) }\\frac{1}{4}\\qquad\\textbf{(C) }2-\\sqrt{3}\\qquad\\textbf{(D) }\\frac{\\sqrt{3}}{6}\\qquad\\textbf{(E) } 1-\\frac{\\sqrt{2}}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $(*)$ Let $ABC$ be a triangle with $\\angle ABC$ obtuse. The $A$-excircle is a circle in the exterior of $\\triangle ABC$ that is tangent to side $\\overline{BC}$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can line $EF$ be tangent to the $A$-excircle?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "No", + "index": "Sky-T1_10k_1621", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $(*)$ Let $ABC$ be a triangle with $\\angle ABC$ obtuse. The $A$-excircle is a circle in the exterior of $\\triangle ABC$ that is tangent to side $\\overline{BC}$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can line $EF$ be tangent to the $A$-excircle?" + } + }, + { + "question": "Return your final response within \\boxed{}. On a certain math exam, $10\\%$ of the students got $70$ points, $25\\%$ got $80$ points, $20\\%$ got $85$ points, $15\\%$ got $90$ points, and the rest got $95$ points. What is the difference between the [mean](https://artofproblemsolving.com/wiki/index.php/Mean) and the [median](https://artofproblemsolving.com/wiki/index.php/Median) score on this exam?\n$\\textbf{(A) }\\ 0 \\qquad \\textbf{(B) }\\ 1 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 4 \\qquad \\textbf{(E) }\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1622", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On a certain math exam, $10\\%$ of the students got $70$ points, $25\\%$ got $80$ points, $20\\%$ got $85$ points, $15\\%$ got $90$ points, and the rest got $95$ points. What is the difference between the [mean](https://artofproblemsolving.com/wiki/index.php/Mean) and the [median](https://artofproblemsolving.com/wiki/index.php/Median) score on this exam?\n$\\textbf{(A) }\\ 0 \\qquad \\textbf{(B) }\\ 1 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 4 \\qquad \\textbf{(E) }\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the following equation, each of the letters represents uniquely a different digit in base ten: \n\\[(YE) \\cdot (ME) = TTT\\]\nThe sum $E+M+T+Y$ equals \n$\\textbf{(A)}\\ 19 \\qquad \\textbf{(B)}\\ 20 \\qquad \\textbf{(C)}\\ 21 \\qquad \\textbf{(D)}\\ 22 \\qquad \\textbf{(E)}\\ 24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1623", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the following equation, each of the letters represents uniquely a different digit in base ten: \n\\[(YE) \\cdot (ME) = TTT\\]\nThe sum $E+M+T+Y$ equals \n$\\textbf{(A)}\\ 19 \\qquad \\textbf{(B)}\\ 20 \\qquad \\textbf{(C)}\\ 21 \\qquad \\textbf{(D)}\\ 22 \\qquad \\textbf{(E)}\\ 24$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(x,y)$ of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) is it true that $0 < x < y < 10^6$ and that the [arithmetic mean](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean) of $x$ and $y$ is exactly $2$ more than the [geometric mean](https://artofproblemsolving.com/wiki/index.php/Geometric_mean) of $x$ and $y$?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "997", + "index": "Sky-T1_10k_1624", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(x,y)$ of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) is it true that $0 < x < y < 10^6$ and that the [arithmetic mean](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean) of $x$ and $y$ is exactly $2$ more than the [geometric mean](https://artofproblemsolving.com/wiki/index.php/Geometric_mean) of $x$ and $y$?" + } + }, + { + "question": "Return your final response within \\boxed{}. When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?\n$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 46 \\qquad \\textbf{(C)}\\ 47 \\qquad \\textbf{(D)}\\ 48 \\qquad \\textbf{(E)}\\ 49$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1625", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?\n$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 46 \\qquad \\textbf{(C)}\\ 47 \\qquad \\textbf{(D)}\\ 48 \\qquad \\textbf{(E)}\\ 49$" + } + }, + { + "question": "Return your final response within \\boxed{}. A wooden [cube](https://artofproblemsolving.com/wiki/index.php/Cube) with edge length $n$ units (where $n$ is an integer $>2$) is painted black all over. By slices parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $n$?\n$\\mathrm{(A)\\ } 5 \\qquad \\mathrm{(B) \\ }6 \\qquad \\mathrm{(C) \\ } 7 \\qquad \\mathrm{(D) \\ } 8 \\qquad \\mathrm{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1626", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A wooden [cube](https://artofproblemsolving.com/wiki/index.php/Cube) with edge length $n$ units (where $n$ is an integer $>2$) is painted black all over. By slices parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $n$?\n$\\mathrm{(A)\\ } 5 \\qquad \\mathrm{(B) \\ }6 \\qquad \\mathrm{(C) \\ } 7 \\qquad \\mathrm{(D) \\ } 8 \\qquad \\mathrm{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$?\n\n$\\text{(A)}\\ \\sqrt{10} \\qquad \\text{(B)}\\ \\sqrt{12} \\qquad \\text{(C)}\\ \\sqrt{13} \\qquad \\text{(D)}\\ \\sqrt{14} \\qquad \\text{(E)}\\ \\sqrt{15}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1627", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$?\n\n$\\text{(A)}\\ \\sqrt{10} \\qquad \\text{(B)}\\ \\sqrt{12} \\qquad \\text{(C)}\\ \\sqrt{13} \\qquad \\text{(D)}\\ \\sqrt{14} \\qquad \\text{(E)}\\ \\sqrt{15}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If 554 is the base $b$ representation of the square of the number whose base $b$ representation is 24, then $b$, when written in base 10, equals \n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 12\\qquad\\textbf{(D)}\\ 14\\qquad\\textbf{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1628", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If 554 is the base $b$ representation of the square of the number whose base $b$ representation is 24, then $b$, when written in base 10, equals \n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 12\\qquad\\textbf{(D)}\\ 14\\qquad\\textbf{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\\angle BAC$, and $BN\\perp AN$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then find $MN$.\n\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ \\dfrac{5}{2}\\qquad\\textbf{(C)}\\ \\dfrac{5}{2}-\\sin\\theta\\qquad\\textbf{(D)}\\ \\dfrac{5}{2}-\\dfrac{1}{2}\\sin\\theta\\qquad\\textbf{(E)}\\ \\dfrac{5}{2}-\\dfrac{1}{2}\\sin\\left(\\dfrac{1}{2}\\theta\\right)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1629", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\\angle BAC$, and $BN\\perp AN$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then find $MN$.\n\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ \\dfrac{5}{2}\\qquad\\textbf{(C)}\\ \\dfrac{5}{2}-\\sin\\theta\\qquad\\textbf{(D)}\\ \\dfrac{5}{2}-\\dfrac{1}{2}\\sin\\theta\\qquad\\textbf{(E)}\\ \\dfrac{5}{2}-\\dfrac{1}{2}\\sin\\left(\\dfrac{1}{2}\\theta\\right)$" + } + }, + { + "question": "Return your final response within \\boxed{}. The repeating decimal $0.ab\\cdots k\\overline{pq\\cdots u}=\\frac mn$, where $m$ and $n$ are relatively prime integers, and there is at least one decimal before the repeating part. Show that $n$ is divisble by 2 or 5 (or both). (For example, $0.011\\overline{36}=0.01136363636\\cdots=\\frac 1{88}$, and 88 is divisible by 2.)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "n is divisible by 2 or 5 (or both)", + "index": "Sky-T1_10k_1630", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The repeating decimal $0.ab\\cdots k\\overline{pq\\cdots u}=\\frac mn$, where $m$ and $n$ are relatively prime integers, and there is at least one decimal before the repeating part. Show that $n$ is divisble by 2 or 5 (or both). (For example, $0.011\\overline{36}=0.01136363636\\cdots=\\frac 1{88}$, and 88 is divisible by 2.)" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$. Find the greatest positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ such that $2^n$ divides $K$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "150", + "index": "Sky-T1_10k_1631", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$. Find the greatest positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ such that $2^n$ divides $K$." + } + }, + { + "question": "Return your final response within \\boxed{}. The shortest distances between an interior [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of a rectangular [parallelepiped](https://artofproblemsolving.com/wiki/index.php/Parallelepiped), $P$, and the edges it does not meet are $2\\sqrt{5}$, $\\frac{30}{\\sqrt{13}}$, and $\\frac{15}{\\sqrt{10}}$. Determine the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) of $P$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "750", + "index": "Sky-T1_10k_1632", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The shortest distances between an interior [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of a rectangular [parallelepiped](https://artofproblemsolving.com/wiki/index.php/Parallelepiped), $P$, and the edges it does not meet are $2\\sqrt{5}$, $\\frac{30}{\\sqrt{13}}$, and $\\frac{15}{\\sqrt{10}}$. Determine the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) of $P$." + } + }, + { + "question": "Return your final response within \\boxed{}. Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\\dfrac mn$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "41", + "index": "Sky-T1_10k_1633", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\\dfrac mn$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\\%$, then $y$ decreases by \n$\\textbf{(A)}\\ p\\%\\qquad \\textbf{(B)}\\ \\frac{p}{1+p}\\%\\qquad \\textbf{(C)}\\ \\frac{100}{p}\\%\\qquad \\textbf{(D)}\\ \\frac{p}{100+p}\\%\\qquad \\textbf{(E)}\\ \\frac{100p}{100+p}\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1634", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\\%$, then $y$ decreases by \n$\\textbf{(A)}\\ p\\%\\qquad \\textbf{(B)}\\ \\frac{p}{1+p}\\%\\qquad \\textbf{(C)}\\ \\frac{100}{p}\\%\\qquad \\textbf{(D)}\\ \\frac{p}{100+p}\\%\\qquad \\textbf{(E)}\\ \\frac{100p}{100+p}\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $x_1, x_2, \\ldots , x_n$ be a sequence of integers such that\n(i) $-1 \\le x_i \\le 2$ for $i = 1,2, \\ldots n$\n(ii) $x_1 + \\cdots + x_n = 19$; and\n(iii) $x_1^2 + x_2^2 + \\cdots + x_n^2 = 99$.\nLet $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \\cdots + x_n^3$, respectively. Then $\\frac Mm =$\n$\\mathrm{(A) \\ }3 \\qquad \\mathrm{(B) \\ }4 \\qquad \\mathrm{(C) \\ }5 \\qquad \\mathrm{(D) \\ }6 \\qquad \\mathrm{(E) \\ }7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1635", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $x_1, x_2, \\ldots , x_n$ be a sequence of integers such that\n(i) $-1 \\le x_i \\le 2$ for $i = 1,2, \\ldots n$\n(ii) $x_1 + \\cdots + x_n = 19$; and\n(iii) $x_1^2 + x_2^2 + \\cdots + x_n^2 = 99$.\nLet $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \\cdots + x_n^3$, respectively. Then $\\frac Mm =$\n$\\mathrm{(A) \\ }3 \\qquad \\mathrm{(B) \\ }4 \\qquad \\mathrm{(C) \\ }5 \\qquad \\mathrm{(D) \\ }6 \\qquad \\mathrm{(E) \\ }7$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 18 \\qquad \\textbf{(D) } 21 \\qquad \\textbf{(E) } 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1636", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 18 \\qquad \\textbf{(D) } 21 \\qquad \\textbf{(E) } 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \\leq 100$ for all $m$ such that $\\frac{1}{2} < m < a$. What is the maximum possible value of $a$?\n$\\textbf{(A)}\\ \\frac{51}{101} \\qquad \\textbf{(B)}\\ \\frac{50}{99} \\qquad \\textbf{(C)}\\ \\frac{51}{100} \\qquad \\textbf{(D)}\\ \\frac{52}{101} \\qquad \\textbf{(E)}\\ \\frac{13}{25}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1637", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \\leq 100$ for all $m$ such that $\\frac{1}{2} < m < a$. What is the maximum possible value of $a$?\n$\\textbf{(A)}\\ \\frac{51}{101} \\qquad \\textbf{(B)}\\ \\frac{50}{99} \\qquad \\textbf{(C)}\\ \\frac{51}{100} \\qquad \\textbf{(D)}\\ \\frac{52}{101} \\qquad \\textbf{(E)}\\ \\frac{13}{25}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning $8$ times, came home by bus in the afternoon $15$ times, and commuted by train (either morning or afternoon) $9$ times. Find $x$.\n$\\textbf{(A)}\\ 19 \\qquad \\textbf{(B)}\\ 18 \\qquad \\textbf{(C)}\\ 17 \\qquad \\textbf{(D)}\\ 16 \\qquad \\\\ \\textbf{(E)}\\ \\text{ not enough information given to solve the problem}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1638", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning $8$ times, came home by bus in the afternoon $15$ times, and commuted by train (either morning or afternoon) $9$ times. Find $x$.\n$\\textbf{(A)}\\ 19 \\qquad \\textbf{(B)}\\ 18 \\qquad \\textbf{(C)}\\ 17 \\qquad \\textbf{(D)}\\ 16 \\qquad \\\\ \\textbf{(E)}\\ \\text{ not enough information given to solve the problem}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The arithmetic mean (average) of the first $n$ positive integers is:\n$\\textbf{(A)}\\ \\frac{n}{2} \\qquad\\textbf{(B)}\\ \\frac{n^2}{2}\\qquad\\textbf{(C)}\\ n\\qquad\\textbf{(D)}\\ \\frac{n-1}{2}\\qquad\\textbf{(E)}\\ \\frac{n+1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1639", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The arithmetic mean (average) of the first $n$ positive integers is:\n$\\textbf{(A)}\\ \\frac{n}{2} \\qquad\\textbf{(B)}\\ \\frac{n^2}{2}\\qquad\\textbf{(C)}\\ n\\qquad\\textbf{(D)}\\ \\frac{n-1}{2}\\qquad\\textbf{(E)}\\ \\frac{n+1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?\n$\\mathrm{(A)}\\ 1 \\qquad \\mathrm{(B)}\\ \\sqrt{2} \\qquad \\mathrm{(C)}\\ \\sqrt{3} \\qquad \\mathrm{(D)}\\ 2 \\qquad \\mathrm{(E)}\\ 2\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1640", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?\n$\\mathrm{(A)}\\ 1 \\qquad \\mathrm{(B)}\\ \\sqrt{2} \\qquad \\mathrm{(C)}\\ \\sqrt{3} \\qquad \\mathrm{(D)}\\ 2 \\qquad \\mathrm{(E)}\\ 2\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A frog is placed at the [origin](https://artofproblemsolving.com/wiki/index.php/Origin) on the [number line](https://artofproblemsolving.com/wiki/index.php/Number_line), and moves according to the following rule: in a given move, the frog advances to either the closest [point](https://artofproblemsolving.com/wiki/index.php/Point) with a greater [integer](https://artofproblemsolving.com/wiki/index.php/Integer) [coordinate](https://artofproblemsolving.com/wiki/index.php/Coordinate) that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of coordinates that correspond to valid moves, beginning with 0 and ending with 39. For example, $0,\\ 3,\\ 6,\\ 13,\\ 15,\\ 26,\\ 39$ is a move sequence. How many move sequences are possible for the frog?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "169", + "index": "Sky-T1_10k_1641", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A frog is placed at the [origin](https://artofproblemsolving.com/wiki/index.php/Origin) on the [number line](https://artofproblemsolving.com/wiki/index.php/Number_line), and moves according to the following rule: in a given move, the frog advances to either the closest [point](https://artofproblemsolving.com/wiki/index.php/Point) with a greater [integer](https://artofproblemsolving.com/wiki/index.php/Integer) [coordinate](https://artofproblemsolving.com/wiki/index.php/Coordinate) that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of coordinates that correspond to valid moves, beginning with 0 and ending with 39. For example, $0,\\ 3,\\ 6,\\ 13,\\ 15,\\ 26,\\ 39$ is a move sequence. How many move sequences are possible for the frog?" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ is real and $4y^2+4xy+x+6=0$, then the complete set of values of $x$ for which $y$ is real, is:\n$\\text{(A) } x\\le-2 \\text{ or } x\\ge3 \\quad \\text{(B) } x\\le2 \\text{ or } x\\ge3 \\quad \\text{(C) } x\\le-3 \\text{ or } x\\ge2 \\quad \\\\ \\text{(D) } -3\\le x\\le2 \\quad \\text{(E) } -2\\le x\\le3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1642", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ is real and $4y^2+4xy+x+6=0$, then the complete set of values of $x$ for which $y$ is real, is:\n$\\text{(A) } x\\le-2 \\text{ or } x\\ge3 \\quad \\text{(B) } x\\le2 \\text{ or } x\\ge3 \\quad \\text{(C) } x\\le-3 \\text{ or } x\\ge2 \\quad \\\\ \\text{(D) } -3\\le x\\le2 \\quad \\text{(E) } -2\\le x\\le3$" + } + }, + { + "question": "Return your final response within \\boxed{}. For $n \\ge 1$ call a finite sequence $(a_1, a_2 \\ldots a_n)$ of positive integers progressive if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \\le i \\le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "47", + "index": "Sky-T1_10k_1643", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For $n \\ge 1$ call a finite sequence $(a_1, a_2 \\ldots a_n)$ of positive integers progressive if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \\le i \\le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$." + } + }, + { + "question": "Return your final response within \\boxed{}. Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is: \n$\\textbf{(A)}\\ 10(q - 1) \\qquad \\textbf{(B)}\\ \\frac {2}{5}(4q - 4) \\qquad \\textbf{(C)}\\ \\frac {2}{5}(q - 1) \\\\ \\textbf{(D)}\\ \\frac{5}{2}(q-1)\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1644", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is: \n$\\textbf{(A)}\\ 10(q - 1) \\qquad \\textbf{(B)}\\ \\frac {2}{5}(4q - 4) \\qquad \\textbf{(C)}\\ \\frac {2}{5}(q - 1) \\\\ \\textbf{(D)}\\ \\frac{5}{2}(q-1)\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be a positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) such that $\\frac 12 + \\frac 13 + \\frac 17 + \\frac 1n$ is an integer. Which of the following statements is not true:\n$\\mathrm{(A)}\\ 2\\ \\text{divides\\ }n \\qquad\\mathrm{(B)}\\ 3\\ \\text{divides\\ }n \\qquad\\mathrm{(C)}$ $\\ 6\\ \\text{divides\\ }n \\qquad\\mathrm{(D)}\\ 7\\ \\text{divides\\ }n \\qquad\\mathrm{(E)}\\ n > 84$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1645", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $n$ be a positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) such that $\\frac 12 + \\frac 13 + \\frac 17 + \\frac 1n$ is an integer. Which of the following statements is not true:\n$\\mathrm{(A)}\\ 2\\ \\text{divides\\ }n \\qquad\\mathrm{(B)}\\ 3\\ \\text{divides\\ }n \\qquad\\mathrm{(C)}$ $\\ 6\\ \\text{divides\\ }n \\qquad\\mathrm{(D)}\\ 7\\ \\text{divides\\ }n \\qquad\\mathrm{(E)}\\ n > 84$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a square, and let $E$ and $F$ be points on $\\overline{AB}$ and $\\overline{BC},$ respectively. The line through $E$ parallel to $\\overline{BC}$ and the line through $F$ parallel to $\\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\\frac{9}{10}$ of the area of square $ABCD.$ Find $\\frac{AE}{EB} + \\frac{EB}{AE}.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "18", + "index": "Sky-T1_10k_1646", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a square, and let $E$ and $F$ be points on $\\overline{AB}$ and $\\overline{BC},$ respectively. The line through $E$ parallel to $\\overline{BC}$ and the line through $F$ parallel to $\\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\\frac{9}{10}$ of the area of square $ABCD.$ Find $\\frac{AE}{EB} + \\frac{EB}{AE}.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Line $l$ in the coordinate plane has equation $3x-5y+40=0$. This line is rotated $45^{\\circ}$ counterclockwise about the point $(20,20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$\n$\\textbf{(A) } 10 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 20 \\qquad \\textbf{(D) } 25 \\qquad \\textbf{(E) } 30$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1647", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Line $l$ in the coordinate plane has equation $3x-5y+40=0$. This line is rotated $45^{\\circ}$ counterclockwise about the point $(20,20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$\n$\\textbf{(A) } 10 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 20 \\qquad \\textbf{(D) } 25 \\qquad \\textbf{(E) } 30$" + } + }, + { + "question": "Return your final response within \\boxed{}. A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?\n$\\mathrm{(A)}\\ 50.2 \\qquad \\mathrm{(B)}\\ 51.5 \\qquad \\mathrm{(C)}\\ 52.4 \\qquad \\mathrm{(D)}\\ 53.8 \\qquad \\mathrm{(E)}\\ 55.2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1648", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?\n$\\mathrm{(A)}\\ 50.2 \\qquad \\mathrm{(B)}\\ 51.5 \\qquad \\mathrm{(C)}\\ 52.4 \\qquad \\mathrm{(D)}\\ 53.8 \\qquad \\mathrm{(E)}\\ 55.2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Positive numbers $x$, $y$, and $z$ satisfy $xyz = 10^{81}$ and $(\\log_{10}x)(\\log_{10} yz) + (\\log_{10}y) (\\log_{10}z) = 468$. Find $\\sqrt {(\\log_{10}x)^2 + (\\log_{10}y)^2 + (\\log_{10}z)^2}$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "75", + "index": "Sky-T1_10k_1649", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Positive numbers $x$, $y$, and $z$ satisfy $xyz = 10^{81}$ and $(\\log_{10}x)(\\log_{10} yz) + (\\log_{10}y) (\\log_{10}z) = 468$. Find $\\sqrt {(\\log_{10}x)^2 + (\\log_{10}y)^2 + (\\log_{10}z)^2}$." + } + }, + { + "question": "Return your final response within \\boxed{}. For what real values of $k$, other than $k = 0$, does the equation $x^2 + kx + k^2 = 0$ have real roots?\n$\\textbf{(A)}\\ {k < 0}\\qquad \\textbf{(B)}\\ {k > 0} \\qquad \\textbf{(C)}\\ {k \\ge 1} \\qquad \\textbf{(D)}\\ \\text{all values of }{k}\\qquad \\textbf{(E)}\\ \\text{no values of }{k}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1650", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For what real values of $k$, other than $k = 0$, does the equation $x^2 + kx + k^2 = 0$ have real roots?\n$\\textbf{(A)}\\ {k < 0}\\qquad \\textbf{(B)}\\ {k > 0} \\qquad \\textbf{(C)}\\ {k \\ge 1} \\qquad \\textbf{(D)}\\ \\text{all values of }{k}\\qquad \\textbf{(E)}\\ \\text{no values of }{k}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is \n$\\text{(A)}\\ 40\\text{ dollars} \\qquad \\text{(B)}\\ 50\\text{ dollars} \\qquad \\text{(C)}\\ 80\\text{ dollars} \\qquad \\text{(D)}\\ 100\\text{ dollars} \\qquad \\text{(E)}\\ 125\\text{ dollars}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1651", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is \n$\\text{(A)}\\ 40\\text{ dollars} \\qquad \\text{(B)}\\ 50\\text{ dollars} \\qquad \\text{(C)}\\ 80\\text{ dollars} \\qquad \\text{(D)}\\ 100\\text{ dollars} \\qquad \\text{(E)}\\ 125\\text{ dollars}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $1-\\frac{4}{x}+\\frac{4}{x^2}=0$, then $\\frac{2}{x}$ equals\n$\\textbf{(A) }-1\\qquad \\textbf{(B) }1\\qquad \\textbf{(C) }2\\qquad \\textbf{(D) }-1\\text{ or }2\\qquad \\textbf{(E) }-1\\text{ or }-2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1652", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $1-\\frac{4}{x}+\\frac{4}{x^2}=0$, then $\\frac{2}{x}$ equals\n$\\textbf{(A) }-1\\qquad \\textbf{(B) }1\\qquad \\textbf{(C) }2\\qquad \\textbf{(D) }-1\\text{ or }2\\qquad \\textbf{(E) }-1\\text{ or }-2$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $\\frac{2a^{-1}+\\frac{a^{-1}}{2}}{a}$ when $a= \\tfrac{1}{2}$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\frac{5}{2}\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1653", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $\\frac{2a^{-1}+\\frac{a^{-1}}{2}}{a}$ when $a= \\tfrac{1}{2}$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\frac{5}{2}\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ is such that $\\frac{1}{x}<2$ and $\\frac{1}{x}>-3$, then:\n$\\text{(A) } -\\frac{1}{3}\\frac{1}{2}\\quad\\\\ \\text{(D) } x>\\frac{1}{2} \\text{ or} -\\frac{1}{3}\\frac{1}{2} \\text{ or } x<-\\frac{1}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1654", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ is such that $\\frac{1}{x}<2$ and $\\frac{1}{x}>-3$, then:\n$\\text{(A) } -\\frac{1}{3}\\frac{1}{2}\\quad\\\\ \\text{(D) } x>\\frac{1}{2} \\text{ or} -\\frac{1}{3}\\frac{1}{2} \\text{ or } x<-\\frac{1}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $$105$, Dorothy paid $$125$, and Sammy paid $$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 20\\qquad\\textbf{(C)}\\ 25\\qquad\\textbf{(D)}\\ 30\\qquad\\textbf{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1655", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $$105$, Dorothy paid $$125$, and Sammy paid $$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 20\\qquad\\textbf{(C)}\\ 25\\qquad\\textbf{(D)}\\ 30\\qquad\\textbf{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a = - 2$, the largest number in the set $\\{ - 3a, 4a, \\frac {24}{a}, a^2, 1\\}$ is\n$\\text{(A)}\\ -3a \\qquad \\text{(B)}\\ 4a \\qquad \\text{(C)}\\ \\frac {24}{a} \\qquad \\text{(D)}\\ a^2 \\qquad \\text{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1656", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a = - 2$, the largest number in the set $\\{ - 3a, 4a, \\frac {24}{a}, a^2, 1\\}$ is\n$\\text{(A)}\\ -3a \\qquad \\text{(B)}\\ 4a \\qquad \\text{(C)}\\ \\frac {24}{a} \\qquad \\text{(D)}\\ a^2 \\qquad \\text{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Walter rolls four standard six-sided dice and finds that the product of the numbers of the upper faces is $144$. Which of he following could not be the sum of the upper four faces? \n$\\mathrm{(A) \\ }14 \\qquad \\mathrm{(B) \\ }15 \\qquad \\mathrm{(C) \\ }16 \\qquad \\mathrm{(D) \\ }17 \\qquad \\mathrm{(E) \\ }18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1657", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Walter rolls four standard six-sided dice and finds that the product of the numbers of the upper faces is $144$. Which of he following could not be the sum of the upper four faces? \n$\\mathrm{(A) \\ }14 \\qquad \\mathrm{(B) \\ }15 \\qquad \\mathrm{(C) \\ }16 \\qquad \\mathrm{(D) \\ }17 \\qquad \\mathrm{(E) \\ }18$" + } + }, + { + "question": "Return your final response within \\boxed{}. $2000(2000^{2000}) = ?$\n$\\textbf{(A)} \\ 2000^{2001} \\qquad \\textbf{(B)} \\ 4000^{2000} \\qquad \\textbf{(C)} \\ 2000^{4000} \\qquad \\textbf{(D)} \\ 4,000,000^{2000} \\qquad \\textbf{(E)} \\ 2000^{4,000,000}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1658", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $2000(2000^{2000}) = ?$\n$\\textbf{(A)} \\ 2000^{2001} \\qquad \\textbf{(B)} \\ 4000^{2000} \\qquad \\textbf{(C)} \\ 2000^{4000} \\qquad \\textbf{(D)} \\ 4,000,000^{2000} \\qquad \\textbf{(E)} \\ 2000^{4,000,000}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\triangle{ABC}$ be a non-equilateral, acute triangle with $\\angle A=60^{\\circ}$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\\triangle{ABC}$, respectively.\n(a) Prove that line $OH$ intersects both segments $AB$ and $AC$.\n(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\left( \\dfrac{4}{5}, 1 \\right)", + "index": "Sky-T1_10k_1659", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\triangle{ABC}$ be a non-equilateral, acute triangle with $\\angle A=60^{\\circ}$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\\triangle{ABC}$, respectively.\n(a) Prove that line $OH$ intersects both segments $AB$ and $AC$.\n(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$." + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$?\n$\\textbf{(A)}\\ \\frac{11}{81} \\qquad \\textbf{(B)}\\ \\frac{13}{81} \\qquad \\textbf{(C)}\\ \\frac{5}{27} \\qquad \\textbf{(D)}\\ \\frac{17}{81} \\qquad \\textbf{(E)}\\ \\frac{19}{81}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1660", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$?\n$\\textbf{(A)}\\ \\frac{11}{81} \\qquad \\textbf{(B)}\\ \\frac{13}{81} \\qquad \\textbf{(C)}\\ \\frac{5}{27} \\qquad \\textbf{(D)}\\ \\frac{17}{81} \\qquad \\textbf{(E)}\\ \\frac{19}{81}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is\n$\\text{(A)}\\ 8 \\qquad \\text{(B)}\\ 9 \\qquad \\text{(C)}\\ 10 \\qquad \\text{(D)}\\ 12 \\qquad \\text{(E)}\\ 18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1661", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is\n$\\text{(A)}\\ 8 \\qquad \\text{(B)}\\ 9 \\qquad \\text{(C)}\\ 10 \\qquad \\text{(D)}\\ 12 \\qquad \\text{(E)}\\ 18$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of these describes the graph of $x^2(x+y+1)=y^2(x+y+1)$ ?\n$\\textbf{(A)}\\ \\text{two parallel lines}\\\\ \\textbf{(B)}\\ \\text{two intersecting lines}\\\\ \\textbf{(C)}\\ \\text{three lines that all pass through a common point}\\\\ \\textbf{(D)}\\ \\text{three lines that do not all pass through a common point}\\\\ \\textbf{(E)}\\ \\text{a line and a parabola}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1662", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of these describes the graph of $x^2(x+y+1)=y^2(x+y+1)$ ?\n$\\textbf{(A)}\\ \\text{two parallel lines}\\\\ \\textbf{(B)}\\ \\text{two intersecting lines}\\\\ \\textbf{(C)}\\ \\text{three lines that all pass through a common point}\\\\ \\textbf{(D)}\\ \\text{three lines that do not all pass through a common point}\\\\ \\textbf{(E)}\\ \\text{a line and a parabola}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?\n$\\textbf{(A)}\\ \\frac{\\sqrt{3}-1}{2}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ \\frac{\\sqrt{5}-1}{2} \\qquad\\textbf{(D)}\\ \\frac{\\sqrt{2}}{2} \\qquad\\textbf{(E)}\\ \\frac{\\sqrt{6}-1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1663", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?\n$\\textbf{(A)}\\ \\frac{\\sqrt{3}-1}{2}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ \\frac{\\sqrt{5}-1}{2} \\qquad\\textbf{(D)}\\ \\frac{\\sqrt{2}}{2} \\qquad\\textbf{(E)}\\ \\frac{\\sqrt{6}-1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $[\\log_{10}(5\\log_{10}100)]^2$?\n$\\textbf{(A)}\\ \\log_{10}50 \\qquad \\textbf{(B)}\\ 25\\qquad \\textbf{(C)}\\ 10 \\qquad \\textbf{(D)}\\ 2\\qquad \\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1664", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $[\\log_{10}(5\\log_{10}100)]^2$?\n$\\textbf{(A)}\\ \\log_{10}50 \\qquad \\textbf{(B)}\\ 25\\qquad \\textbf{(C)}\\ 10 \\qquad \\textbf{(D)}\\ 2\\qquad \\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=$\n$\\text{(A)}\\ -1 \\qquad \\text{(B)}\\ 1 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1665", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=$\n$\\text{(A)}\\ -1 \\qquad \\text{(B)}\\ 1 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$\n$\\textbf{(A)} ~47 \\qquad\\textbf{(B)} ~94 \\qquad\\textbf{(C)} ~227 \\qquad\\textbf{(D)} ~471 \\qquad\\textbf{(E)} ~542$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1666", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$\n$\\textbf{(A)} ~47 \\qquad\\textbf{(B)} ~94 \\qquad\\textbf{(C)} ~227 \\qquad\\textbf{(D)} ~471 \\qquad\\textbf{(E)} ~542$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is the value of $\\sqrt{\\log_2{6}+\\log_3{6}}?$\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } \\sqrt{\\log_5{6}} \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } \\sqrt{\\log_2{3}}+\\sqrt{\\log_3{2}} \\qquad\\textbf{(E) } \\sqrt{\\log_2{6}}+\\sqrt{\\log_3{6}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1667", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is the value of $\\sqrt{\\log_2{6}+\\log_3{6}}?$\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } \\sqrt{\\log_5{6}} \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } \\sqrt{\\log_2{3}}+\\sqrt{\\log_3{2}} \\qquad\\textbf{(E) } \\sqrt{\\log_2{6}}+\\sqrt{\\log_3{6}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period?\n$\\textbf{(A)}\\ 50 \\qquad\\textbf{(B)}\\ 55 \\qquad\\textbf{(C)}\\ 60 \\qquad\\textbf{(D)}\\ 65 \\qquad\\textbf{(E)}\\ 70$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1668", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period?\n$\\textbf{(A)}\\ 50 \\qquad\\textbf{(B)}\\ 55 \\qquad\\textbf{(C)}\\ 60 \\qquad\\textbf{(D)}\\ 65 \\qquad\\textbf{(E)}\\ 70$" + } + }, + { + "question": "Return your final response within \\boxed{}. A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are $(3,17)$ and $(48,281)$? (Include both endpoints of the segment in your count.)\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 16\\qquad\\textbf{(E)}\\ 46$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1669", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are $(3,17)$ and $(48,281)$? (Include both endpoints of the segment in your count.)\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 16\\qquad\\textbf{(E)}\\ 46$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many ordered triples $(x,y,z)$ of positive integers satisfy $\\text{lcm}(x,y) = 72, \\text{lcm}(x,z) = 600 \\text{ and lcm}(y,z)=900$?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 27\\qquad\\textbf{(E)}\\ 64$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1670", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many ordered triples $(x,y,z)$ of positive integers satisfy $\\text{lcm}(x,y) = 72, \\text{lcm}(x,z) = 600 \\text{ and lcm}(y,z)=900$?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 27\\qquad\\textbf{(E)}\\ 64$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define \\[P(x) =(x-1^2)(x-2^2)\\cdots(x-100^2).\\] How many integers $n$ are there such that $P(n)\\leq 0$?\n$\\textbf{(A) } 4900 \\qquad \\textbf{(B) } 4950\\qquad \\textbf{(C) } 5000\\qquad \\textbf{(D) } 5050 \\qquad \\textbf{(E) } 5100$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1671", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define \\[P(x) =(x-1^2)(x-2^2)\\cdots(x-100^2).\\] How many integers $n$ are there such that $P(n)\\leq 0$?\n$\\textbf{(A) } 4900 \\qquad \\textbf{(B) } 4950\\qquad \\textbf{(C) } 5000\\qquad \\textbf{(D) } 5050 \\qquad \\textbf{(E) } 5100$" + } + }, + { + "question": "Return your final response within \\boxed{}. (Zuming Feng) Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "n is composite and not the product of two distinct primes", + "index": "Sky-T1_10k_1672", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. (Zuming Feng) Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime." + } + }, + { + "question": "Return your final response within \\boxed{}. The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is\n$\\textbf{(A)}\\ \\frac{1}{4} \\qquad \\textbf{(B)}\\ \\frac{1}{3} \\qquad \\textbf{(C)}\\ \\frac{1}{2} \\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ 2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1673", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is\n$\\textbf{(A)}\\ \\frac{1}{4} \\qquad \\textbf{(B)}\\ \\frac{1}{3} \\qquad \\textbf{(C)}\\ \\frac{1}{2} \\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ 2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Of the following sets of data the only one that does not determine the shape of a triangle is: \n$\\textbf{(A)}\\ \\text{the ratio of two sides and the inc{}luded angle}\\\\ \\qquad\\textbf{(B)}\\ \\text{the ratios of the three altitudes}\\\\ \\qquad\\textbf{(C)}\\ \\text{the ratios of the three medians}\\\\ \\qquad\\textbf{(D)}\\ \\text{the ratio of the altitude to the corresponding base}\\\\ \\qquad\\textbf{(E)}\\ \\text{two angles}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1674", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Of the following sets of data the only one that does not determine the shape of a triangle is: \n$\\textbf{(A)}\\ \\text{the ratio of two sides and the inc{}luded angle}\\\\ \\qquad\\textbf{(B)}\\ \\text{the ratios of the three altitudes}\\\\ \\qquad\\textbf{(C)}\\ \\text{the ratios of the three medians}\\\\ \\qquad\\textbf{(D)}\\ \\text{the ratio of the altitude to the corresponding base}\\\\ \\qquad\\textbf{(E)}\\ \\text{two angles}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to\n$\\textbf{(A) }-1\\qquad \\textbf{(B) }\\textstyle\\frac{1}{2}\\qquad \\textbf{(C) }\\textstyle\\frac{3}{2}\\qquad \\textbf{(D) }2\\qquad \\textbf{(E) }\\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1675", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to\n$\\textbf{(A) }-1\\qquad \\textbf{(B) }\\textstyle\\frac{1}{2}\\qquad \\textbf{(C) }\\textstyle\\frac{3}{2}\\qquad \\textbf{(D) }2\\qquad \\textbf{(E) }\\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Determine whether or not there are any positive integral solutions of the simultaneous equations \n\\begin{align*} x_1^2 +x_2^2 +\\cdots +x_{1985}^2 & = y^3,\\\\ x_1^3 +x_2^3 +\\cdots +x_{1985}^3 & = z^2 \\end{align*}\nwith distinct integers $x_1,x_2,\\cdots,x_{1985}$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "Yes", + "index": "Sky-T1_10k_1676", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Determine whether or not there are any positive integral solutions of the simultaneous equations \n\\begin{align*} x_1^2 +x_2^2 +\\cdots +x_{1985}^2 & = y^3,\\\\ x_1^3 +x_2^3 +\\cdots +x_{1985}^3 & = z^2 \\end{align*}\nwith distinct integers $x_1,x_2,\\cdots,x_{1985}$." + } + }, + { + "question": "Return your final response within \\boxed{}. Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "41", + "index": "Sky-T1_10k_1677", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is" + } + }, + { + "question": "Return your final response within \\boxed{}. A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, \nwith each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents? \n$\\text{(A)} \\ \\frac{37}{924} \\qquad \\text{(B)} \\ \\frac{91}{924} \\qquad \\text{(C)} \\ \\frac{127}{924} \\qquad \\text{(D)}\\ \\frac{132}{924}\\qquad \\text{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1678", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, \nwith each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents? \n$\\text{(A)} \\ \\frac{37}{924} \\qquad \\text{(B)} \\ \\frac{91}{924} \\qquad \\text{(C)} \\ \\frac{127}{924} \\qquad \\text{(D)}\\ \\frac{132}{924}\\qquad \\text{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Chloe chooses a real number uniformly at random from the interval $[ 0,2017 ]$. Independently, Laurent chooses a real number uniformly at random from the interval $[ 0 , 4034 ]$. What is the probability that Laurent's number is greater than Chloe's number? \n$\\textbf{(A)}\\ \\dfrac{1}{2} \\qquad\\textbf{(B)}\\ \\dfrac{2}{3} \\qquad\\textbf{(C)}\\ \\dfrac{3}{4} \\qquad\\textbf{(D)}\\ \\dfrac{5}{6} \\qquad\\textbf{(E)}\\ \\dfrac{7}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1679", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Chloe chooses a real number uniformly at random from the interval $[ 0,2017 ]$. Independently, Laurent chooses a real number uniformly at random from the interval $[ 0 , 4034 ]$. What is the probability that Laurent's number is greater than Chloe's number? \n$\\textbf{(A)}\\ \\dfrac{1}{2} \\qquad\\textbf{(B)}\\ \\dfrac{2}{3} \\qquad\\textbf{(C)}\\ \\dfrac{3}{4} \\qquad\\textbf{(D)}\\ \\dfrac{5}{6} \\qquad\\textbf{(E)}\\ \\dfrac{7}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\lfloor x \\rfloor$ be the greatest integer less than or equal to $x$. Then the number of real solutions to $4x^2-40\\lfloor x \\rfloor +51=0$ is\n$\\mathrm{(A)\\ } 0 \\qquad \\mathrm{(B) \\ }1 \\qquad \\mathrm{(C) \\ } 2 \\qquad \\mathrm{(D) \\ } 3 \\qquad \\mathrm{(E) \\ }4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1680", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\lfloor x \\rfloor$ be the greatest integer less than or equal to $x$. Then the number of real solutions to $4x^2-40\\lfloor x \\rfloor +51=0$ is\n$\\mathrm{(A)\\ } 0 \\qquad \\mathrm{(B) \\ }1 \\qquad \\mathrm{(C) \\ } 2 \\qquad \\mathrm{(D) \\ } 3 \\qquad \\mathrm{(E) \\ }4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Elmer's new car gives $50\\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?\n$\\textbf{(A) } 20\\% \\qquad \\textbf{(B) } 26\\tfrac23\\% \\qquad \\textbf{(C) } 27\\tfrac79\\% \\qquad \\textbf{(D) } 33\\tfrac13\\% \\qquad \\textbf{(E) } 66\\tfrac23\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1681", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Elmer's new car gives $50\\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?\n$\\textbf{(A) } 20\\% \\qquad \\textbf{(B) } 26\\tfrac23\\% \\qquad \\textbf{(C) } 27\\tfrac79\\% \\qquad \\textbf{(D) } 33\\tfrac13\\% \\qquad \\textbf{(E) } 66\\tfrac23\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\\clubsuit(8)=8$ and $\\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\\clubsuit(\\clubsuit(x))=3$?\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 6 \\qquad\\textbf{(D) } 9 \\qquad\\textbf{(E) } 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1682", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\\clubsuit(8)=8$ and $\\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\\clubsuit(\\clubsuit(x))=3$?\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 6 \\qquad\\textbf{(D) } 9 \\qquad\\textbf{(E) } 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$\n$\\textbf{(A) }5\\qquad\\textbf{(B) }7\\qquad\\textbf{(C) }9\\qquad\\textbf{(D) }11\\qquad \\textbf{(E) }12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1683", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$\n$\\textbf{(A) }5\\qquad\\textbf{(B) }7\\qquad\\textbf{(C) }9\\qquad\\textbf{(D) }11\\qquad \\textbf{(E) }12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? \n$\\textbf{(A) }\\ \\frac15 \\qquad\\textbf{(B) }\\ \\frac13 \\qquad\\textbf{(C) }\\ \\frac25 \\qquad\\textbf{(D) }\\ \\frac23 \\qquad\\textbf{(E) }\\ \\frac45$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1684", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? \n$\\textbf{(A) }\\ \\frac15 \\qquad\\textbf{(B) }\\ \\frac13 \\qquad\\textbf{(C) }\\ \\frac25 \\qquad\\textbf{(D) }\\ \\frac23 \\qquad\\textbf{(E) }\\ \\frac45$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $\\frac{2a^{-1}+\\frac{a^{-1}}{2}}{a}$ when $a= \\frac{1}{2}$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\frac{5}{2}\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1685", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $\\frac{2a^{-1}+\\frac{a^{-1}}{2}}{a}$ when $a= \\frac{1}{2}$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ \\frac{5}{2}\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. The $120$ permutations of $AHSME$ are arranged in dictionary order as if each were an ordinary five-letter word. \nThe last letter of the $86$th word in this list is:\n$\\textbf{(A)}\\ \\text{A} \\qquad \\textbf{(B)}\\ \\text{H} \\qquad \\textbf{(C)}\\ \\text{S} \\qquad \\textbf{(D)}\\ \\text{M}\\qquad \\textbf{(E)}\\ \\text{E}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1686", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The $120$ permutations of $AHSME$ are arranged in dictionary order as if each were an ordinary five-letter word. \nThe last letter of the $86$th word in this list is:\n$\\textbf{(A)}\\ \\text{A} \\qquad \\textbf{(B)}\\ \\text{H} \\qquad \\textbf{(C)}\\ \\text{S} \\qquad \\textbf{(D)}\\ \\text{M}\\qquad \\textbf{(E)}\\ \\text{E}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let S be the set of values assumed by the fraction $\\frac{2x+3}{x+2}$.\nWhen $x$ is any member of the interval $x \\ge 0$. If there exists a number $M$ such that no number of the set $S$ is greater than $M$, \nthen $M$ is an upper bound of $S$. If there exists a number $m$ such that such that no number of the set $S$ is less than $m$, \nthen $m$ is a lower bound of $S$. We may then say:\n$\\textbf{(A)}\\ \\text{m is in S, but M is not in S}\\qquad\\\\ \\textbf{(B)}\\ \\text{M is in S, but m is not in S}\\qquad\\\\ \\textbf{(C)}\\ \\text{Both m and M are in S}\\qquad\\\\ \\textbf{(D)}\\ \\text{Neither m nor M are in S}\\qquad\\\\ \\textbf{(E)}\\ \\text{M does not exist either in or outside S}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1687", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let S be the set of values assumed by the fraction $\\frac{2x+3}{x+2}$.\nWhen $x$ is any member of the interval $x \\ge 0$. If there exists a number $M$ such that no number of the set $S$ is greater than $M$, \nthen $M$ is an upper bound of $S$. If there exists a number $m$ such that such that no number of the set $S$ is less than $m$, \nthen $m$ is a lower bound of $S$. We may then say:\n$\\textbf{(A)}\\ \\text{m is in S, but M is not in S}\\qquad\\\\ \\textbf{(B)}\\ \\text{M is in S, but m is not in S}\\qquad\\\\ \\textbf{(C)}\\ \\text{Both m and M are in S}\\qquad\\\\ \\textbf{(D)}\\ \\text{Neither m nor M are in S}\\qquad\\\\ \\textbf{(E)}\\ \\text{M does not exist either in or outside S}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\frac{a}{10^x-1}+\\frac{b}{10^x+2}=\\frac{2 \\cdot 10^x+3}{(10^x-1)(10^x+2)}$ is an identity for positive rational values of $x$, then the value of $a-b$ is:\n$\\textbf{(A)}\\ \\frac{4}{3} \\qquad \\textbf{(B)}\\ \\frac{5}{3} \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ \\frac{11}{4} \\qquad \\textbf{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1688", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\frac{a}{10^x-1}+\\frac{b}{10^x+2}=\\frac{2 \\cdot 10^x+3}{(10^x-1)(10^x+2)}$ is an identity for positive rational values of $x$, then the value of $a-b$ is:\n$\\textbf{(A)}\\ \\frac{4}{3} \\qquad \\textbf{(B)}\\ \\frac{5}{3} \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ \\frac{11}{4} \\qquad \\textbf{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. The 2-digit integers from 19 to 92 are written consecutively to form the integer $N=192021\\cdots9192$. Suppose that $3^k$ is the highest power of 3 that is a factor of $N$. What is $k$?\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } 3\\quad \\text{(E) more than } 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1689", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The 2-digit integers from 19 to 92 are written consecutively to form the integer $N=192021\\cdots9192$. Suppose that $3^k$ is the highest power of 3 that is a factor of $N$. What is $k$?\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } 3\\quad \\text{(E) more than } 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$\n$\\textbf{(A) } 80 \\qquad \\textbf{(B) } 100 \\qquad \\textbf{(C) } 125 \\qquad \\textbf{(D) } 200 \\qquad \\textbf{(E) } 500$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1690", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$\n$\\textbf{(A) } 80 \\qquad \\textbf{(B) } 100 \\qquad \\textbf{(C) } 125 \\qquad \\textbf{(D) } 200 \\qquad \\textbf{(E) } 500$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\sqrt{x+2}=2$, then $(x+2)^2$ equals:\n$\\textbf{(A)}\\ \\sqrt{2}\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1691", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\sqrt{x+2}=2$, then $(x+2)^2$ equals:\n$\\textbf{(A)}\\ \\sqrt{2}\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. From time $t=0$ to time $t=1$ a population increased by $i\\%$, and from time $t=1$ to time $t=2$ the population increased by $j\\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by\n$\\text{(A) (i+j)\\%} \\quad \\text{(B) } ij\\%\\quad \\text{(C) } (i+ij)\\%\\quad \\text{(D) } \\left(i+j+\\frac{ij}{100}\\right)\\%\\quad \\text{(E) } \\left(i+j+\\frac{i+j}{100}\\right)\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1692", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. From time $t=0$ to time $t=1$ a population increased by $i\\%$, and from time $t=1$ to time $t=2$ the population increased by $j\\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by\n$\\text{(A) (i+j)\\%} \\quad \\text{(B) } ij\\%\\quad \\text{(C) } (i+ij)\\%\\quad \\text{(D) } \\left(i+j+\\frac{ij}{100}\\right)\\%\\quad \\text{(E) } \\left(i+j+\\frac{i+j}{100}\\right)\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\log_{b^2}x+\\log_{x^2}b=1, b>0, b \\neq 1, x \\neq 1$, then $x$ equals:\n$\\textbf{(A)}\\ 1/b^2 \\qquad \\textbf{(B)}\\ 1/b \\qquad \\textbf{(C)}\\ b^2 \\qquad \\textbf{(D)}\\ b \\qquad \\textbf{(E)}\\ \\sqrt{b}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1693", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\log_{b^2}x+\\log_{x^2}b=1, b>0, b \\neq 1, x \\neq 1$, then $x$ equals:\n$\\textbf{(A)}\\ 1/b^2 \\qquad \\textbf{(B)}\\ 1/b \\qquad \\textbf{(C)}\\ b^2 \\qquad \\textbf{(D)}\\ b \\qquad \\textbf{(E)}\\ \\sqrt{b}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $as^2\\qquad\\\\ \\textbf{(B) }\\text{contains only three points if }a=2s^2\\text{ and is a circle if }a>2s^2\\qquad\\\\ \\textbf{(C) }\\text{is a circle with positive radius only if }s^2s^2\\qquad\\\\ \\textbf{(B) }\\text{contains only three points if }a=2s^2\\text{ and is a circle if }a>2s^2\\qquad\\\\ \\textbf{(C) }\\text{is a circle with positive radius only if }s^20$) and one secant line. The maximum number of non-overlapping \nareas into which the disk can be divided is\n$\\textbf{(A) }2n+1\\qquad \\textbf{(B) }2n+2\\qquad \\textbf{(C) }3n-1\\qquad \\textbf{(D) }3n\\qquad \\textbf{(E) }3n+1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1774", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A circular disk is divided by $2n$ equally spaced radii($n>0$) and one secant line. The maximum number of non-overlapping \nareas into which the disk can be divided is\n$\\textbf{(A) }2n+1\\qquad \\textbf{(B) }2n+2\\qquad \\textbf{(C) }3n-1\\qquad \\textbf{(D) }3n\\qquad \\textbf{(E) }3n+1$" + } + }, + { + "question": "Return your final response within \\boxed{}. The points $(6,12)$ and $(0,-6)$ are connected by a straight line. Another point on this line is:\n$\\textbf{(A) \\ }(3,3) \\qquad \\textbf{(B) \\ }(2,1) \\qquad \\textbf{(C) \\ }(7,16) \\qquad \\textbf{(D) \\ }(-1,-4) \\qquad \\textbf{(E) \\ }(-3,-8)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1775", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The points $(6,12)$ and $(0,-6)$ are connected by a straight line. Another point on this line is:\n$\\textbf{(A) \\ }(3,3) \\qquad \\textbf{(B) \\ }(2,1) \\qquad \\textbf{(C) \\ }(7,16) \\qquad \\textbf{(D) \\ }(-1,-4) \\qquad \\textbf{(E) \\ }(-3,-8)$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x =\\sqrt{1+\\sqrt{1+\\sqrt{1+\\sqrt{1+\\cdots}}}}$, then:\n$\\textbf{(A)}\\ x = 1\\qquad\\textbf{(B)}\\ 0 < x < 1\\qquad\\textbf{(C)}\\ 1 < x < 2\\qquad\\textbf{(D)}\\ x\\text{ is infinite}$\n$\\textbf{(E)}\\ x > 2\\text{ but finite}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1776", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x =\\sqrt{1+\\sqrt{1+\\sqrt{1+\\sqrt{1+\\cdots}}}}$, then:\n$\\textbf{(A)}\\ x = 1\\qquad\\textbf{(B)}\\ 0 < x < 1\\qquad\\textbf{(C)}\\ 1 < x < 2\\qquad\\textbf{(D)}\\ x\\text{ is infinite}$\n$\\textbf{(E)}\\ x > 2\\text{ but finite}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?\n$\\textbf{(A)}\\ 16\\qquad\\textbf{(B)}\\ 17\\qquad\\textbf{(C)}\\ 25\\qquad\\textbf{(D)}\\ 26\\qquad\\textbf{(E)}\\ 27$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1777", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?\n$\\textbf{(A)}\\ 16\\qquad\\textbf{(B)}\\ 17\\qquad\\textbf{(C)}\\ 25\\qquad\\textbf{(D)}\\ 26\\qquad\\textbf{(E)}\\ 27$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three vertices of parallelogram $PQRS$ are $P(-3,-2), Q(1,-5), R(9,1)$ with $P$ and $R$ diagonally opposite. \nThe sum of the coordinates of vertex $S$ is:\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 11 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1778", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three vertices of parallelogram $PQRS$ are $P(-3,-2), Q(1,-5), R(9,1)$ with $P$ and $R$ diagonally opposite. \nThe sum of the coordinates of vertex $S$ is:\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 11 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. The reciprocal of $\\left( \\frac{1}{2}+\\frac{1}{3}\\right)$ is \n$\\text{(A)}\\ \\frac{1}{6} \\qquad \\text{(B)}\\ \\frac{2}{5} \\qquad \\text{(C)}\\ \\frac{6}{5} \\qquad \\text{(D)}\\ \\frac{5}{2} \\qquad \\text{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1779", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The reciprocal of $\\left( \\frac{1}{2}+\\frac{1}{3}\\right)$ is \n$\\text{(A)}\\ \\frac{1}{6} \\qquad \\text{(B)}\\ \\frac{2}{5} \\qquad \\text{(C)}\\ \\frac{6}{5} \\qquad \\text{(D)}\\ \\frac{5}{2} \\qquad \\text{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place?\n$\\textbf{(A) } \\text{David} \\qquad\\textbf{(B) } \\text{Hikmet} \\qquad\\textbf{(C) } \\text{Jack} \\qquad\\textbf{(D) } \\text{Rand} \\qquad\\textbf{(E) } \\text{Todd}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1780", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place?\n$\\textbf{(A) } \\text{David} \\qquad\\textbf{(B) } \\text{Hikmet} \\qquad\\textbf{(C) } \\text{Jack} \\qquad\\textbf{(D) } \\text{Rand} \\qquad\\textbf{(E) } \\text{Todd}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "173", + "index": "Sky-T1_10k_1781", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $A,B,C,D$ denote four points in space and $AB$ the distance between $A$ and $B$, and so on. Show that \n\\[AC^2+BD^2+AD^2+BC^2\\ge AB^2+CD^2.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "AC^2 + BD^2 + AD^2 + BC^2 \\ge AB^2 + CD^2", + "index": "Sky-T1_10k_1782", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $A,B,C,D$ denote four points in space and $AB$ the distance between $A$ and $B$, and so on. Show that \n\\[AC^2+BD^2+AD^2+BC^2\\ge AB^2+CD^2.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "816", + "index": "Sky-T1_10k_1783", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?" + } + }, + { + "question": "Return your final response within \\boxed{}. John ordered $4$ pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by $50\\%$. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was:\n$\\textbf{(A)}\\ 4:1 \\qquad \\textbf{(B)}\\ 2:1 \\qquad \\textbf{(C)}\\ 1:4 \\qquad \\textbf{(D)}\\ 1:2 \\qquad \\textbf{(E)}\\ 1:8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1784", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. John ordered $4$ pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by $50\\%$. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was:\n$\\textbf{(A)}\\ 4:1 \\qquad \\textbf{(B)}\\ 2:1 \\qquad \\textbf{(C)}\\ 1:4 \\qquad \\textbf{(D)}\\ 1:2 \\qquad \\textbf{(E)}\\ 1:8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ryan got $80\\%$ of the problems correct on a $25$-problem test, $90\\%$ on a $40$-problem test, and $70\\%$ on a $10$-problem test. What percent of all the problems did Ryan answer correctly? \n$\\textbf{(A)}\\ 64 \\qquad\\textbf{(B)}\\ 75\\qquad\\textbf{(C)}\\ 80\\qquad\\textbf{(D)}\\ 84\\qquad\\textbf{(E)}\\ 86$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1785", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ryan got $80\\%$ of the problems correct on a $25$-problem test, $90\\%$ on a $40$-problem test, and $70\\%$ on a $10$-problem test. What percent of all the problems did Ryan answer correctly? \n$\\textbf{(A)}\\ 64 \\qquad\\textbf{(B)}\\ 75\\qquad\\textbf{(C)}\\ 80\\qquad\\textbf{(D)}\\ 84\\qquad\\textbf{(E)}\\ 86$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $1+3+5+\\cdots+2017+2019-2-4-6-\\cdots-2016-2018$?\n$\\textbf{(A) }-1010\\qquad\\textbf{(B) }-1009\\qquad\\textbf{(C) }1008\\qquad\\textbf{(D) }1009\\qquad \\textbf{(E) }1010$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1786", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $1+3+5+\\cdots+2017+2019-2-4-6-\\cdots-2016-2018$?\n$\\textbf{(A) }-1010\\qquad\\textbf{(B) }-1009\\qquad\\textbf{(C) }1008\\qquad\\textbf{(D) }1009\\qquad \\textbf{(E) }1010$" + } + }, + { + "question": "Return your final response within \\boxed{}. What value of $x$ satisfies\n\\[x- \\frac{3}{4} = \\frac{5}{12} - \\frac{1}{3}?\\]\n$\\textbf{(A)}\\ {-}\\frac{2}{3}\\qquad\\textbf{(B)}\\ \\frac{7}{36}\\qquad\\textbf{(C)}\\ \\frac{7}{12}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{5}{6}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1787", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What value of $x$ satisfies\n\\[x- \\frac{3}{4} = \\frac{5}{12} - \\frac{1}{3}?\\]\n$\\textbf{(A)}\\ {-}\\frac{2}{3}\\qquad\\textbf{(B)}\\ \\frac{7}{36}\\qquad\\textbf{(C)}\\ \\frac{7}{12}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{5}{6}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the sum of the first $10$ terms and the sum of the first $100$ terms of a given arithmetic progression are $100$ and $10$, \nrespectively, then the sum of first $110$ terms is:\n$\\text{(A)} \\ 90 \\qquad \\text{(B)} \\ -90 \\qquad \\text{(C)} \\ 110 \\qquad \\text{(D)} \\ -110 \\qquad \\text{(E)} \\ -100$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1788", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the sum of the first $10$ terms and the sum of the first $100$ terms of a given arithmetic progression are $100$ and $10$, \nrespectively, then the sum of first $110$ terms is:\n$\\text{(A)} \\ 90 \\qquad \\text{(B)} \\ -90 \\qquad \\text{(C)} \\ 110 \\qquad \\text{(D)} \\ -110 \\qquad \\text{(E)} \\ -100$" + } + }, + { + "question": "Return your final response within \\boxed{}. Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\\frac{PQ}{RS}$?\n$\\textbf{(A) } 3 \\qquad \\textbf{(B) } 5 \\qquad \\textbf{(C) } 3\\sqrt{5} \\qquad \\textbf{(D) } 7 \\qquad \\textbf{(E) } 5\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1789", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\\frac{PQ}{RS}$?\n$\\textbf{(A) } 3 \\qquad \\textbf{(B) } 5 \\qquad \\textbf{(C) } 3\\sqrt{5} \\qquad \\textbf{(D) } 7 \\qquad \\textbf{(E) } 5\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\\lfloor\\log_2x\\rfloor=\\lfloor\\log_2y\\rfloor$?\n$\\textbf{(A)}\\ \\frac{1}{8}\\qquad\\textbf{(B)}\\ \\frac{1}{6}\\qquad\\textbf{(C)}\\ \\frac{1}{4}\\qquad\\textbf{(D)}\\ \\frac{1}{3}\\qquad\\textbf{(E)}\\ \\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1790", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\\lfloor\\log_2x\\rfloor=\\lfloor\\log_2y\\rfloor$?\n$\\textbf{(A)}\\ \\frac{1}{8}\\qquad\\textbf{(B)}\\ \\frac{1}{6}\\qquad\\textbf{(C)}\\ \\frac{1}{4}\\qquad\\textbf{(D)}\\ \\frac{1}{3}\\qquad\\textbf{(E)}\\ \\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?\n$\\textbf{(A)} \\: 105 \\qquad\\textbf{(B)} \\: 120 \\qquad\\textbf{(C)} \\: 135 \\qquad\\textbf{(D)} \\: 150 \\qquad\\textbf{(E)} \\: 165$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1791", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?\n$\\textbf{(A)} \\: 105 \\qquad\\textbf{(B)} \\: 120 \\qquad\\textbf{(C)} \\: 135 \\qquad\\textbf{(D)} \\: 150 \\qquad\\textbf{(E)} \\: 165$" + } + }, + { + "question": "Return your final response within \\boxed{}. The bottom, side, and front areas of a rectangular box are known. The product of these areas\nis equal to:\n$\\textbf{(A)}\\ \\text{the volume of the box} \\qquad\\textbf{(B)}\\ \\text{the square root of the volume} \\qquad\\textbf{(C)}\\ \\text{twice the volume}$\n$\\textbf{(D)}\\ \\text{the square of the volume} \\qquad\\textbf{(E)}\\ \\text{the cube of the volume}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1792", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The bottom, side, and front areas of a rectangular box are known. The product of these areas\nis equal to:\n$\\textbf{(A)}\\ \\text{the volume of the box} \\qquad\\textbf{(B)}\\ \\text{the square root of the volume} \\qquad\\textbf{(C)}\\ \\text{twice the volume}$\n$\\textbf{(D)}\\ \\text{the square of the volume} \\qquad\\textbf{(E)}\\ \\text{the cube of the volume}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?\n$\\text{(A)}\\ 60\\qquad\\text{(B)}\\ 170\\qquad\\text{(C)}\\ 290\\qquad\\text{(D)}\\ 320\\qquad\\text{(E)}\\ 660$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1793", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?\n$\\text{(A)}\\ 60\\qquad\\text{(B)}\\ 170\\qquad\\text{(C)}\\ 290\\qquad\\text{(D)}\\ 320\\qquad\\text{(E)}\\ 660$" + } + }, + { + "question": "Return your final response within \\boxed{}. The volume of a rectangular solid each of whose side, front, and bottom faces are $12\\text{ in}^{2}$, $8\\text{ in}^{2}$, and $6\\text{ in}^{2}$ respectively is:\n$\\textbf{(A)}\\ 576\\text{ in}^{3}\\qquad\\textbf{(B)}\\ 24\\text{ in}^{3}\\qquad\\textbf{(C)}\\ 9\\text{ in}^{3}\\qquad\\textbf{(D)}\\ 104\\text{ in}^{3}\\qquad\\textbf{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1794", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The volume of a rectangular solid each of whose side, front, and bottom faces are $12\\text{ in}^{2}$, $8\\text{ in}^{2}$, and $6\\text{ in}^{2}$ respectively is:\n$\\textbf{(A)}\\ 576\\text{ in}^{3}\\qquad\\textbf{(B)}\\ 24\\text{ in}^{3}\\qquad\\textbf{(C)}\\ 9\\text{ in}^{3}\\qquad\\textbf{(D)}\\ 104\\text{ in}^{3}\\qquad\\textbf{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the sum of the digits of the square of $\\text 111111111$?\n$\\mathrm{(A)}\\ 18\\qquad\\mathrm{(B)}\\ 27\\qquad\\mathrm{(C)}\\ 45\\qquad\\mathrm{(D)}\\ 63\\qquad\\mathrm{(E)}\\ 81$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1795", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the sum of the digits of the square of $\\text 111111111$?\n$\\mathrm{(A)}\\ 18\\qquad\\mathrm{(B)}\\ 27\\qquad\\mathrm{(C)}\\ 45\\qquad\\mathrm{(D)}\\ 63\\qquad\\mathrm{(E)}\\ 81$" + } + }, + { + "question": "Return your final response within \\boxed{}. A $4\\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?\n$\\begin{tabular}[t]{|c|c|c|c|} \\multicolumn{4}{c}{}\\\\\\hline 1&2&3&4\\\\\\hline 8&9&10&11\\\\\\hline 15&16&17&18\\\\\\hline 22&23&24&25\\\\\\hline \\end{tabular}$\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 8 \\qquad \\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1796", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A $4\\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?\n$\\begin{tabular}[t]{|c|c|c|c|} \\multicolumn{4}{c}{}\\\\\\hline 1&2&3&4\\\\\\hline 8&9&10&11\\\\\\hline 15&16&17&18\\\\\\hline 22&23&24&25\\\\\\hline \\end{tabular}$\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 8 \\qquad \\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. The base of isosceles $\\triangle ABC$ is $24$ and its area is $60$. What is the length of one\nof the congruent sides?\n$\\mathrm{(A)}\\ 5 \\qquad \\mathrm{(B)}\\ 8 \\qquad \\mathrm{(C)}\\ 13 \\qquad \\mathrm{(D)}\\ 14 \\qquad \\mathrm{(E)}\\ 18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1797", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The base of isosceles $\\triangle ABC$ is $24$ and its area is $60$. What is the length of one\nof the congruent sides?\n$\\mathrm{(A)}\\ 5 \\qquad \\mathrm{(B)}\\ 8 \\qquad \\mathrm{(C)}\\ 13 \\qquad \\mathrm{(D)}\\ 14 \\qquad \\mathrm{(E)}\\ 18$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $xy = a, xz =b,$ and $yz = c$, and none of these quantities is $0$, then $x^2+y^2+z^2$ equals\n$\\textbf{(A)}\\ \\frac{ab+ac+bc}{abc}\\qquad \\textbf{(B)}\\ \\frac{a^2+b^2+c^2}{abc}\\qquad \\textbf{(C)}\\ \\frac{(a+b+c)^2}{abc}\\qquad \\textbf{(D)}\\ \\frac{(ab+ac+bc)^2}{abc}\\qquad \\textbf{(E)}\\ \\frac{(ab)^2+(ac)^2+(bc)^2}{abc}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1798", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $xy = a, xz =b,$ and $yz = c$, and none of these quantities is $0$, then $x^2+y^2+z^2$ equals\n$\\textbf{(A)}\\ \\frac{ab+ac+bc}{abc}\\qquad \\textbf{(B)}\\ \\frac{a^2+b^2+c^2}{abc}\\qquad \\textbf{(C)}\\ \\frac{(a+b+c)^2}{abc}\\qquad \\textbf{(D)}\\ \\frac{(ab+ac+bc)^2}{abc}\\qquad \\textbf{(E)}\\ \\frac{(ab)^2+(ac)^2+(bc)^2}{abc}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{123456789}{2^{26}\\cdot 5^4}$ as a decimal?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 22\\qquad\\textbf{(C)}\\ 26\\qquad\\textbf{(D)}\\ 30\\qquad\\textbf{(E)}\\ 104$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1799", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{123456789}{2^{26}\\cdot 5^4}$ as a decimal?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 22\\qquad\\textbf{(C)}\\ 26\\qquad\\textbf{(D)}\\ 30\\qquad\\textbf{(E)}\\ 104$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?\n$\\textbf{(A)}\\ -\\textdollar 1.06\\qquad\\textbf{(B)}\\ -\\textdollar 0.53 \\qquad\\textbf{(C)}\\ \\textdollar 0\\qquad\\textbf{(D)}\\ \\textdollar 0.53\\qquad\\textbf{(E)}\\ \\textdollar 1.06$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1800", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?\n$\\textbf{(A)}\\ -\\textdollar 1.06\\qquad\\textbf{(B)}\\ -\\textdollar 0.53 \\qquad\\textbf{(C)}\\ \\textdollar 0\\qquad\\textbf{(D)}\\ \\textdollar 0.53\\qquad\\textbf{(E)}\\ \\textdollar 1.06$" + } + }, + { + "question": "Return your final response within \\boxed{}. Five runners, $P$, $Q$, $R$, $S$, $T$, have a race, and $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ and before $Q$. Who could NOT have finished third in the race?\n$\\text{(A)}\\ P\\text{ and }Q \\qquad \\text{(B)}\\ P\\text{ and }R \\qquad \\text{(C)}\\ P\\text{ and }S \\qquad \\text{(D)}\\ P\\text{ and }T \\qquad \\text{(E)}\\ P,S\\text{ and }T$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1801", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five runners, $P$, $Q$, $R$, $S$, $T$, have a race, and $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ and before $Q$. Who could NOT have finished third in the race?\n$\\text{(A)}\\ P\\text{ and }Q \\qquad \\text{(B)}\\ P\\text{ and }R \\qquad \\text{(C)}\\ P\\text{ and }S \\qquad \\text{(D)}\\ P\\text{ and }T \\qquad \\text{(E)}\\ P,S\\text{ and }T$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\theta$ is an acute angle, and $\\sin 2\\theta=a$, then $\\sin\\theta+\\cos\\theta$ equals\n$\\textbf{(A) }\\sqrt{a+1}\\qquad \\textbf{(B) }(\\sqrt{2}-1)a+1\\qquad \\textbf{(C) }\\sqrt{a+1}-\\sqrt{a^2-a}\\qquad\\\\ \\textbf{(D) }\\sqrt{a+1}+\\sqrt{a^2-a}\\qquad \\textbf{(E) }\\sqrt{a+1}+a^2-a$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1802", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\theta$ is an acute angle, and $\\sin 2\\theta=a$, then $\\sin\\theta+\\cos\\theta$ equals\n$\\textbf{(A) }\\sqrt{a+1}\\qquad \\textbf{(B) }(\\sqrt{2}-1)a+1\\qquad \\textbf{(C) }\\sqrt{a+1}-\\sqrt{a^2-a}\\qquad\\\\ \\textbf{(D) }\\sqrt{a+1}+\\sqrt{a^2-a}\\qquad \\textbf{(E) }\\sqrt{a+1}+a^2-a$" + } + }, + { + "question": "Return your final response within \\boxed{}. A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square?\n$\\mathrm{(A)}\\ 12.5\\qquad\\mathrm{(B)}\\ 25\\qquad\\mathrm{(C)}\\ 50\\qquad\\mathrm{(D)}\\ 75\\qquad\\mathrm{(E)}\\ 87.5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1803", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square?\n$\\mathrm{(A)}\\ 12.5\\qquad\\mathrm{(B)}\\ 25\\qquad\\mathrm{(C)}\\ 50\\qquad\\mathrm{(D)}\\ 75\\qquad\\mathrm{(E)}\\ 87.5$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a$, $b$, $c$, and $d$ are the solutions of the equation $x^4 - bx - 3 = 0$, then an equation whose solutions are \n\\[\\dfrac {a + b + c}{d^2}, \\dfrac {a + b + d}{c^2}, \\dfrac {a + c + d}{b^2}, \\dfrac {b + c + d}{a^2}\\]is\n$\\textbf{(A)}\\ 3x^4 + bx + 1 = 0\\qquad \\textbf{(B)}\\ 3x^4 - bx + 1 = 0\\qquad \\textbf{(C)}\\ 3x^4 + bx^3 - 1 = 0\\qquad \\\\\\textbf{(D)}\\ 3x^4 - bx^3 - 1 = 0\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1804", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a$, $b$, $c$, and $d$ are the solutions of the equation $x^4 - bx - 3 = 0$, then an equation whose solutions are \n\\[\\dfrac {a + b + c}{d^2}, \\dfrac {a + b + d}{c^2}, \\dfrac {a + c + d}{b^2}, \\dfrac {b + c + d}{a^2}\\]is\n$\\textbf{(A)}\\ 3x^4 + bx + 1 = 0\\qquad \\textbf{(B)}\\ 3x^4 - bx + 1 = 0\\qquad \\textbf{(C)}\\ 3x^4 + bx^3 - 1 = 0\\qquad \\\\\\textbf{(D)}\\ 3x^4 - bx^3 - 1 = 0\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$?\n$\\textbf{(A) }24\\qquad\\textbf{(B) }29\\qquad\\textbf{(C) }43\\qquad\\textbf{(D) }48\\qquad \\textbf{(E) }57$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1805", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$?\n$\\textbf{(A) }24\\qquad\\textbf{(B) }29\\qquad\\textbf{(C) }43\\qquad\\textbf{(D) }48\\qquad \\textbf{(E) }57$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $x,$ $y,$ and $z$ be positive real numbers that satisfy\n\\[2\\log_{x}(2y) = 2\\log_{2x}(4z) = \\log_{2x^4}(8yz) \\ne 0.\\]\nThe value of $xy^5z$ can be expressed in the form $\\frac{1}{2^{p/q}},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "49", + "index": "Sky-T1_10k_1806", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $x,$ $y,$ and $z$ be positive real numbers that satisfy\n\\[2\\log_{x}(2y) = 2\\log_{2x}(4z) = \\log_{2x^4}(8yz) \\ne 0.\\]\nThe value of $xy^5z$ can be expressed in the form $\\frac{1}{2^{p/q}},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$" + } + }, + { + "question": "Return your final response within \\boxed{}. The difference of the roots of $x^2-7x-9=0$ is:\n$\\textbf{(A) \\ }+7 \\qquad \\textbf{(B) \\ }+\\frac{7}{2} \\qquad \\textbf{(C) \\ }+9 \\qquad \\textbf{(D) \\ }2\\sqrt{85} \\qquad \\textbf{(E) \\ }\\sqrt{85}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1807", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The difference of the roots of $x^2-7x-9=0$ is:\n$\\textbf{(A) \\ }+7 \\qquad \\textbf{(B) \\ }+\\frac{7}{2} \\qquad \\textbf{(C) \\ }+9 \\qquad \\textbf{(D) \\ }2\\sqrt{85} \\qquad \\textbf{(E) \\ }\\sqrt{85}$" + } + }, + { + "question": "Return your final response within \\boxed{}. While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?\n$\\mathrm{(A)}\\ 2\\qquad\\mathrm{(B)}\\ 4\\qquad\\mathrm{(C)}\\ 6\\qquad\\mathrm{(D)}\\ 8\\qquad\\mathrm{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1808", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?\n$\\mathrm{(A)}\\ 2\\qquad\\mathrm{(B)}\\ 4\\qquad\\mathrm{(C)}\\ 6\\qquad\\mathrm{(D)}\\ 8\\qquad\\mathrm{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?\n$\\textbf{(A) }6\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }11\\qquad\\textbf{(E) }15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1809", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?\n$\\textbf{(A) }6\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }11\\qquad\\textbf{(E) }15$" + } + }, + { + "question": "Return your final response within \\boxed{}. A barn with a roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:\n$\\mathrm{(A) \\ } 360 \\qquad \\mathrm{(B) \\ } 460 \\qquad \\mathrm{(C) \\ } 490 \\qquad \\mathrm{(D) \\ } 590 \\qquad \\mathrm{(E) \\ } 720$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1810", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A barn with a roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:\n$\\mathrm{(A) \\ } 360 \\qquad \\mathrm{(B) \\ } 460 \\qquad \\mathrm{(C) \\ } 490 \\qquad \\mathrm{(D) \\ } 590 \\qquad \\mathrm{(E) \\ } 720$" + } + }, + { + "question": "Return your final response within \\boxed{}. All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have?\n$\\textbf{(A) }1\\qquad\\textbf{(B) }2\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }4\\qquad\\textbf{(E) }5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1811", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have?\n$\\textbf{(A) }1\\qquad\\textbf{(B) }2\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }4\\qquad\\textbf{(E) }5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\\overline{CD}$. For $i=1,2,\\dots,$ let $P_i$ be the intersection of $\\overline{AQ_i}$ and $\\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\\overline{CD}$. What is \n\\[\\sum_{i=1}^{\\infty} \\text{Area of } \\triangle DQ_i P_i \\, ?\\]\n$\\textbf{(A)}\\ \\frac{1}{6} \\qquad \\textbf{(B)}\\ \\frac{1}{4} \\qquad \\textbf{(C)}\\ \\frac{1}{3} \\qquad \\textbf{(D)}\\ \\frac{1}{2} \\qquad \\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1812", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\\overline{CD}$. For $i=1,2,\\dots,$ let $P_i$ be the intersection of $\\overline{AQ_i}$ and $\\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\\overline{CD}$. What is \n\\[\\sum_{i=1}^{\\infty} \\text{Area of } \\triangle DQ_i P_i \\, ?\\]\n$\\textbf{(A)}\\ \\frac{1}{6} \\qquad \\textbf{(B)}\\ \\frac{1}{4} \\qquad \\textbf{(C)}\\ \\frac{1}{3} \\qquad \\textbf{(D)}\\ \\frac{1}{2} \\qquad \\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two right circular cylinders have the same volume. The radius of the second cylinder is $10\\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?\n$\\textbf{(A)}\\ \\text{The second height is } 10\\% \\text{ less than the first.} \\\\ \\textbf{(B)}\\ \\text{The first height is } 10\\% \\text{ more than the second.}\\\\ \\textbf{(C)}\\ \\text{The second height is } 21\\% \\text{ less than the first.} \\\\ \\textbf{(D)}\\ \\text{The first height is } 21\\% \\text{ more than the second.}\\\\ \\textbf{(E)}\\ \\text{The second height is } 80\\% \\text{ of the first.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1813", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two right circular cylinders have the same volume. The radius of the second cylinder is $10\\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?\n$\\textbf{(A)}\\ \\text{The second height is } 10\\% \\text{ less than the first.} \\\\ \\textbf{(B)}\\ \\text{The first height is } 10\\% \\text{ more than the second.}\\\\ \\textbf{(C)}\\ \\text{The second height is } 21\\% \\text{ less than the first.} \\\\ \\textbf{(D)}\\ \\text{The first height is } 21\\% \\text{ more than the second.}\\\\ \\textbf{(E)}\\ \\text{The second height is } 80\\% \\text{ of the first.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of functions $f(x)$ from $\\{1, 2, 3, 4, 5\\}$ to $\\{1, 2, 3, 4, 5\\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\\{1, 2, 3, 4, 5\\}$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "756", + "index": "Sky-T1_10k_1815", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of functions $f(x)$ from $\\{1, 2, 3, 4, 5\\}$ to $\\{1, 2, 3, 4, 5\\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\\{1, 2, 3, 4, 5\\}$." + } + }, + { + "question": "Return your final response within \\boxed{}. Five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$. The sum of the two lowest test scores is\n$\\text{(A)}\\ 170 \\qquad \\text{(B)}\\ 171 \\qquad \\text{(C)}\\ 176 \\qquad \\text{(D)}\\ 177 \\qquad \\text{(E)}\\ \\text{not determined by the information given}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1816", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$. The sum of the two lowest test scores is\n$\\text{(A)}\\ 170 \\qquad \\text{(B)}\\ 171 \\qquad \\text{(C)}\\ 176 \\qquad \\text{(D)}\\ 177 \\qquad \\text{(E)}\\ \\text{not determined by the information given}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Some checkers placed on an $n\\times n$ checkerboard satisfy the following conditions:\n(a) every square that does not contain a checker shares a side with one that does;\n(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.\nProve that at least $(n^{2}-2)/3$ checkers have been placed on the board.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\frac{n^2 - 2}{3}", + "index": "Sky-T1_10k_1817", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Some checkers placed on an $n\\times n$ checkerboard satisfy the following conditions:\n(a) every square that does not contain a checker shares a side with one that does;\n(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.\nProve that at least $(n^{2}-2)/3$ checkers have been placed on the board." + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?\n$\\textbf{(A)}\\ P^2Q \\qquad \\textbf{(B)}\\ P^nQ^m \\qquad \\textbf{(C)}\\ P^nQ^{2m} \\qquad \\textbf{(D)}\\ P^{2m}Q^n \\qquad \\textbf{(E)}\\ P^{2n}Q^m$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1818", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?\n$\\textbf{(A)}\\ P^2Q \\qquad \\textbf{(B)}\\ P^nQ^m \\qquad \\textbf{(C)}\\ P^nQ^{2m} \\qquad \\textbf{(D)}\\ P^{2m}Q^n \\qquad \\textbf{(E)}\\ P^{2n}Q^m$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many values of $\\theta$ in the interval $0<\\theta\\le 2\\pi$ satisfy \\[1-3\\sin\\theta+5\\cos3\\theta = 0?\\]\n$\\textbf{(A) }2 \\qquad \\textbf{(B) }4 \\qquad \\textbf{(C) }5\\qquad \\textbf{(D) }6 \\qquad \\textbf{(E) }8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1819", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many values of $\\theta$ in the interval $0<\\theta\\le 2\\pi$ satisfy \\[1-3\\sin\\theta+5\\cos3\\theta = 0?\\]\n$\\textbf{(A) }2 \\qquad \\textbf{(B) }4 \\qquad \\textbf{(C) }5\\qquad \\textbf{(D) }6 \\qquad \\textbf{(E) }8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Call a real-valued [function](https://artofproblemsolving.com/wiki/index.php/Function) $f$ very convex if\n\\[\\frac {f(x) + f(y)}{2} \\ge f\\left(\\frac {x + y}{2}\\right) + |x - y|\\]\nholds for all real numbers $x$ and $y$. Prove that no very convex function exists.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "No very convex function exists.", + "index": "Sky-T1_10k_1820", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Call a real-valued [function](https://artofproblemsolving.com/wiki/index.php/Function) $f$ very convex if\n\\[\\frac {f(x) + f(y)}{2} \\ge f\\left(\\frac {x + y}{2}\\right) + |x - y|\\]\nholds for all real numbers $x$ and $y$. Prove that no very convex function exists." + } + }, + { + "question": "Return your final response within \\boxed{}. A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \\frac{3}{7}$, and\n\\[a_n=\\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\\]for all $n \\geq 3$ Then $a_{2019}$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$\n$\\textbf{(A) } 2020 \\qquad\\textbf{(B) } 4039 \\qquad\\textbf{(C) } 6057 \\qquad\\textbf{(D) } 6061 \\qquad\\textbf{(E) } 8078$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1821", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \\frac{3}{7}$, and\n\\[a_n=\\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\\]for all $n \\geq 3$ Then $a_{2019}$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$\n$\\textbf{(A) } 2020 \\qquad\\textbf{(B) } 4039 \\qquad\\textbf{(C) } 6057 \\qquad\\textbf{(D) } 6061 \\qquad\\textbf{(E) } 8078$" + } + }, + { + "question": "Return your final response within \\boxed{}. Circle $I$ passes through the center of, and is tangent to, circle $II$. The area of circle $I$ is $4$ square inches. \nThen the area of circle $II$, in square inches, is:\n$\\textbf{(A) }8\\qquad \\textbf{(B) }8\\sqrt{2}\\qquad \\textbf{(C) }8\\sqrt{\\pi}\\qquad \\textbf{(D) }16\\qquad \\textbf{(E) }16\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1822", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Circle $I$ passes through the center of, and is tangent to, circle $II$. The area of circle $I$ is $4$ square inches. \nThen the area of circle $II$, in square inches, is:\n$\\textbf{(A) }8\\qquad \\textbf{(B) }8\\sqrt{2}\\qquad \\textbf{(C) }8\\sqrt{\\pi}\\qquad \\textbf{(D) }16\\qquad \\textbf{(E) }16\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. \"If a whole number $n$ is not prime, then the whole number $n-2$ is not prime.\" A value of $n$ which shows this statement to be false is\n$\\textbf{(A)}\\ 9 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 13 \\qquad \\textbf{(D)}\\ 16 \\qquad \\textbf{(E)}\\ 23$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1823", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. \"If a whole number $n$ is not prime, then the whole number $n-2$ is not prime.\" A value of $n$ which shows this statement to be false is\n$\\textbf{(A)}\\ 9 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 13 \\qquad \\textbf{(D)}\\ 16 \\qquad \\textbf{(E)}\\ 23$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is:\n$\\textbf{(A)}\\ 21\\frac{1}{3}\\pi-32\\sqrt{3}\\qquad \\textbf{(B)}\\ 32\\sqrt{3}+21\\frac{1}{3}\\pi\\qquad \\textbf{(C)}\\ 32\\sqrt{3}+42\\frac{2}{3}\\pi \\qquad\\\\ \\textbf{(D)}\\ 16\\sqrt {3} + 42\\frac {2}{3}\\pi \\qquad \\textbf{(E)}\\ 42\\frac {2}{3}\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1824", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is:\n$\\textbf{(A)}\\ 21\\frac{1}{3}\\pi-32\\sqrt{3}\\qquad \\textbf{(B)}\\ 32\\sqrt{3}+21\\frac{1}{3}\\pi\\qquad \\textbf{(C)}\\ 32\\sqrt{3}+42\\frac{2}{3}\\pi \\qquad\\\\ \\textbf{(D)}\\ 16\\sqrt {3} + 42\\frac {2}{3}\\pi \\qquad \\textbf{(E)}\\ 42\\frac {2}{3}\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50) = 12$ because the prime factorization of $50$ is $2 \\times 5^{2}$, and $2 + 5 + 5 = 12$. What is the range of the function $r$, $\\{r(n): n \\text{ is a composite positive integer}\\}$ ?\n$\\textbf{(A)}\\; \\text{the set of positive integers} \\\\ \\textbf{(B)}\\; \\text{the set of composite positive integers} \\\\ \\textbf{(C)}\\; \\text{the set of even positive integers} \\\\ \\textbf{(D)}\\; \\text{the set of integers greater than 3} \\\\ \\textbf{(E)}\\; \\text{the set of integers greater than 4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1825", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50) = 12$ because the prime factorization of $50$ is $2 \\times 5^{2}$, and $2 + 5 + 5 = 12$. What is the range of the function $r$, $\\{r(n): n \\text{ is a composite positive integer}\\}$ ?\n$\\textbf{(A)}\\; \\text{the set of positive integers} \\\\ \\textbf{(B)}\\; \\text{the set of composite positive integers} \\\\ \\textbf{(C)}\\; \\text{the set of even positive integers} \\\\ \\textbf{(D)}\\; \\text{the set of integers greater than 3} \\\\ \\textbf{(E)}\\; \\text{the set of integers greater than 4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1826", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ lies in the first quadrant. Points $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is not always true?\n$\\textbf{(A) }$ Triangle $A'B'C'$ lies in the first quadrant.\n$\\textbf{(B) }$ Triangles $ABC$ and $A'B'C'$ have the same area.\n$\\textbf{(C) }$ The slope of line $AA'$ is $-1$.\n$\\textbf{(D) }$ The slopes of lines $AA'$ and $CC'$ are the same.\n$\\textbf{(E) }$ Lines $AB$ and $A'B'$ are perpendicular to each other.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1827", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ lies in the first quadrant. Points $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is not always true?\n$\\textbf{(A) }$ Triangle $A'B'C'$ lies in the first quadrant.\n$\\textbf{(B) }$ Triangles $ABC$ and $A'B'C'$ have the same area.\n$\\textbf{(C) }$ The slope of line $AA'$ is $-1$.\n$\\textbf{(D) }$ The slopes of lines $AA'$ and $CC'$ are the same.\n$\\textbf{(E) }$ Lines $AB$ and $A'B'$ are perpendicular to each other." + } + }, + { + "question": "Return your final response within \\boxed{}. In the country of East Westmore, statisticians estimate there is a baby born every $8$ hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?\n$\\textbf{(A)}\\hspace{.05in}600\\qquad\\textbf{(B)}\\hspace{.05in}700\\qquad\\textbf{(C)}\\hspace{.05in}800\\qquad\\textbf{(D)}\\hspace{.05in}900\\qquad\\textbf{(E)}\\hspace{.05in}1000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1828", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the country of East Westmore, statisticians estimate there is a baby born every $8$ hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?\n$\\textbf{(A)}\\hspace{.05in}600\\qquad\\textbf{(B)}\\hspace{.05in}700\\qquad\\textbf{(C)}\\hspace{.05in}800\\qquad\\textbf{(D)}\\hspace{.05in}900\\qquad\\textbf{(E)}\\hspace{.05in}1000$" + } + }, + { + "question": "Return your final response within \\boxed{}. An integer $N$ is selected at random in the range $1\\leq N \\leq 2020$ . What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?\n$\\textbf{(A)}\\ \\frac{1}{5}\\qquad\\textbf{(B)}\\ \\frac{2}{5}\\qquad\\textbf{(C)}\\ \\frac{3}{5}\\qquad\\textbf{(D)}\\ \\frac{4}{5}\\qquad\\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1829", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An integer $N$ is selected at random in the range $1\\leq N \\leq 2020$ . What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?\n$\\textbf{(A)}\\ \\frac{1}{5}\\qquad\\textbf{(B)}\\ \\frac{2}{5}\\qquad\\textbf{(C)}\\ \\frac{3}{5}\\qquad\\textbf{(D)}\\ \\frac{4}{5}\\qquad\\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: \n$\\textbf{(A)}\\ 108\\qquad\\textbf{(B)}\\ 90\\qquad\\textbf{(C)}\\ 72\\qquad\\textbf{(D)}\\ 54\\qquad\\textbf{(E)}\\ 36$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1830", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: \n$\\textbf{(A)}\\ 108\\qquad\\textbf{(B)}\\ 90\\qquad\\textbf{(C)}\\ 72\\qquad\\textbf{(D)}\\ 54\\qquad\\textbf{(E)}\\ 36$" + } + }, + { + "question": "Return your final response within \\boxed{}. Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 30\\qquad\\textbf{(C)}\\ 40\\qquad\\textbf{(D)}\\ 50\\qquad\\textbf{(E)}\\ 60$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1831", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 30\\qquad\\textbf{(C)}\\ 40\\qquad\\textbf{(D)}\\ 50\\qquad\\textbf{(E)}\\ 60$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a_{1}, a_{2}, \\dots, a_{n}$ ($n > 3$) be real numbers such that\n\\[a_{1} + a_{2} + \\cdots + a_{n} \\geq n \\qquad \\mbox{and} \\qquad a_{1}^{2} + a_{2}^{2} + \\cdots + a_{n}^{2} \\geq n^{2}.\\]\nProve that $\\max(a_{1}, a_{2}, \\dots, a_{n}) \\geq 2$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\max(a_{1}, a_{2}, \\dots, a_{n}) \\geq 2", + "index": "Sky-T1_10k_1832", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a_{1}, a_{2}, \\dots, a_{n}$ ($n > 3$) be real numbers such that\n\\[a_{1} + a_{2} + \\cdots + a_{n} \\geq n \\qquad \\mbox{and} \\qquad a_{1}^{2} + a_{2}^{2} + \\cdots + a_{n}^{2} \\geq n^{2}.\\]\nProve that $\\max(a_{1}, a_{2}, \\dots, a_{n}) \\geq 2$." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \\cdot 3^x \\cdot 5^y \\cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal?\n$\\textbf{(A) } 21\\qquad\\textbf{(B) }25\\qquad\\textbf{(C) }27\\qquad\\textbf{(D) }35\\qquad\\textbf{(E) }56$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1833", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \\cdot 3^x \\cdot 5^y \\cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal?\n$\\textbf{(A) } 21\\qquad\\textbf{(B) }25\\qquad\\textbf{(C) }27\\qquad\\textbf{(D) }35\\qquad\\textbf{(E) }56$" + } + }, + { + "question": "Return your final response within \\boxed{}. Equilateral $\\triangle ABC$ has side length $2$, $M$ is the midpoint of $\\overline{AC}$, and $C$ is the midpoint of $\\overline{BD}$. What is the area of $\\triangle CDM$?\n\n$\\textbf{(A) }\\ \\frac {\\sqrt {2}}{2}\\qquad \\textbf{(B) }\\ \\frac {3}{4}\\qquad \\textbf{(C) }\\ \\frac {\\sqrt {3}}{2}\\qquad \\textbf{(D) }\\ 1\\qquad \\textbf{(E) }\\ \\sqrt {2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1834", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Equilateral $\\triangle ABC$ has side length $2$, $M$ is the midpoint of $\\overline{AC}$, and $C$ is the midpoint of $\\overline{BD}$. What is the area of $\\triangle CDM$?\n\n$\\textbf{(A) }\\ \\frac {\\sqrt {2}}{2}\\qquad \\textbf{(B) }\\ \\frac {3}{4}\\qquad \\textbf{(C) }\\ \\frac {\\sqrt {3}}{2}\\qquad \\textbf{(D) }\\ 1\\qquad \\textbf{(E) }\\ \\sqrt {2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and \nsaved a distance equal to $\\frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was: \n$\\textbf{(A)}\\ \\frac{1}{2} \\qquad \\textbf{(B)}\\ \\frac{2}{3} \\qquad \\textbf{(C)}\\ \\frac{1}{4} \\qquad \\textbf{(D)}\\ \\frac{3}{4}\\qquad \\textbf{(E)}\\ \\frac{2}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1835", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and \nsaved a distance equal to $\\frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was: \n$\\textbf{(A)}\\ \\frac{1}{2} \\qquad \\textbf{(B)}\\ \\frac{2}{3} \\qquad \\textbf{(C)}\\ \\frac{1}{4} \\qquad \\textbf{(D)}\\ \\frac{3}{4}\\qquad \\textbf{(E)}\\ \\frac{2}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The logarithm of $27\\sqrt[4]{9}\\sqrt[3]{9}$ to the base $3$ is: \n$\\textbf{(A)}\\ 8\\frac{1}{2} \\qquad \\textbf{(B)}\\ 4\\frac{1}{6} \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1836", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The logarithm of $27\\sqrt[4]{9}\\sqrt[3]{9}$ to the base $3$ is: \n$\\textbf{(A)}\\ 8\\frac{1}{2} \\qquad \\textbf{(B)}\\ 4\\frac{1}{6} \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 3 \\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "61", + "index": "Sky-T1_10k_1837", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?" + } + }, + { + "question": "Return your final response within \\boxed{}. When you simplify $\\left[ \\sqrt [3]{\\sqrt [6]{a^9}} \\right]^4\\left[ \\sqrt [6]{\\sqrt [3]{a^9}} \\right]^4$, the result is: \n$\\textbf{(A)}\\ a^{16} \\qquad\\textbf{(B)}\\ a^{12} \\qquad\\textbf{(C)}\\ a^8 \\qquad\\textbf{(D)}\\ a^4 \\qquad\\textbf{(E)}\\ a^2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1838", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When you simplify $\\left[ \\sqrt [3]{\\sqrt [6]{a^9}} \\right]^4\\left[ \\sqrt [6]{\\sqrt [3]{a^9}} \\right]^4$, the result is: \n$\\textbf{(A)}\\ a^{16} \\qquad\\textbf{(B)}\\ a^{12} \\qquad\\textbf{(C)}\\ a^8 \\qquad\\textbf{(D)}\\ a^4 \\qquad\\textbf{(E)}\\ a^2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $m/n$, in lowest terms, be the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that a randomly chosen positive [divisor](https://artofproblemsolving.com/wiki/index.php/Divisor) of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "634", + "index": "Sky-T1_10k_1839", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $m/n$, in lowest terms, be the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that a randomly chosen positive [divisor](https://artofproblemsolving.com/wiki/index.php/Divisor) of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. Twenty percent less than 60 is one-third more than what number?\n$\\mathrm{(A)}\\ 16\\qquad \\mathrm{(B)}\\ 30\\qquad \\mathrm{(C)}\\ 32\\qquad \\mathrm{(D)}\\ 36\\qquad \\mathrm{(E)}\\ 48$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1840", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Twenty percent less than 60 is one-third more than what number?\n$\\mathrm{(A)}\\ 16\\qquad \\mathrm{(B)}\\ 30\\qquad \\mathrm{(C)}\\ 32\\qquad \\mathrm{(D)}\\ 36\\qquad \\mathrm{(E)}\\ 48$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of two prime numbers is $85$. What is the product of these two prime numbers?\n$\\textbf{(A) }85\\qquad\\textbf{(B) }91\\qquad\\textbf{(C) }115\\qquad\\textbf{(D) }133\\qquad \\textbf{(E) }166$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1841", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of two prime numbers is $85$. What is the product of these two prime numbers?\n$\\textbf{(A) }85\\qquad\\textbf{(B) }91\\qquad\\textbf{(C) }115\\qquad\\textbf{(D) }133\\qquad \\textbf{(E) }166$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that \n$DF=\\frac{1}{3}DA$. What is the ratio of the area of $\\triangle DFE$ to the area of quadrilateral $ABEF$?\n$\\textbf{(A)}\\ 1:2 \\qquad \\textbf{(B)}\\ 1:3 \\qquad \\textbf{(C)}\\ 1:5 \\qquad \\textbf{(D)}\\ 1:6 \\qquad \\textbf{(E)}\\ 1:7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1842", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that \n$DF=\\frac{1}{3}DA$. What is the ratio of the area of $\\triangle DFE$ to the area of quadrilateral $ABEF$?\n$\\textbf{(A)}\\ 1:2 \\qquad \\textbf{(B)}\\ 1:3 \\qquad \\textbf{(C)}\\ 1:5 \\qquad \\textbf{(D)}\\ 1:6 \\qquad \\textbf{(E)}\\ 1:7$" + } + }, + { + "question": "Return your final response within \\boxed{}. A straight concrete sidewalk is to be $3$ feet wide, $60$ feet long, and $3$ inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 20 \\qquad \\text{(E)}\\ \\text{more than 20}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1843", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A straight concrete sidewalk is to be $3$ feet wide, $60$ feet long, and $3$ inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 20 \\qquad \\text{(E)}\\ \\text{more than 20}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the adjoining figure triangle $ABC$ is such that $AB = 4$ and $AC = 8$. IF $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BC$?\n\n$\\textbf{(A)}\\ 2\\sqrt{26} \\qquad \\textbf{(B)}\\ 2\\sqrt{31} \\qquad \\textbf{(C)}\\ 9 \\qquad \\textbf{(D)}\\ 4 + 2\\sqrt{13}\\\\ \\textbf{(E)}\\ \\text{not enough information given to solve the problem}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1844", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the adjoining figure triangle $ABC$ is such that $AB = 4$ and $AC = 8$. IF $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BC$?\n\n$\\textbf{(A)}\\ 2\\sqrt{26} \\qquad \\textbf{(B)}\\ 2\\sqrt{31} \\qquad \\textbf{(C)}\\ 9 \\qquad \\textbf{(D)}\\ 4 + 2\\sqrt{13}\\\\ \\textbf{(E)}\\ \\text{not enough information given to solve the problem}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the statements: \n$\\textbf{(1)}\\ \\text{p and q are both true}\\qquad\\textbf{(2)}\\ \\text{p is true and q is false}\\qquad\\textbf{(3)}\\ \\text{p is false and q is true}\\qquad\\textbf{(4)}\\ \\text{p is false and q is false.}$\nHow many of these imply the negative of the statement \"p and q are both true?\" \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1845", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the statements: \n$\\textbf{(1)}\\ \\text{p and q are both true}\\qquad\\textbf{(2)}\\ \\text{p is true and q is false}\\qquad\\textbf{(3)}\\ \\text{p is false and q is true}\\qquad\\textbf{(4)}\\ \\text{p is false and q is false.}$\nHow many of these imply the negative of the statement \"p and q are both true?\" \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. $P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals:\n$\\textbf{(A) }2\\sqrt{3}\\qquad\\textbf{(B) }3\\sqrt{2}\\qquad\\textbf{(C) }3\\sqrt{3}\\qquad\\textbf{(D) }4\\sqrt{2}\\qquad \\textbf{(E) }2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1846", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals:\n$\\textbf{(A) }2\\sqrt{3}\\qquad\\textbf{(B) }3\\sqrt{2}\\qquad\\textbf{(C) }3\\sqrt{3}\\qquad\\textbf{(D) }4\\sqrt{2}\\qquad \\textbf{(E) }2$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a town of $351$ adults, every adult owns a car, motorcycle, or both. If $331$ adults own cars and $45$ adults own motorcycles, how many of the car owners do not own a motorcycle?\n$\\textbf{(A)}\\ 20 \\qquad \\textbf{(B)}\\ 25 \\qquad \\textbf{(C)}\\ 45 \\qquad \\textbf{(D)}\\ 306 \\qquad \\textbf{(E)}\\ 351$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1847", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a town of $351$ adults, every adult owns a car, motorcycle, or both. If $331$ adults own cars and $45$ adults own motorcycles, how many of the car owners do not own a motorcycle?\n$\\textbf{(A)}\\ 20 \\qquad \\textbf{(B)}\\ 25 \\qquad \\textbf{(C)}\\ 45 \\qquad \\textbf{(D)}\\ 306 \\qquad \\textbf{(E)}\\ 351$" + } + }, + { + "question": "Return your final response within \\boxed{}. A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \\le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "373", + "index": "Sky-T1_10k_1848", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \\le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?\n$\\textbf{(A)}\\ 15 \\qquad \\textbf{(B)}\\ 20 \\qquad \\textbf{(C)}\\ 25 \\qquad \\textbf{(D)}\\ 30 \\qquad \\textbf{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1849", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?\n$\\textbf{(A)}\\ 15 \\qquad \\textbf{(B)}\\ 20 \\qquad \\textbf{(C)}\\ 25 \\qquad \\textbf{(D)}\\ 30 \\qquad \\textbf{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\\frac{4}{7}$, then the number common to both sets of four numbers is\n$\\text{(A)}\\ 5\\frac{3}{7}\\qquad\\text{(B)}\\ 6\\qquad\\text{(C)}\\ 6\\frac{4}{7}\\qquad\\text{(D)}\\ 7\\qquad\\text{(E)}\\ 7\\frac{3}{7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1850", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\\frac{4}{7}$, then the number common to both sets of four numbers is\n$\\text{(A)}\\ 5\\frac{3}{7}\\qquad\\text{(B)}\\ 6\\qquad\\text{(C)}\\ 6\\frac{4}{7}\\qquad\\text{(D)}\\ 7\\qquad\\text{(E)}\\ 7\\frac{3}{7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year?\n$\\textbf{(A)}\\ 150\\qquad\\textbf{(B)}\\ 200\\qquad\\textbf{(C)}\\ 250\\qquad\\textbf{(D)}\\ 300\\qquad\\textbf{(E)}\\ 400$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1851", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year?\n$\\textbf{(A)}\\ 150\\qquad\\textbf{(B)}\\ 200\\qquad\\textbf{(C)}\\ 250\\qquad\\textbf{(D)}\\ 300\\qquad\\textbf{(E)}\\ 400$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?\n$\\textbf{(A) } -5 \\qquad\\textbf{(B) } 0 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } \\frac{15}{4} \\qquad\\textbf{(E) } \\frac{35}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1852", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?\n$\\textbf{(A) } -5 \\qquad\\textbf{(B) } 0 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } \\frac{15}{4} \\qquad\\textbf{(E) } \\frac{35}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $p$ and $q$ are primes and $x^2-px+q=0$ has distinct positive integral roots, then which of the following statements are true? \n$I.\\ \\text{The difference of the roots is odd.} \\\\ II.\\ \\text{At least one root is prime.} \\\\ III.\\ p^2-q\\ \\text{is prime}. \\\\ IV.\\ p+q\\ \\text{is prime}$\n$\\\\ \\textbf{(A)}\\ I\\ \\text{only} \\qquad \\textbf{(B)}\\ II\\ \\text{only} \\qquad \\textbf{(C)}\\ II\\ \\text{and}\\ III\\ \\text{only} \\\\ \\textbf{(D)}\\ I, II, \\text{and}\\ IV\\ \\text{only}\\ \\qquad \\textbf{(E)}\\ \\text{All are true.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1853", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $p$ and $q$ are primes and $x^2-px+q=0$ has distinct positive integral roots, then which of the following statements are true? \n$I.\\ \\text{The difference of the roots is odd.} \\\\ II.\\ \\text{At least one root is prime.} \\\\ III.\\ p^2-q\\ \\text{is prime}. \\\\ IV.\\ p+q\\ \\text{is prime}$\n$\\\\ \\textbf{(A)}\\ I\\ \\text{only} \\qquad \\textbf{(B)}\\ II\\ \\text{only} \\qquad \\textbf{(C)}\\ II\\ \\text{and}\\ III\\ \\text{only} \\\\ \\textbf{(D)}\\ I, II, \\text{and}\\ IV\\ \\text{only}\\ \\qquad \\textbf{(E)}\\ \\text{All are true.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three balls marked $1,2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6$, what is the probability that the ball numbered $2$ was drawn all three times? \n$\\textbf{(A)} \\ \\frac{1}{27} \\qquad \\textbf{(B)} \\ \\frac{1}{8} \\qquad \\textbf{(C)} \\ \\frac{1}{7} \\qquad \\textbf{(D)} \\ \\frac{1}{6} \\qquad \\textbf{(E)}\\ \\frac{1}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1854", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three balls marked $1,2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6$, what is the probability that the ball numbered $2$ was drawn all three times? \n$\\textbf{(A)} \\ \\frac{1}{27} \\qquad \\textbf{(B)} \\ \\frac{1}{8} \\qquad \\textbf{(C)} \\ \\frac{1}{7} \\qquad \\textbf{(D)} \\ \\frac{1}{6} \\qquad \\textbf{(E)}\\ \\frac{1}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the $xy$-plane, how many lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer pass through the point $(4,3)$?\n$\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ 1 \\qquad\\textbf{(C)}\\ 2 \\qquad\\textbf{(D)}\\ 3 \\qquad\\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1855", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the $xy$-plane, how many lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer pass through the point $(4,3)$?\n$\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ 1 \\qquad\\textbf{(C)}\\ 2 \\qquad\\textbf{(D)}\\ 3 \\qquad\\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x,y>0, \\log_y(x)+\\log_x(y)=\\frac{10}{3} \\text{ and } xy=144,\\text{ then }\\frac{x+y}{2}=$\n$\\text{(A) } 12\\sqrt{2}\\quad \\text{(B) } 13\\sqrt{3}\\quad \\text{(C) } 24\\quad \\text{(D) } 30\\quad \\text{(E) } 36$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1856", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x,y>0, \\log_y(x)+\\log_x(y)=\\frac{10}{3} \\text{ and } xy=144,\\text{ then }\\frac{x+y}{2}=$\n$\\text{(A) } 12\\sqrt{2}\\quad \\text{(B) } 13\\sqrt{3}\\quad \\text{(C) } 24\\quad \\text{(D) } 30\\quad \\text{(E) } 36$" + } + }, + { + "question": "Return your final response within \\boxed{}. An $a \\times b \\times c$ rectangular box is built from $a \\cdot b \\cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \\times b \\times c$ parallel to the $(b \\times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \\times 1 \\times c$ parallel to the $(a \\times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "180", + "index": "Sky-T1_10k_1857", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An $a \\times b \\times c$ rectangular box is built from $a \\cdot b \\cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \\times b \\times c$ parallel to the $(b \\times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \\times 1 \\times c$ parallel to the $(a \\times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box." + } + }, + { + "question": "Return your final response within \\boxed{}. For what value of $x$ does $10^{x}\\cdot 100^{2x}=1000^{5}$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1858", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For what value of $x$ does $10^{x}\\cdot 100^{2x}=1000^{5}$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?\n$\\textbf{(A)}\\ 2.0\\qquad\\textbf{(B)}\\ 2.2\\qquad\\textbf{(C)}\\ 2.8\\qquad\\textbf{(D)}\\ 3.4\\qquad\\textbf{(E)}\\ 4.4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1859", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?\n$\\textbf{(A)}\\ 2.0\\qquad\\textbf{(B)}\\ 2.2\\qquad\\textbf{(C)}\\ 2.8\\qquad\\textbf{(D)}\\ 3.4\\qquad\\textbf{(E)}\\ 4.4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mia is \"helping\" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?\n$\\textbf{(A)}\\ 13.5\\qquad\\textbf{(B)}\\ 14\\qquad\\textbf{(C)}\\ 14.5\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 15.5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1860", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mia is \"helping\" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?\n$\\textbf{(A)}\\ 13.5\\qquad\\textbf{(B)}\\ 14\\qquad\\textbf{(C)}\\ 14.5\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 15.5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which one of the following combinations of given parts does not determine the indicated triangle? \n$\\textbf{(A)}\\ \\text{base angle and vertex angle; isosceles triangle} \\\\ \\textbf{(B)}\\ \\text{vertex angle and the base; isosceles triangle} \\\\ \\textbf{(C)}\\ \\text{the radius of the circumscribed circle; equilateral triangle} \\\\ \\textbf{(D)}\\ \\text{one arm and the radius of the inscribed circle; right triangle} \\\\ \\textbf{(E)}\\ \\text{two angles and a side opposite one of them; scalene triangle}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1861", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which one of the following combinations of given parts does not determine the indicated triangle? \n$\\textbf{(A)}\\ \\text{base angle and vertex angle; isosceles triangle} \\\\ \\textbf{(B)}\\ \\text{vertex angle and the base; isosceles triangle} \\\\ \\textbf{(C)}\\ \\text{the radius of the circumscribed circle; equilateral triangle} \\\\ \\textbf{(D)}\\ \\text{one arm and the radius of the inscribed circle; right triangle} \\\\ \\textbf{(E)}\\ \\text{two angles and a side opposite one of them; scalene triangle}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?\n$\\begin{tabular}{rr}&\\ \\texttt{6 4 1}\\\\ &\\texttt{8 5 2}\\\\ &+\\texttt{9 7 3}\\\\ \\hline &\\texttt{2 4 5 6}\\end{tabular}$\n$\\text{(A)}\\ 4\\qquad\\text{(B)}\\ 5\\qquad\\text{(C)}\\ 6\\qquad\\text{(D)}\\ 7\\qquad\\text{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1862", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?\n$\\begin{tabular}{rr}&\\ \\texttt{6 4 1}\\\\ &\\texttt{8 5 2}\\\\ &+\\texttt{9 7 3}\\\\ \\hline &\\texttt{2 4 5 6}\\end{tabular}$\n$\\text{(A)}\\ 4\\qquad\\text{(B)}\\ 5\\qquad\\text{(C)}\\ 6\\qquad\\text{(D)}\\ 7\\qquad\\text{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is\n$\\textbf{(A) }.05\\qquad \\textbf{(B) }.065\\qquad \\textbf{(C) }.08\\qquad \\textbf{(D) }.09\\qquad \\textbf{(E) }.1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1863", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is\n$\\textbf{(A) }.05\\qquad \\textbf{(B) }.065\\qquad \\textbf{(C) }.08\\qquad \\textbf{(D) }.09\\qquad \\textbf{(E) }.1$" + } + }, + { + "question": "Return your final response within \\boxed{}. The Tigers beat the Sharks 2 out of the 3 times they played. They then played $N$ more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for $N$?\n$\\textbf{(A)}\\; 35 \\qquad \\textbf{(B)}\\; 37 \\qquad \\textbf{(C)}\\; 39 \\qquad \\textbf{(D)}\\; 41 \\qquad \\textbf{(E)}\\; 43$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1864", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The Tigers beat the Sharks 2 out of the 3 times they played. They then played $N$ more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for $N$?\n$\\textbf{(A)}\\; 35 \\qquad \\textbf{(B)}\\; 37 \\qquad \\textbf{(C)}\\; 39 \\qquad \\textbf{(D)}\\; 41 \\qquad \\textbf{(E)}\\; 43$" + } + }, + { + "question": "Return your final response within \\boxed{}. In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?\n$\\textbf{(A)} ~10\\qquad\\textbf{(B)} ~18\\qquad\\textbf{(C)} ~24 \\qquad\\textbf{(D)} ~32 \\qquad\\textbf{(E)} ~44$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1865", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?\n$\\textbf{(A)} ~10\\qquad\\textbf{(B)} ~18\\qquad\\textbf{(C)} ~24 \\qquad\\textbf{(D)} ~32 \\qquad\\textbf{(E)} ~44$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\\overline{AC}$ and $\\overline{AB}$, respectively, so that $\\overline{BD}$ and $\\overline{CE}$ are angle bisectors, intersecting $\\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?\n\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ \\frac{5}{8}\\sqrt{3} \\qquad \\textbf{(C)}\\ \\frac{4}{5}\\sqrt{2} \\qquad \\textbf{(D)}\\ \\frac{8}{15}\\sqrt{5} \\qquad \\textbf{(E)}\\ \\frac{6}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1866", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\\overline{AC}$ and $\\overline{AB}$, respectively, so that $\\overline{BD}$ and $\\overline{CE}$ are angle bisectors, intersecting $\\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?\n\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ \\frac{5}{8}\\sqrt{3} \\qquad \\textbf{(C)}\\ \\frac{4}{5}\\sqrt{2} \\qquad \\textbf{(D)}\\ \\frac{8}{15}\\sqrt{5} \\qquad \\textbf{(E)}\\ \\frac{6}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. There is a positive integer $n$ such that $(n+1)! + (n+2)! = n! \\cdot 440$. What is the sum of the digits of $n$?\n$\\textbf{(A) }3\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }11\\qquad\\textbf{(E) }12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1867", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There is a positive integer $n$ such that $(n+1)! + (n+2)! = n! \\cdot 440$. What is the sum of the digits of $n$?\n$\\textbf{(A) }3\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }11\\qquad\\textbf{(E) }12$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: \n\\[x=x^2+y^2 \\ \\ y=2xy\\] \nis\n$\\textbf{(A) }0\\qquad \\textbf{(B) }1\\qquad \\textbf{(C) }2\\qquad \\textbf{(D) }3\\qquad \\textbf{(E) }4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1868", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: \n\\[x=x^2+y^2 \\ \\ y=2xy\\] \nis\n$\\textbf{(A) }0\\qquad \\textbf{(B) }1\\qquad \\textbf{(C) }2\\qquad \\textbf{(D) }3\\qquad \\textbf{(E) }4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:\ni. The actual attendance in Atlanta is within $10 \\%$ of Anita's estimate.\nii. Bob's estimate is within $10 \\%$ of the actual attendance in Boston.\nTo the nearest 1,000, the largest possible difference between the numbers attending the two games is\n$\\mathrm{(A) \\ 10000 } \\qquad \\mathrm{(B) \\ 11000 } \\qquad \\mathrm{(C) \\ 20000 } \\qquad \\mathrm{(D) \\ 21000 } \\qquad \\mathrm{(E) \\ 22000 }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1869", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:\ni. The actual attendance in Atlanta is within $10 \\%$ of Anita's estimate.\nii. Bob's estimate is within $10 \\%$ of the actual attendance in Boston.\nTo the nearest 1,000, the largest possible difference between the numbers attending the two games is\n$\\mathrm{(A) \\ 10000 } \\qquad \\mathrm{(B) \\ 11000 } \\qquad \\mathrm{(C) \\ 20000 } \\qquad \\mathrm{(D) \\ 21000 } \\qquad \\mathrm{(E) \\ 22000 }$" + } + }, + { + "question": "Return your final response within \\boxed{}. A set of $25$ square blocks is arranged into a $5 \\times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?\n$\\textbf{(A) } 100 \\qquad\\textbf{(B) } 125 \\qquad\\textbf{(C) } 600 \\qquad\\textbf{(D) } 2300 \\qquad\\textbf{(E) } 3600$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1870", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A set of $25$ square blocks is arranged into a $5 \\times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?\n$\\textbf{(A) } 100 \\qquad\\textbf{(B) } 125 \\qquad\\textbf{(C) } 600 \\qquad\\textbf{(D) } 2300 \\qquad\\textbf{(E) } 3600$" + } + }, + { + "question": "Return your final response within \\boxed{}. Pick two consecutive positive integers whose sum is less than $100$. Square both\nof those integers and then find the difference of the squares. Which of the\nfollowing could be the difference?\n$\\mathrm{(A)}\\ 2 \\qquad \\mathrm{(B)}\\ 64 \\qquad \\mathrm{(C)}\\ 79 \\qquad \\mathrm{(D)}\\ 96 \\qquad \\mathrm{(E)}\\ 131$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1871", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Pick two consecutive positive integers whose sum is less than $100$. Square both\nof those integers and then find the difference of the squares. Which of the\nfollowing could be the difference?\n$\\mathrm{(A)}\\ 2 \\qquad \\mathrm{(B)}\\ 64 \\qquad \\mathrm{(C)}\\ 79 \\qquad \\mathrm{(D)}\\ 96 \\qquad \\mathrm{(E)}\\ 131$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the series $2+1+\\frac {1}{2}+\\frac {1}{4}+\\cdots$ and the following five statements:\n\n(1) the sum increases without limit\n(2) the sum decreases without limit\n(3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small\n(4) the difference between the sum and 4 can be made less than any positive quantity no matter how small\n(5) the sum approaches a limit\nOf these statments, the correct ones are:\n$\\textbf{(A)}\\ \\text{Only }3 \\text{ and }4\\qquad \\textbf{(B)}\\ \\text{Only }5 \\qquad \\textbf{(C)}\\ \\text{Only }2\\text{ and }4 \\qquad \\textbf{(D)}\\ \\text{Only }2,3\\text{ and }4 \\qquad \\textbf{(E)}\\ \\text{Only }4\\text{ and }5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1872", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the series $2+1+\\frac {1}{2}+\\frac {1}{4}+\\cdots$ and the following five statements:\n\n(1) the sum increases without limit\n(2) the sum decreases without limit\n(3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small\n(4) the difference between the sum and 4 can be made less than any positive quantity no matter how small\n(5) the sum approaches a limit\nOf these statments, the correct ones are:\n$\\textbf{(A)}\\ \\text{Only }3 \\text{ and }4\\qquad \\textbf{(B)}\\ \\text{Only }5 \\qquad \\textbf{(C)}\\ \\text{Only }2\\text{ and }4 \\qquad \\textbf{(D)}\\ \\text{Only }2,3\\text{ and }4 \\qquad \\textbf{(E)}\\ \\text{Only }4\\text{ and }5$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is $\\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$?\n$\\textbf {(A) } 16 \\qquad \\textbf {(B) } 24 \\qquad \\textbf {(C) } 32 \\qquad \\textbf {(D) } 48 \\qquad \\textbf {(E) } 64$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1873", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is $\\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$?\n$\\textbf {(A) } 16 \\qquad \\textbf {(B) } 24 \\qquad \\textbf {(C) } 32 \\qquad \\textbf {(D) } 48 \\qquad \\textbf {(E) } 64$" + } + }, + { + "question": "Return your final response within \\boxed{}. Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?\n$\\textbf{(A)}\\ \\frac{3}{2}\\qquad\\textbf{(B)}\\ \\frac{5}{3}\\qquad\\textbf{(C)}\\ \\frac{7}{4}\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ \\frac{13}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1874", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?\n$\\textbf{(A)}\\ \\frac{3}{2}\\qquad\\textbf{(B)}\\ \\frac{5}{3}\\qquad\\textbf{(C)}\\ \\frac{7}{4}\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ \\frac{13}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?\n$\\textbf{(A) }{-}88 \\qquad \\textbf{(B) }{-}80 \\qquad \\textbf{(C) }{-}64 \\qquad \\textbf{(D) }{-}41\\qquad \\textbf{(E) }{-}40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1875", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?\n$\\textbf{(A) }{-}88 \\qquad \\textbf{(B) }{-}80 \\qquad \\textbf{(C) }{-}64 \\qquad \\textbf{(D) }{-}41\\qquad \\textbf{(E) }{-}40$" + } + }, + { + "question": "Return your final response within \\boxed{}. For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\\frac{7}{51}$ is $0.\\overline{23}_k = 0.232323..._k$. What is $k$?\n$\\textbf{(A) } 13 \\qquad\\textbf{(B) } 14 \\qquad\\textbf{(C) } 15 \\qquad\\textbf{(D) } 16 \\qquad\\textbf{(E) } 17$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1876", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\\frac{7}{51}$ is $0.\\overline{23}_k = 0.232323..._k$. What is $k$?\n$\\textbf{(A) } 13 \\qquad\\textbf{(B) } 14 \\qquad\\textbf{(C) } 15 \\qquad\\textbf{(D) } 16 \\qquad\\textbf{(E) } 17$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $N = 34 \\cdot 34 \\cdot 63 \\cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?\n$\\textbf{(A)} ~1 : 16 \\qquad\\textbf{(B)} ~1 : 15 \\qquad\\textbf{(C)} ~1 : 14 \\qquad\\textbf{(D)} ~1 : 8 \\qquad\\textbf{(E)} ~1 : 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1877", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $N = 34 \\cdot 34 \\cdot 63 \\cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?\n$\\textbf{(A)} ~1 : 16 \\qquad\\textbf{(B)} ~1 : 15 \\qquad\\textbf{(C)} ~1 : 14 \\qquad\\textbf{(D)} ~1 : 8 \\qquad\\textbf{(E)} ~1 : 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T$. The equation of the line is:\n$\\textbf{(A)}\\ 2Tx+a^2y+2aT=0 \\qquad \\textbf{(B)}\\ 2Tx-a^2y+2aT=0 \\qquad \\textbf{(C)}\\ 2Tx+a^2y-2aT=0 \\qquad \\\\ \\textbf{(D)}\\ 2Tx-a^2y-2aT=0 \\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1878", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T$. The equation of the line is:\n$\\textbf{(A)}\\ 2Tx+a^2y+2aT=0 \\qquad \\textbf{(B)}\\ 2Tx-a^2y+2aT=0 \\qquad \\textbf{(C)}\\ 2Tx+a^2y-2aT=0 \\qquad \\\\ \\textbf{(D)}\\ 2Tx-a^2y-2aT=0 \\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Boris has an incredible coin-changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?\n$\\textbf{(A)} $3.63 \\qquad \\textbf{(B)} $5.13 \\qquad \\textbf{(C)}$6.30 \\qquad \\textbf{(D)} $7.45 \\qquad \\textbf{(E)} $9.07$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1879", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Boris has an incredible coin-changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?\n$\\textbf{(A)} $3.63 \\qquad \\textbf{(B)} $5.13 \\qquad \\textbf{(C)}$6.30 \\qquad \\textbf{(D)} $7.45 \\qquad \\textbf{(E)} $9.07$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\log 125$ equals: \n$\\textbf{(A)}\\ 100 \\log 1.25 \\qquad \\textbf{(B)}\\ 5 \\log 3 \\qquad \\textbf{(C)}\\ 3 \\log 25 \\\\ \\textbf{(D)}\\ 3 - 3\\log 2 \\qquad \\textbf{(E)}\\ (\\log 25)(\\log 5)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1880", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\log 125$ equals: \n$\\textbf{(A)}\\ 100 \\log 1.25 \\qquad \\textbf{(B)}\\ 5 \\log 3 \\qquad \\textbf{(C)}\\ 3 \\log 25 \\\\ \\textbf{(D)}\\ 3 - 3\\log 2 \\qquad \\textbf{(E)}\\ (\\log 25)(\\log 5)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the smallest integer $k$ for which the conditions\n(1) $a_1,a_2,a_3\\cdots$ is a nondecreasing sequence of positive integers\n(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$\n(3) $a_9=k$\nare satisfied by more than one sequence.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "748", + "index": "Sky-T1_10k_1881", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the smallest integer $k$ for which the conditions\n(1) $a_1,a_2,a_3\\cdots$ is a nondecreasing sequence of positive integers\n(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$\n(3) $a_9=k$\nare satisfied by more than one sequence." + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following represents the result when the figure shown below is rotated clockwise $120^\\circ$ about its center?\n[asy] unitsize(6); draw(circle((0,0),5)); draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); [/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1882", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following represents the result when the figure shown below is rotated clockwise $120^\\circ$ about its center?\n[asy] unitsize(6); draw(circle((0,0),5)); draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); [/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. A $3 \\times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\\,^{\\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?\n$\\textbf{(A)}\\ \\frac{49}{512}\\qquad\\textbf{(B)}\\ \\frac{7}{64}\\qquad\\textbf{(C)}\\ \\frac{121}{1024}\\qquad\\textbf{(D)}\\ \\frac{81}{512}\\qquad\\textbf{(E)}\\ \\frac{9}{32}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1883", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A $3 \\times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\\,^{\\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?\n$\\textbf{(A)}\\ \\frac{49}{512}\\qquad\\textbf{(B)}\\ \\frac{7}{64}\\qquad\\textbf{(C)}\\ \\frac{121}{1024}\\qquad\\textbf{(D)}\\ \\frac{81}{512}\\qquad\\textbf{(E)}\\ \\frac{9}{32}$" + } + }, + { + "question": "Return your final response within \\boxed{}. When the repeating decimal $0.363636\\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is: \n$\\textbf{(A)}\\ 15 \\qquad \\textbf{(B) }\\ 45 \\qquad \\textbf{(C) }\\ 114 \\qquad \\textbf{(D) }\\ 135 \\qquad \\textbf{(E) }\\ 150$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1884", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When the repeating decimal $0.363636\\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is: \n$\\textbf{(A)}\\ 15 \\qquad \\textbf{(B) }\\ 45 \\qquad \\textbf{(C) }\\ 114 \\qquad \\textbf{(D) }\\ 135 \\qquad \\textbf{(E) }\\ 150$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of teeth in three meshed gears $A$, $B$, and $C$ are $x$, $y$, and $z$, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of $A$, $B$, and $C$ are in the proportion\n$\\text{(A)} \\ x: y: z ~~\\text{(B)} \\ z: y: x ~~ \\text{(C)} \\ y: z: x~~ \\text{(D)} \\ yz: xz: xy ~~ \\text{(E)} \\ xz: yx: zy$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1885", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of teeth in three meshed gears $A$, $B$, and $C$ are $x$, $y$, and $z$, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of $A$, $B$, and $C$ are in the proportion\n$\\text{(A)} \\ x: y: z ~~\\text{(B)} \\ z: y: x ~~ \\text{(C)} \\ y: z: x~~ \\text{(D)} \\ yz: xz: xy ~~ \\text{(E)} \\ xz: yx: zy$" + } + }, + { + "question": "Return your final response within \\boxed{}. The fraction $\\frac{\\sqrt{a^2+x^2}-\\frac{x^2-a^2}{\\sqrt{a^2+x^2}}}{a^2+x^2}$ reduces to: \n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ \\frac{2a^2}{a^2+x^2} \\qquad \\textbf{(C)}\\ \\frac{2x^2}{(a^2+x^2)^{\\frac{3}{2}}}\\qquad \\textbf{(D)}\\ \\frac{2a^2}{(a^2+x^2)^{\\frac{3}{2}}}\\qquad \\textbf{(E)}\\ \\frac{2x^2}{a^2+x^2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1886", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The fraction $\\frac{\\sqrt{a^2+x^2}-\\frac{x^2-a^2}{\\sqrt{a^2+x^2}}}{a^2+x^2}$ reduces to: \n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ \\frac{2a^2}{a^2+x^2} \\qquad \\textbf{(C)}\\ \\frac{2x^2}{(a^2+x^2)^{\\frac{3}{2}}}\\qquad \\textbf{(D)}\\ \\frac{2a^2}{(a^2+x^2)^{\\frac{3}{2}}}\\qquad \\textbf{(E)}\\ \\frac{2x^2}{a^2+x^2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\\frac{1}{6}$, independent of the outcome of any other toss.) \n$\\textbf{(A) } \\frac{1}{3} \\textbf{(B) } \\frac{2}{9} \\textbf{(C) } \\frac{5}{18} \\textbf{(D) } \\frac{25}{91} \\textbf{(E) } \\frac{36}{91}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1887", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\\frac{1}{6}$, independent of the outcome of any other toss.) \n$\\textbf{(A) } \\frac{1}{3} \\textbf{(B) } \\frac{2}{9} \\textbf{(C) } \\frac{5}{18} \\textbf{(D) } \\frac{25}{91} \\textbf{(E) } \\frac{36}{91}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Randy's trip?\n$\\textbf {(A) } 30 \\qquad \\textbf {(B) } \\frac{400}{11} \\qquad \\textbf {(C) } \\frac{75}{2} \\qquad \\textbf {(D) } 40 \\qquad \\textbf {(E) } \\frac{300}{7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1888", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Randy's trip?\n$\\textbf {(A) } 30 \\qquad \\textbf {(B) } \\frac{400}{11} \\qquad \\textbf {(C) } \\frac{75}{2} \\qquad \\textbf {(D) } 40 \\qquad \\textbf {(E) } \\frac{300}{7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?\n[asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,-1)--(5.5,-1));[/asy]\nNote: 1 mile = 5280 feet\n$\\textbf{(A) }\\frac{\\pi}{11}\\qquad\\textbf{(B) }\\frac{\\pi}{10}\\qquad\\textbf{(C) }\\frac{\\pi}{5}\\qquad\\textbf{(D) }\\frac{2\\pi}{5}\\qquad\\textbf{(E) }\\frac{2\\pi}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1889", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?\n[asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,-1)--(5.5,-1));[/asy]\nNote: 1 mile = 5280 feet\n$\\textbf{(A) }\\frac{\\pi}{11}\\qquad\\textbf{(B) }\\frac{\\pi}{10}\\qquad\\textbf{(C) }\\frac{\\pi}{5}\\qquad\\textbf{(D) }\\frac{2\\pi}{5}\\qquad\\textbf{(E) }\\frac{2\\pi}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly $10$ feet away from each other?\n$\\textbf{(A) } A\\ \\text{east, } B\\ \\text{west} \\qquad \\textbf{(B) } A\\ \\text{north, } B\\ \\text{south} \\qquad \\textbf{(C) } A\\ \\text{north, } B\\ \\text{west} \\qquad \\textbf{(D) } A\\ \\text{up, } B\\ \\text{south} \\qquad \\textbf{(E) } A\\ \\text{up, } B\\ \\text{west}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1890", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly $10$ feet away from each other?\n$\\textbf{(A) } A\\ \\text{east, } B\\ \\text{west} \\qquad \\textbf{(B) } A\\ \\text{north, } B\\ \\text{south} \\qquad \\textbf{(C) } A\\ \\text{north, } B\\ \\text{west} \\qquad \\textbf{(D) } A\\ \\text{up, } B\\ \\text{south} \\qquad \\textbf{(E) } A\\ \\text{up, } B\\ \\text{west}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A large bag of coins contains pennies, dimes and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is\n$\\mathrm{(A)\\ } $306 \\qquad \\mathrm{(B) \\ } $333 \\qquad \\mathrm{(C)\\ } $342 \\qquad \\mathrm{(D) \\ } $348 \\qquad \\mathrm{(E) \\ } $360$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1891", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A large bag of coins contains pennies, dimes and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is\n$\\mathrm{(A)\\ } $306 \\qquad \\mathrm{(B) \\ } $333 \\qquad \\mathrm{(C)\\ } $342 \\qquad \\mathrm{(D) \\ } $348 \\qquad \\mathrm{(E) \\ } $360$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?\n$\\textbf{(A)}\\; 3 \\qquad\\textbf{(B)}\\; 4 \\qquad\\textbf{(C)}\\; 5 \\qquad\\textbf{(D)}\\; 6 \\qquad\\textbf{(E)}\\; 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1892", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?\n$\\textbf{(A)}\\; 3 \\qquad\\textbf{(B)}\\; 4 \\qquad\\textbf{(C)}\\; 5 \\qquad\\textbf{(D)}\\; 6 \\qquad\\textbf{(E)}\\; 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, \none at the rate of $3$ feet per second, the other at $2$ feet per second. \nThey swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other. \n$\\textbf{(A)}\\ 24\\qquad \\textbf{(B)}\\ 21\\qquad \\textbf{(C)}\\ 20\\qquad \\textbf{(D)}\\ 19\\qquad \\textbf{(E)}\\ 18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1893", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, \none at the rate of $3$ feet per second, the other at $2$ feet per second. \nThey swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other. \n$\\textbf{(A)}\\ 24\\qquad \\textbf{(B)}\\ 21\\qquad \\textbf{(C)}\\ 20\\qquad \\textbf{(D)}\\ 19\\qquad \\textbf{(E)}\\ 18$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "IO intersects segments AB and BC.", + "index": "Sky-T1_10k_1894", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y )$ satisfy all of the following conditions:\n$\\text{(i) }\\frac{a}{2}\\le x\\le 2a\\qquad \\text{(ii) }\\frac{a}{2}\\le y\\le 2a\\qquad \\text{(iii) }x+y\\ge a\\\\ \\\\ \\qquad \\text{(iv) }x+a\\ge y\\qquad \\text{(v) }y+a\\ge x$\nThe boundary of set $S$ is a polygon with\n$\\textbf{(A) }3\\text{ sides}\\qquad \\textbf{(B) }4\\text{ sides}\\qquad \\textbf{(C) }5\\text{ sides}\\qquad \\textbf{(D) }6\\text{ sides}\\qquad \\textbf{(E) }7\\text{ sides}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1895", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y )$ satisfy all of the following conditions:\n$\\text{(i) }\\frac{a}{2}\\le x\\le 2a\\qquad \\text{(ii) }\\frac{a}{2}\\le y\\le 2a\\qquad \\text{(iii) }x+y\\ge a\\\\ \\\\ \\qquad \\text{(iv) }x+a\\ge y\\qquad \\text{(v) }y+a\\ge x$\nThe boundary of set $S$ is a polygon with\n$\\textbf{(A) }3\\text{ sides}\\qquad \\textbf{(B) }4\\text{ sides}\\qquad \\textbf{(C) }5\\text{ sides}\\qquad \\textbf{(D) }6\\text{ sides}\\qquad \\textbf{(E) }7\\text{ sides}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A man has $\\textdollar{10,000 }$ to invest. He invests $\\textdollar{4000}$ at 5% and $\\textdollar{3500}$ at 4%. \nIn order to have a yearly income of $\\textdollar{500}$, he must invest the remainder at: \n$\\textbf{(A)}\\ 6\\%\\qquad\\textbf{(B)}\\ 6.1\\%\\qquad\\textbf{(C)}\\ 6.2\\%\\qquad\\textbf{(D)}\\ 6.3\\%\\qquad\\textbf{(E)}\\ 6.4\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1896", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A man has $\\textdollar{10,000 }$ to invest. He invests $\\textdollar{4000}$ at 5% and $\\textdollar{3500}$ at 4%. \nIn order to have a yearly income of $\\textdollar{500}$, he must invest the remainder at: \n$\\textbf{(A)}\\ 6\\%\\qquad\\textbf{(B)}\\ 6.1\\%\\qquad\\textbf{(C)}\\ 6.2\\%\\qquad\\textbf{(D)}\\ 6.3\\%\\qquad\\textbf{(E)}\\ 6.4\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which pair of numbers does NOT have a product equal to $36$?\n$\\text{(A)}\\ \\{-4,-9\\}\\qquad\\text{(B)}\\ \\{-3,-12\\}\\qquad\\text{(C)}\\ \\left\\{\\frac{1}{2},-72\\right\\}\\qquad\\text{(D)}\\ \\{ 1,36\\}\\qquad\\text{(E)}\\ \\left\\{\\frac{3}{2},24\\right\\}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1897", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which pair of numbers does NOT have a product equal to $36$?\n$\\text{(A)}\\ \\{-4,-9\\}\\qquad\\text{(B)}\\ \\{-3,-12\\}\\qquad\\text{(C)}\\ \\left\\{\\frac{1}{2},-72\\right\\}\\qquad\\text{(D)}\\ \\{ 1,36\\}\\qquad\\text{(E)}\\ \\left\\{\\frac{3}{2},24\\right\\}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the length of a [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of a [square](https://artofproblemsolving.com/wiki/index.php/Square) is $a + b$, then the area of the square is:\n$\\mathrm{(A) \\ (a+b)^2 } \\qquad \\mathrm{(B) \\ \\frac{1}{2}(a+b)^2 } \\qquad \\mathrm{(C) \\ a^2+b^2 } \\qquad \\mathrm{(D) \\ \\frac {1}{2}(a^2+b^2) } \\qquad \\mathrm{(E) \\ \\text{none of these} }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1898", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the length of a [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of a [square](https://artofproblemsolving.com/wiki/index.php/Square) is $a + b$, then the area of the square is:\n$\\mathrm{(A) \\ (a+b)^2 } \\qquad \\mathrm{(B) \\ \\frac{1}{2}(a+b)^2 } \\qquad \\mathrm{(C) \\ a^2+b^2 } \\qquad \\mathrm{(D) \\ \\frac {1}{2}(a^2+b^2) } \\qquad \\mathrm{(E) \\ \\text{none of these} }$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three $\\Delta$'s and a $\\diamondsuit$ will balance nine $\\bullet$'s. One $\\Delta$ will balance a $\\diamondsuit$ and a $\\bullet$.\n\nHow many $\\bullet$'s will balance the two $\\diamondsuit$'s in this balance?\n\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 3 \\qquad \\text{(D)}\\ 4 \\qquad \\text{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1899", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three $\\Delta$'s and a $\\diamondsuit$ will balance nine $\\bullet$'s. One $\\Delta$ will balance a $\\diamondsuit$ and a $\\bullet$.\n\nHow many $\\bullet$'s will balance the two $\\diamondsuit$'s in this balance?\n\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 3 \\qquad \\text{(D)}\\ 4 \\qquad \\text{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $A=20^\\circ$ and $B=25^\\circ$, then the value of $(1+\\tan A)(1+\\tan B)$ is\n$\\mathrm{(A)\\ } \\sqrt{3} \\qquad \\mathrm{(B) \\ }2 \\qquad \\mathrm{(C) \\ } 1+\\sqrt{2} \\qquad \\mathrm{(D) \\ } 2(\\tan A+\\tan B) \\qquad \\mathrm{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1900", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $A=20^\\circ$ and $B=25^\\circ$, then the value of $(1+\\tan A)(1+\\tan B)$ is\n$\\mathrm{(A)\\ } \\sqrt{3} \\qquad \\mathrm{(B) \\ }2 \\qquad \\mathrm{(C) \\ } 1+\\sqrt{2} \\qquad \\mathrm{(D) \\ } 2(\\tan A+\\tan B) \\qquad \\mathrm{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:\n$\\textbf{(A)}\\ \\text{rhombus} \\qquad\\textbf{(B)}\\ \\text{rectangles} \\qquad\\textbf{(C)}\\ \\text{square} \\qquad\\textbf{(D)}\\ \\text{isosceles trapezoid}\\qquad\\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1901", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:\n$\\textbf{(A)}\\ \\text{rhombus} \\qquad\\textbf{(B)}\\ \\text{rectangles} \\qquad\\textbf{(C)}\\ \\text{square} \\qquad\\textbf{(D)}\\ \\text{isosceles trapezoid}\\qquad\\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $(2^0-1+5^2-0)^{-1}\\times5?$\n$\\textbf{(A)}\\ -125\\qquad\\textbf{(B)}\\ -120\\qquad\\textbf{(C)}\\ \\frac{1}{5}\\qquad\\textbf{(D)}\\ \\frac{5}{24}\\qquad\\textbf{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1902", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $(2^0-1+5^2-0)^{-1}\\times5?$\n$\\textbf{(A)}\\ -125\\qquad\\textbf{(B)}\\ -120\\qquad\\textbf{(C)}\\ \\frac{1}{5}\\qquad\\textbf{(D)}\\ \\frac{5}{24}\\qquad\\textbf{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?\n$\\mathrm{(A)}\\ {{{0}}} \\qquad \\mathrm{(B)}\\ {{{1}}} \\qquad \\mathrm{(C)}\\ {{{2}}} \\qquad \\mathrm{(D)}\\ {{{3}}} \\qquad \\mathrm{(E)}\\ {{{4}}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1903", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?\n$\\mathrm{(A)}\\ {{{0}}} \\qquad \\mathrm{(B)}\\ {{{1}}} \\qquad \\mathrm{(C)}\\ {{{2}}} \\qquad \\mathrm{(D)}\\ {{{3}}} \\qquad \\mathrm{(E)}\\ {{{4}}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?\n$\\textbf{(A)}\\ \\frac{1}{21}\\qquad\\textbf{(B)}\\ \\frac{1}{19}\\qquad\\textbf{(C)}\\ \\frac{1}{18}\\qquad\\textbf{(D)}\\ \\frac{1}{2}\\qquad\\textbf{(E)}\\ \\frac{11}{21}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1904", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?\n$\\textbf{(A)}\\ \\frac{1}{21}\\qquad\\textbf{(B)}\\ \\frac{1}{19}\\qquad\\textbf{(C)}\\ \\frac{1}{18}\\qquad\\textbf{(D)}\\ \\frac{1}{2}\\qquad\\textbf{(E)}\\ \\frac{11}{21}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?\n$\\textbf{(A)} \\text{ Bela will always win.} \\qquad \\textbf{(B)} \\text{ Jenn will always win.} \\qquad \\textbf{(C)} \\text{ Bela will win if and only if }n \\text{ is odd.} \\\\ \\textbf{(D)} \\text{ Jenn will win if and only if }n \\text{ is odd.} \\qquad \\textbf{(E)} \\text { Jenn will win if and only if } n>8.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1905", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?\n$\\textbf{(A)} \\text{ Bela will always win.} \\qquad \\textbf{(B)} \\text{ Jenn will always win.} \\qquad \\textbf{(C)} \\text{ Bela will win if and only if }n \\text{ is odd.} \\\\ \\textbf{(D)} \\text{ Jenn will win if and only if }n \\text{ is odd.} \\qquad \\textbf{(E)} \\text { Jenn will win if and only if } n>8.$" + } + }, + { + "question": "Return your final response within \\boxed{}. On a $\\textdollar{10000}$ order a merchant has a choice between three successive discounts of $20$%, $20$%, and $10$% and \nthree successive discounts of $40$%, $5$%, and $5$%. By choosing the better offer, he can save: \n$\\textbf{(A)}\\ \\text{nothing at all}\\qquad\\textbf{(B)}\\ $440\\qquad\\textbf{(C)}\\ $330\\qquad\\textbf{(D)}\\ $345\\qquad\\textbf{(E)}\\ $360$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1906", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On a $\\textdollar{10000}$ order a merchant has a choice between three successive discounts of $20$%, $20$%, and $10$% and \nthree successive discounts of $40$%, $5$%, and $5$%. By choosing the better offer, he can save: \n$\\textbf{(A)}\\ \\text{nothing at all}\\qquad\\textbf{(B)}\\ $440\\qquad\\textbf{(C)}\\ $330\\qquad\\textbf{(D)}\\ $345\\qquad\\textbf{(E)}\\ $360$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a cyclic quadrilateral. Prove that \\[|AB - CD| + |AD - BC| \\geq 2|AC - BD|.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "|AB - CD| + |AD - BC| \\geq 2|AC - BD|", + "index": "Sky-T1_10k_1907", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a cyclic quadrilateral. Prove that \\[|AB - CD| + |AD - BC| \\geq 2|AC - BD|.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?\n$\\textbf{(A)}\\ \\text{ If Lewis did not receive an A, then he got all of the multiple choice questions wrong.} \\\\ \\qquad\\textbf{(B)}\\ \\text{ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.} \\\\ \\qquad\\textbf{(C)}\\ \\text{ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.} \\\\ \\qquad\\textbf{(D)}\\ \\text{ If Lewis received an A, then he got all of the multiple choice questions right.} \\\\ \\qquad\\textbf{(E)}\\ \\text{ If Lewis received an A, then he got at least one of the multiple choice questions right.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1908", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?\n$\\textbf{(A)}\\ \\text{ If Lewis did not receive an A, then he got all of the multiple choice questions wrong.} \\\\ \\qquad\\textbf{(B)}\\ \\text{ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.} \\\\ \\qquad\\textbf{(C)}\\ \\text{ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.} \\\\ \\qquad\\textbf{(D)}\\ \\text{ If Lewis received an A, then he got all of the multiple choice questions right.} \\\\ \\qquad\\textbf{(E)}\\ \\text{ If Lewis received an A, then he got at least one of the multiple choice questions right.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $3(4x+5\\pi)=P$ then $6(8x+10\\pi)=$\n$\\text{(A) } 2P\\quad \\text{(B) } 4P\\quad \\text{(C) } 6P\\quad \\text{(D) } 8P\\quad \\text{(E) } 18P$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1909", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $3(4x+5\\pi)=P$ then $6(8x+10\\pi)=$\n$\\text{(A) } 2P\\quad \\text{(B) } 4P\\quad \\text{(C) } 6P\\quad \\text{(D) } 8P\\quad \\text{(E) } 18P$" + } + }, + { + "question": "Return your final response within \\boxed{}. One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?\n$\\textbf{(A) }2.5\\qquad\\textbf{(B) }3.0\\qquad\\textbf{(C) }3.5\\qquad\\textbf{(D) }4.0\\qquad \\textbf{(E) }4.5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1910", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?\n$\\textbf{(A) }2.5\\qquad\\textbf{(B) }3.0\\qquad\\textbf{(C) }3.5\\qquad\\textbf{(D) }4.0\\qquad \\textbf{(E) }4.5$" + } + }, + { + "question": "Return your final response within \\boxed{}. [Diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) $DB$ of [rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is divided into three segments of length $1$ by [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) lines $L$ and $L'$ that pass through $A$ and $C$ and are [perpendicular](https://artofproblemsolving.com/wiki/index.php/Perpendicular) to $DB$. The area of $ABCD$, rounded to the one decimal place, is \n\n$\\mathrm{(A)\\ } 4.1 \\qquad \\mathrm{(B) \\ }4.2 \\qquad \\mathrm{(C) \\ } 4.3 \\qquad \\mathrm{(D) \\ } 4.4 \\qquad \\mathrm{(E) \\ }4.5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1911", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [Diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) $DB$ of [rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is divided into three segments of length $1$ by [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) lines $L$ and $L'$ that pass through $A$ and $C$ and are [perpendicular](https://artofproblemsolving.com/wiki/index.php/Perpendicular) to $DB$. The area of $ABCD$, rounded to the one decimal place, is \n\n$\\mathrm{(A)\\ } 4.1 \\qquad \\mathrm{(B) \\ }4.2 \\qquad \\mathrm{(C) \\ } 4.3 \\qquad \\mathrm{(D) \\ } 4.4 \\qquad \\mathrm{(E) \\ }4.5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?\n$\\mathrm{(A)}\\ 10\\qquad \\mathrm{(B)}\\ 12\\qquad \\mathrm{(C)}\\ 15\\qquad \\mathrm{(D)}\\ 18\\qquad \\mathrm{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1912", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?\n$\\mathrm{(A)}\\ 10\\qquad \\mathrm{(B)}\\ 12\\qquad \\mathrm{(C)}\\ 15\\qquad \\mathrm{(D)}\\ 18\\qquad \\mathrm{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?\n$\\textbf{(A) } \\frac{1}{64}\\qquad\\textbf{(B) } \\frac{1}{6}\\qquad\\textbf{(C) } \\frac{1}{4}\\qquad\\textbf{(D) } \\frac{5}{16}\\qquad\\textbf{(E) }\\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1913", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?\n$\\textbf{(A) } \\frac{1}{64}\\qquad\\textbf{(B) } \\frac{1}{6}\\qquad\\textbf{(C) } \\frac{1}{4}\\qquad\\textbf{(D) } \\frac{5}{16}\\qquad\\textbf{(E) }\\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 22\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 25\\qquad\\textbf{(E)}\\ 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1914", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 22\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 25\\qquad\\textbf{(E)}\\ 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. Gary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only $2/9$ as much beverage. Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "37", + "index": "Sky-T1_10k_1915", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Gary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only $2/9$ as much beverage. Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. A [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "173", + "index": "Sky-T1_10k_1916", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. For how many positive integer values of $N$ is the expression $\\dfrac{36}{N+2}$ an integer?\n$\\text{(A)}\\ 7 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 10 \\qquad \\text{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1917", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many positive integer values of $N$ is the expression $\\dfrac{36}{N+2}$ an integer?\n$\\text{(A)}\\ 7 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 10 \\qquad \\text{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?\n$\\textbf{(A) }2\\sqrt{3}\\qquad \\textbf{(B) }4\\qquad \\textbf{(C) }3\\sqrt{2}\\qquad \\textbf{(D) }2\\sqrt{5}\\qquad \\textbf{(E) }5\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1918", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?\n$\\textbf{(A) }2\\sqrt{3}\\qquad \\textbf{(B) }4\\qquad \\textbf{(C) }3\\sqrt{2}\\qquad \\textbf{(D) }2\\sqrt{5}\\qquad \\textbf{(E) }5\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. The values of $a$ in the equation: $\\log_{10}(a^2 - 15a) = 2$ are:\n$\\textbf{(A)}\\ \\frac {15\\pm\\sqrt {233}}{2} \\qquad\\textbf{(B)}\\ 20, - 5 \\qquad\\textbf{(C)}\\ \\frac {15 \\pm \\sqrt {305}}{2}$\n$\\textbf{(D)}\\ \\pm20 \\qquad\\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1919", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The values of $a$ in the equation: $\\log_{10}(a^2 - 15a) = 2$ are:\n$\\textbf{(A)}\\ \\frac {15\\pm\\sqrt {233}}{2} \\qquad\\textbf{(B)}\\ 20, - 5 \\qquad\\textbf{(C)}\\ \\frac {15 \\pm \\sqrt {305}}{2}$\n$\\textbf{(D)}\\ \\pm20 \\qquad\\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,36,37\\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?\n$\\textbf{(A) }5 \\qquad \\textbf{(B) }7 \\qquad \\textbf{(C) }8\\qquad \\textbf{(D) }9 \\qquad \\textbf{(E) }10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1920", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,36,37\\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?\n$\\textbf{(A) }5 \\qquad \\textbf{(B) }7 \\qquad \\textbf{(C) }8\\qquad \\textbf{(D) }9 \\qquad \\textbf{(E) }10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?\n[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill((6,0)--(8,0)--(8,26)--(6,26)--cycle,gray); fill((12,0)--(14,0)--(14,26)--(12,26)--cycle,gray); fill((18,0)--(20,0)--(20,26)--(18,26)--cycle,gray); fill((24,0)--(26,0)--(26,26)--(24,26)--cycle,gray); fill((0,0)--(26,0)--(26,2)--(0,2)--cycle,gray); fill((0,12)--(26,12)--(26,14)--(0,14)--cycle,gray); fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); [/asy]\n$\\textbf{(A)}\\ 26\\qquad\\textbf{(B)}\\ 28\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 32\\qquad\\textbf{(E)}\\ 34$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1921", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?\n[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill((6,0)--(8,0)--(8,26)--(6,26)--cycle,gray); fill((12,0)--(14,0)--(14,26)--(12,26)--cycle,gray); fill((18,0)--(20,0)--(20,26)--(18,26)--cycle,gray); fill((24,0)--(26,0)--(26,26)--(24,26)--cycle,gray); fill((0,0)--(26,0)--(26,2)--(0,2)--cycle,gray); fill((0,12)--(26,12)--(26,14)--(0,14)--cycle,gray); fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); [/asy]\n$\\textbf{(A)}\\ 26\\qquad\\textbf{(B)}\\ 28\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 32\\qquad\\textbf{(E)}\\ 34$" + } + }, + { + "question": "Return your final response within \\boxed{}. An unfair coin has probability $p$ of coming up heads on a single toss. \nLet $w$ be the probability that, in $5$ independent toss of this coin, \nheads come up exactly $3$ times. If $w = 144 / 625$, then \n$\\textbf{(A)}\\ p\\text{ must be }\\tfrac{2}{5}\\qquad \\textbf{(B)}\\ p\\text{ must be }\\tfrac{3}{5}\\qquad\\\\ \\textbf{(C)}\\ p\\text{ must be greater than }\\tfrac{3}{5}\\qquad \\textbf{(D)}\\ p\\text{ is not uniquely determined}\\qquad\\\\ \\textbf{(E)}\\ \\text{there is no value of } p \\text{ for which }w =\\tfrac{144}{625}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1922", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An unfair coin has probability $p$ of coming up heads on a single toss. \nLet $w$ be the probability that, in $5$ independent toss of this coin, \nheads come up exactly $3$ times. If $w = 144 / 625$, then \n$\\textbf{(A)}\\ p\\text{ must be }\\tfrac{2}{5}\\qquad \\textbf{(B)}\\ p\\text{ must be }\\tfrac{3}{5}\\qquad\\\\ \\textbf{(C)}\\ p\\text{ must be greater than }\\tfrac{3}{5}\\qquad \\textbf{(D)}\\ p\\text{ is not uniquely determined}\\qquad\\\\ \\textbf{(E)}\\ \\text{there is no value of } p \\text{ for which }w =\\tfrac{144}{625}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The angle formed by the hands of a clock at $2:15$ is:\n$\\textbf{(A)}\\ 30^\\circ \\qquad \\textbf{(B)}\\ 27\\frac{1}{2}^\\circ\\qquad \\textbf{(C)}\\ 157\\frac{1}{2}^\\circ\\qquad \\textbf{(D)}\\ 172\\frac{1}{2}^\\circ\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1923", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The angle formed by the hands of a clock at $2:15$ is:\n$\\textbf{(A)}\\ 30^\\circ \\qquad \\textbf{(B)}\\ 27\\frac{1}{2}^\\circ\\qquad \\textbf{(C)}\\ 157\\frac{1}{2}^\\circ\\qquad \\textbf{(D)}\\ 172\\frac{1}{2}^\\circ\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the smallest positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "44", + "index": "Sky-T1_10k_1924", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the smallest positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms." + } + }, + { + "question": "Return your final response within \\boxed{}. For every $n$ the sum of n terms of an arithmetic progression is $2n + 3n^2$. The $r$th term is: \n$\\textbf{(A)}\\ 3r^2 \\qquad \\textbf{(B) }\\ 3r^2 + 2r \\qquad \\textbf{(C) }\\ 6r - 1 \\qquad \\textbf{(D) }\\ 5r + 5 \\qquad \\textbf{(E) }\\ 6r+2\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1925", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For every $n$ the sum of n terms of an arithmetic progression is $2n + 3n^2$. The $r$th term is: \n$\\textbf{(A)}\\ 3r^2 \\qquad \\textbf{(B) }\\ 3r^2 + 2r \\qquad \\textbf{(C) }\\ 6r - 1 \\qquad \\textbf{(D) }\\ 5r + 5 \\qquad \\textbf{(E) }\\ 6r+2\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade? \n$\\textbf{(A) } 129\\qquad \\textbf{(B) } 137\\qquad \\textbf{(C) } 174\\qquad \\textbf{(D) } 233\\qquad \\textbf{(E) } 411$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1926", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade? \n$\\textbf{(A) } 129\\qquad \\textbf{(B) } 137\\qquad \\textbf{(C) } 174\\qquad \\textbf{(D) } 233\\qquad \\textbf{(E) } 411$" + } + }, + { + "question": "Return your final response within \\boxed{}. Figure $ABCD$ is a trapezoid with $AB||DC$, $AB=5$, $BC=3\\sqrt{2}$, $\\angle BCD=45^\\circ$, and $\\angle CDA=60^\\circ$. The length of $DC$ is\n$\\mathrm{(A) \\ }7+\\frac{2}{3}\\sqrt{3} \\qquad \\mathrm{(B) \\ }8 \\qquad \\mathrm{(C) \\ } 9 \\frac{1}{2} \\qquad \\mathrm{(D) \\ }8+\\sqrt{3} \\qquad \\mathrm{(E) \\ } 8+3\\sqrt{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1927", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Figure $ABCD$ is a trapezoid with $AB||DC$, $AB=5$, $BC=3\\sqrt{2}$, $\\angle BCD=45^\\circ$, and $\\angle CDA=60^\\circ$. The length of $DC$ is\n$\\mathrm{(A) \\ }7+\\frac{2}{3}\\sqrt{3} \\qquad \\mathrm{(B) \\ }8 \\qquad \\mathrm{(C) \\ } 9 \\frac{1}{2} \\qquad \\mathrm{(D) \\ }8+\\sqrt{3} \\qquad \\mathrm{(E) \\ } 8+3\\sqrt{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A set of $n$ numbers has the sum $s$. Each number of the set is increased by $20$, then multiplied by $5$, and then decreased by $20$. The sum of the numbers in the new set thus obtained is:\n$\\textbf{(A)}\\ s \\plus{} 20n\\qquad \n\\textbf{(B)}\\ 5s \\plus{} 80n\\qquad \n\\textbf{(C)}\\ s\\qquad \n\\textbf{(D)}\\ 5s\\qquad \n\\textbf{(E)}\\ 5s \\plus{} 4n$ (Error compiling LaTeX. Unknown error_msg)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1928", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A set of $n$ numbers has the sum $s$. Each number of the set is increased by $20$, then multiplied by $5$, and then decreased by $20$. The sum of the numbers in the new set thus obtained is:\n$\\textbf{(A)}\\ s \\plus{} 20n\\qquad \n\\textbf{(B)}\\ 5s \\plus{} 80n\\qquad \n\\textbf{(C)}\\ s\\qquad \n\\textbf{(D)}\\ 5s\\qquad \n\\textbf{(E)}\\ 5s \\plus{} 4n$ (Error compiling LaTeX. Unknown error_msg)" + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is \n\n$\\mathrm{(A) \\ \\text{500 thousand} } \\qquad \\mathrm{(B) \\ \\text{5 million} } \\qquad \\mathrm{(C) \\ \\text{50 million} } \\qquad \\mathrm{(D) \\ \\text{500 million} } \\qquad \\mathrm{(E) \\ \\text{5 billion} }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1929", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is \n\n$\\mathrm{(A) \\ \\text{500 thousand} } \\qquad \\mathrm{(B) \\ \\text{5 million} } \\qquad \\mathrm{(C) \\ \\text{50 million} } \\qquad \\mathrm{(D) \\ \\text{500 million} } \\qquad \\mathrm{(E) \\ \\text{5 billion} }$" + } + }, + { + "question": "Return your final response within \\boxed{}. The teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$, and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_i$ wins is $\\frac{i}{i+j}$, and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "781", + "index": "Sky-T1_10k_1930", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$, and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_i$ wins is $\\frac{i}{i+j}$, and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$." + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?\n$\\textbf{(A)}\\ 8\\sqrt{3}\\qquad\\textbf{(B)}\\ 10\\sqrt{2}\\qquad\\textbf{(C)}\\ 16\\sqrt{3}\\qquad\\textbf{(D)}\\ 20\\sqrt{2}\\qquad\\textbf{(E)}\\ 40\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1931", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?\n$\\textbf{(A)}\\ 8\\sqrt{3}\\qquad\\textbf{(B)}\\ 10\\sqrt{2}\\qquad\\textbf{(C)}\\ 16\\sqrt{3}\\qquad\\textbf{(D)}\\ 20\\sqrt{2}\\qquad\\textbf{(E)}\\ 40\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ten-letter code $\\text{BEST OF LUCK}$ represents the ten digits $0-9$, in order. What 4-digit number is represented by the code word $\\text{CLUE}$?\n$\\textbf{(A)}\\ 8671 \\qquad \\textbf{(B)}\\ 8672 \\qquad \\textbf{(C)}\\ 9781 \\qquad \\textbf{(D)}\\ 9782 \\qquad \\textbf{(E)}\\ 9872$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1932", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ten-letter code $\\text{BEST OF LUCK}$ represents the ten digits $0-9$, in order. What 4-digit number is represented by the code word $\\text{CLUE}$?\n$\\textbf{(A)}\\ 8671 \\qquad \\textbf{(B)}\\ 8672 \\qquad \\textbf{(C)}\\ 9781 \\qquad \\textbf{(D)}\\ 9782 \\qquad \\textbf{(E)}\\ 9872$" + } + }, + { + "question": "Return your final response within \\boxed{}. The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\\tfrac{3}{2}$ and center $(0,\\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?\n\n$\\textbf{(A) } \\dfrac{4\\pi}{5} \\qquad\\textbf{(B) } \\dfrac{9\\pi}{8} \\qquad\\textbf{(C) } \\dfrac{4\\pi}{3} \\qquad\\textbf{(D) } \\dfrac{7\\pi}{5} \\qquad\\textbf{(E) } \\dfrac{3\\pi}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1933", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\\tfrac{3}{2}$ and center $(0,\\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?\n\n$\\textbf{(A) } \\dfrac{4\\pi}{5} \\qquad\\textbf{(B) } \\dfrac{9\\pi}{8} \\qquad\\textbf{(C) } \\dfrac{4\\pi}{3} \\qquad\\textbf{(D) } \\dfrac{7\\pi}{5} \\qquad\\textbf{(E) } \\dfrac{3\\pi}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?\n$\\textbf{(A)}\\ 30\\%\\qquad\\textbf{(B)}\\ 40\\%\\qquad\\textbf{(C)}\\ 50\\%\\qquad\\textbf{(D)}\\ 60\\%\\qquad\\textbf{(E)}\\ 70\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1934", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?\n$\\textbf{(A)}\\ 30\\%\\qquad\\textbf{(B)}\\ 40\\%\\qquad\\textbf{(C)}\\ 50\\%\\qquad\\textbf{(D)}\\ 60\\%\\qquad\\textbf{(E)}\\ 70\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $n\\heartsuit m=n^3m^2$, what is $\\frac{2\\heartsuit 4}{4\\heartsuit 2}$?\n$\\textbf{(A)}\\ \\frac{1}{4}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ 1\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1935", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $n\\heartsuit m=n^3m^2$, what is $\\frac{2\\heartsuit 4}{4\\heartsuit 2}$?\n$\\textbf{(A)}\\ \\frac{1}{4}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ 1\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is closest to $\\sqrt{65}-\\sqrt{63}$?\n$\\textbf{(A)}\\ .12 \\qquad \\textbf{(B)}\\ .13 \\qquad \\textbf{(C)}\\ .14 \\qquad \\textbf{(D)}\\ .15 \\qquad \\textbf{(E)}\\ .16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1936", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is closest to $\\sqrt{65}-\\sqrt{63}$?\n$\\textbf{(A)}\\ .12 \\qquad \\textbf{(B)}\\ .13 \\qquad \\textbf{(C)}\\ .14 \\qquad \\textbf{(D)}\\ .15 \\qquad \\textbf{(E)}\\ .16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four siblings ordered an extra large pizza. Alex ate $\\frac15$, Beth $\\frac13$, and Cyril $\\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?\n$\\textbf{(A) } \\text{Alex, Beth, Cyril, Dan}$ \n$\\textbf{(B) } \\text{Beth, Cyril, Alex, Dan}$ \n$\\textbf{(C) } \\text{Beth, Cyril, Dan, Alex}$ \n$\\textbf{(D) } \\text{Beth, Dan, Cyril, Alex}$ \n$\\textbf{(E) } \\text{Dan, Beth, Cyril, Alex}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1937", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four siblings ordered an extra large pizza. Alex ate $\\frac15$, Beth $\\frac13$, and Cyril $\\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?\n$\\textbf{(A) } \\text{Alex, Beth, Cyril, Dan}$ \n$\\textbf{(B) } \\text{Beth, Cyril, Alex, Dan}$ \n$\\textbf{(C) } \\text{Beth, Cyril, Dan, Alex}$ \n$\\textbf{(D) } \\text{Beth, Dan, Cyril, Alex}$ \n$\\textbf{(E) } \\text{Dan, Beth, Cyril, Alex}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $R$ be a rectangle. How many circles in the plane of $R$ have a diameter both of whose endpoints are vertices of $R$?\n$\\mathrm{(A) \\ }1 \\qquad \\mathrm{(B) \\ }2 \\qquad \\mathrm{(C) \\ }4 \\qquad \\mathrm{(D) \\ }5 \\qquad \\mathrm{(E) \\ }6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1938", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $R$ be a rectangle. How many circles in the plane of $R$ have a diameter both of whose endpoints are vertices of $R$?\n$\\mathrm{(A) \\ }1 \\qquad \\mathrm{(B) \\ }2 \\qquad \\mathrm{(C) \\ }4 \\qquad \\mathrm{(D) \\ }5 \\qquad \\mathrm{(E) \\ }6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the equations $x^2+kx+6=0$ and $x^2-kx+6=0$. If, when the roots of the equation are suitably listed, \neach root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals:\n$\\textbf{(A)}\\ 5 \\qquad \\textbf{(B)}\\ -5 \\qquad \\textbf{(C)}\\ 7 \\qquad \\textbf{(D)}\\ -7 \\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1939", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the equations $x^2+kx+6=0$ and $x^2-kx+6=0$. If, when the roots of the equation are suitably listed, \neach root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals:\n$\\textbf{(A)}\\ 5 \\qquad \\textbf{(B)}\\ -5 \\qquad \\textbf{(C)}\\ 7 \\qquad \\textbf{(D)}\\ -7 \\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?\n$\\textbf{(A)}\\ 4 \\qquad\\textbf{(B)}\\ 5 \\qquad\\textbf{(C)}\\ 6 \\qquad\\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 13$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1940", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?\n$\\textbf{(A)}\\ 4 \\qquad\\textbf{(B)}\\ 5 \\qquad\\textbf{(C)}\\ 6 \\qquad\\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 13$" + } + }, + { + "question": "Return your final response within \\boxed{}. A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?\n$\\textbf{(A)}\\ \\frac{1}{36} \\qquad \\textbf{(B)}\\ \\frac{1}{12} \\qquad \\textbf{(C)}\\ \\frac{1}{6} \\qquad \\textbf{(D)}\\ \\frac{1}{4} \\qquad \\textbf{(E)}\\ \\frac{5}{18}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1941", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?\n$\\textbf{(A)}\\ \\frac{1}{36} \\qquad \\textbf{(B)}\\ \\frac{1}{12} \\qquad \\textbf{(C)}\\ \\frac{1}{6} \\qquad \\textbf{(D)}\\ \\frac{1}{4} \\qquad \\textbf{(E)}\\ \\frac{5}{18}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Niki usually leaves her cell phone on. If her cell phone is on but\nshe is not actually using it, the battery will last for $24$ hours. If\nshe is using it constantly, the battery will last for only $3$ hours.\nSince the last recharge, her phone has been on $9$ hours, and during\nthat time she has used it for $60$ minutes. If she doesn’t use it any\nmore but leaves the phone on, how many more hours will the battery last?\n$\\textbf{(A)}\\ 7 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 11 \\qquad \\textbf{(D)}\\ 14 \\qquad \\textbf{(E)}\\ 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1942", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Niki usually leaves her cell phone on. If her cell phone is on but\nshe is not actually using it, the battery will last for $24$ hours. If\nshe is using it constantly, the battery will last for only $3$ hours.\nSince the last recharge, her phone has been on $9$ hours, and during\nthat time she has used it for $60$ minutes. If she doesn’t use it any\nmore but leaves the phone on, how many more hours will the battery last?\n$\\textbf{(A)}\\ 7 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 11 \\qquad \\textbf{(D)}\\ 14 \\qquad \\textbf{(E)}\\ 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. A shop advertises everything is \"half price in today's sale.\" In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price?\n$\\textbf{(A)}\\hspace{.05in}10\\qquad\\textbf{(B)}\\hspace{.05in}33\\qquad\\textbf{(C)}\\hspace{.05in}40\\qquad\\textbf{(D)}\\hspace{.05in}60\\qquad\\textbf{(E)}\\hspace{.05in}70$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1943", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A shop advertises everything is \"half price in today's sale.\" In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price?\n$\\textbf{(A)}\\hspace{.05in}10\\qquad\\textbf{(B)}\\hspace{.05in}33\\qquad\\textbf{(C)}\\hspace{.05in}40\\qquad\\textbf{(D)}\\hspace{.05in}60\\qquad\\textbf{(E)}\\hspace{.05in}70$" + } + }, + { + "question": "Return your final response within \\boxed{}. For a given value of $k$ the product of the roots of $x^2-3kx+2k^2-1=0$\nis $7$. The roots may be characterized as:\n$\\textbf{(A) }\\text{integral and positive} \\qquad\\textbf{(B) }\\text{integral and negative} \\qquad \\\\ \\textbf{(C) }\\text{rational, but not integral} \\qquad\\textbf{(D) }\\text{irrational} \\qquad\\textbf{(E) } \\text{imaginary}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1944", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a given value of $k$ the product of the roots of $x^2-3kx+2k^2-1=0$\nis $7$. The roots may be characterized as:\n$\\textbf{(A) }\\text{integral and positive} \\qquad\\textbf{(B) }\\text{integral and negative} \\qquad \\\\ \\textbf{(C) }\\text{rational, but not integral} \\qquad\\textbf{(D) }\\text{irrational} \\qquad\\textbf{(E) } \\text{imaginary}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose?\n$\\text{(A)}\\ 24\\% \\qquad \\text{(B)}\\ 27\\% \\qquad \\text{(C)}\\ 36\\% \\qquad \\text{(D)}\\ 45\\% \\qquad \\text{(E)}\\ 73\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1945", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose?\n$\\text{(A)}\\ 24\\% \\qquad \\text{(B)}\\ 27\\% \\qquad \\text{(C)}\\ 36\\% \\qquad \\text{(D)}\\ 45\\% \\qquad \\text{(E)}\\ 73\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "All points in the plane are assigned the same real number.", + "index": "Sky-T1_10k_1946", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number." + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following numbers has the largest prime factor?\n$\\text{(A)}\\ 39\\qquad\\text{(B)}\\ 51\\qquad\\text{(C)}\\ 77\\qquad\\text{(D)}\\ 91\\qquad\\text{(E)}\\ 121$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1947", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following numbers has the largest prime factor?\n$\\text{(A)}\\ 39\\qquad\\text{(B)}\\ 51\\qquad\\text{(C)}\\ 77\\qquad\\text{(D)}\\ 91\\qquad\\text{(E)}\\ 121$" + } + }, + { + "question": "Return your final response within \\boxed{}. In an isosceles trapezoid, the parallel bases have lengths $\\log 3$ and $\\log 192$, and the altitude to these bases has length $\\log 16$. The perimeter of the trapezoid can be written in the form $\\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "18", + "index": "Sky-T1_10k_1948", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In an isosceles trapezoid, the parallel bases have lengths $\\log 3$ and $\\log 192$, and the altitude to these bases has length $\\log 16$. The perimeter of the trapezoid can be written in the form $\\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$." + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair $(a,b)$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1949", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair $(a,b)$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?\n$\\mathrm{(A)}\\ 10 \\qquad \\mathrm{(B)}\\ 11 \\qquad \\mathrm{(C)}\\ 12 \\qquad \\mathrm{(D)}\\ 13 \\qquad \\mathrm{(E)}\\ 14$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1950", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?\n$\\mathrm{(A)}\\ 10 \\qquad \\mathrm{(B)}\\ 11 \\qquad \\mathrm{(C)}\\ 12 \\qquad \\mathrm{(D)}\\ 13 \\qquad \\mathrm{(E)}\\ 14$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number $\\sqrt {2}$ is equal to: \n$\\textbf{(A)}\\ \\text{a rational fraction} \\qquad \\textbf{(B)}\\ \\text{a finite decimal} \\qquad \\textbf{(C)}\\ 1.41421 \\\\ \\textbf{(D)}\\ \\text{an infinite repeating decimal} \\qquad \\textbf{(E)}\\ \\text{an infinite non - repeating decimal}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1951", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number $\\sqrt {2}$ is equal to: \n$\\textbf{(A)}\\ \\text{a rational fraction} \\qquad \\textbf{(B)}\\ \\text{a finite decimal} \\qquad \\textbf{(C)}\\ 1.41421 \\\\ \\textbf{(D)}\\ \\text{an infinite repeating decimal} \\qquad \\textbf{(E)}\\ \\text{an infinite non - repeating decimal}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC, \\overline{CA} = \\overline{CB}$. On $CB$ square $BCDE$ is constructed away from the triangle. If $x$ is the number of degrees in $\\angle DAB$, then\n$\\textbf{(A)}\\ x\\text{ depends upon }\\triangle ABC \\qquad \\textbf{(B)}\\ x\\text{ is independent of the triangle} \\\\ \\textbf{(C)}\\ x\\text{ may equal }\\angle CAD \\qquad \\\\ \\textbf{(D)}\\ x\\text{ can never equal }\\angle CAB \\\\ \\textbf{(E)}\\ x\\text{ is greater than }45^{\\circ}\\text{ but less than }90^{\\circ}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1952", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC, \\overline{CA} = \\overline{CB}$. On $CB$ square $BCDE$ is constructed away from the triangle. If $x$ is the number of degrees in $\\angle DAB$, then\n$\\textbf{(A)}\\ x\\text{ depends upon }\\triangle ABC \\qquad \\textbf{(B)}\\ x\\text{ is independent of the triangle} \\\\ \\textbf{(C)}\\ x\\text{ may equal }\\angle CAD \\qquad \\\\ \\textbf{(D)}\\ x\\text{ can never equal }\\angle CAB \\\\ \\textbf{(E)}\\ x\\text{ is greater than }45^{\\circ}\\text{ but less than }90^{\\circ}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the [distance](https://artofproblemsolving.com/wiki/index.php/Distance) in feet between the start of the walkway and the middle person.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "52", + "index": "Sky-T1_10k_1953", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the [distance](https://artofproblemsolving.com/wiki/index.php/Distance) in feet between the start of the walkway and the middle person." + } + }, + { + "question": "Return your final response within \\boxed{}. Positive real numbers $x \\neq 1$ and $y \\neq 1$ satisfy $\\log_2{x} = \\log_y{16}$ and $xy = 64$. What is $(\\log_2{\\tfrac{x}{y}})^2$?\n$\\textbf{(A) } \\frac{25}{2} \\qquad\\textbf{(B) } 20 \\qquad\\textbf{(C) } \\frac{45}{2} \\qquad\\textbf{(D) } 25 \\qquad\\textbf{(E) } 32$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1954", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Positive real numbers $x \\neq 1$ and $y \\neq 1$ satisfy $\\log_2{x} = \\log_y{16}$ and $xy = 64$. What is $(\\log_2{\\tfrac{x}{y}})^2$?\n$\\textbf{(A) } \\frac{25}{2} \\qquad\\textbf{(B) } 20 \\qquad\\textbf{(C) } \\frac{45}{2} \\qquad\\textbf{(D) } 25 \\qquad\\textbf{(E) } 32$" + } + }, + { + "question": "Return your final response within \\boxed{}. In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is:\n$\\text{(A) } 16n \\quad \\text{(B) } 18n \\quad \\text{(C) } 21n \\quad \\text{(D) } 24n \\quad \\text{(E) } 27n$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1955", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is:\n$\\text{(A) } 16n \\quad \\text{(B) } 18n \\quad \\text{(C) } 21n \\quad \\text{(D) } 24n \\quad \\text{(E) } 27n$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are $n$ students standing in a circle, one behind the\nother. The students have heights $h_1 < h_2 < \\ldots < h_n$. If a\nstudent with height $h_k$ is standing directly behind a student\nwith height $h_{k-2}$ or less, the two students are permitted to\nswitch places. Prove that it is not possible to make more than\n$\\binom{n}{3}$ such switches before reaching a position in which\nno further switches are possible.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\binom{n}{3}", + "index": "Sky-T1_10k_1956", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are $n$ students standing in a circle, one behind the\nother. The students have heights $h_1 < h_2 < \\ldots < h_n$. If a\nstudent with height $h_k$ is standing directly behind a student\nwith height $h_{k-2}$ or less, the two students are permitted to\nswitch places. Prove that it is not possible to make more than\n$\\binom{n}{3}$ such switches before reaching a position in which\nno further switches are possible." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that\n\\[\\left\\{ \\dfrac{ra}{p} \\right\\} + \\left\\{ \\dfrac{rb}{p} \\right\\} + \\left\\{ \\dfrac{rc}{p} \\right\\} + \\left\\{ \\dfrac{rd}{p} \\right\\} = 2\\]\nfor any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$.\n(Note: $\\{x\\} = x - \\lfloor x \\rfloor$ denotes the fractional part of $x$.)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "At least two of the numbers a+b, a+c, a+d, b+c, b+d, c+d are divisible by p.", + "index": "Sky-T1_10k_1957", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that\n\\[\\left\\{ \\dfrac{ra}{p} \\right\\} + \\left\\{ \\dfrac{rb}{p} \\right\\} + \\left\\{ \\dfrac{rc}{p} \\right\\} + \\left\\{ \\dfrac{rd}{p} \\right\\} = 2\\]\nfor any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$.\n(Note: $\\{x\\} = x - \\lfloor x \\rfloor$ denotes the fractional part of $x$.)" + } + }, + { + "question": "Return your final response within \\boxed{}. [asy] size((400)); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1)); draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1)); draw((10,0)--(10,5)--(11,7), linewidth(1)); draw((15,0)--(15,5)--(16,7), linewidth(1)); draw((20,0)--(20,5)--(21,7), linewidth(1)); draw((0,5)--(1,7)--(6,7), linewidth(1)); draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1)); draw((8.5,7)--(9.5,9), linewidth(1)); draw((13.5,7)--(14.5,9), linewidth(1)); draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1)); draw((12,9)--(13,11), linewidth(1)); draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1)); draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed); draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed); draw((35,0)--(35,5)--(36,7), dashed); draw((40,0)--(40,5)--(41,7), dashed); draw((45,0)--(45,5)--(46,7), dashed); draw((25,5)--(26,7)--(31,7), dashed); draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed); draw((33.5,7)--(34.5,9), dashed); draw((38.5,7)--(39.5,9), dashed); draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed); draw((37,9)--(38,11), dashed); draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed); draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed); draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed); draw((60,0)--(60,5)--(61,7), dashed); draw((65,0)--(65,5)--(66,7), dashed); draw((70,0)--(70,5)--(71,7), dashed); draw((50,5)--(51,7)--(56,7), dashed); draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed); draw((58.5,7)--(59.5,9), dashed); draw((63.5,7)--(64.5,9), dashed); draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed); draw((62,9)--(63,11), dashed); draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed); draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed); draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed); draw((85,0)--(85,5)--(86,7), dashed); draw((90,0)--(90,5)--(91,7), dashed); draw((95,0)--(95,5)--(96,7), dashed); draw((75,5)--(76,7)--(81,7), dashed); draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed); draw((83.5,7)--(84.5,9), dashed); draw((88.5,7)--(89.5,9), dashed); draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed); draw((87,9)--(88,11), dashed); draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed); draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1)); draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1)); draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1)); draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1)); draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1)); draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed); draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed); draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed); draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed); draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed); draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed); draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed); draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed); draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed); draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed); draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1)); draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1)); draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1)); draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed); draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed); draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed); draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));[/asy]\nTwenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.\n$\\text{(A) } 55\\quad \\text{(B) } 83\\quad \\text{(C) } 114\\quad \\text{(D) } 137\\quad \\text{(E) } 144$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1958", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [asy] size((400)); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1)); draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1)); draw((10,0)--(10,5)--(11,7), linewidth(1)); draw((15,0)--(15,5)--(16,7), linewidth(1)); draw((20,0)--(20,5)--(21,7), linewidth(1)); draw((0,5)--(1,7)--(6,7), linewidth(1)); draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1)); draw((8.5,7)--(9.5,9), linewidth(1)); draw((13.5,7)--(14.5,9), linewidth(1)); draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1)); draw((12,9)--(13,11), linewidth(1)); draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1)); draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed); draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed); draw((35,0)--(35,5)--(36,7), dashed); draw((40,0)--(40,5)--(41,7), dashed); draw((45,0)--(45,5)--(46,7), dashed); draw((25,5)--(26,7)--(31,7), dashed); draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed); draw((33.5,7)--(34.5,9), dashed); draw((38.5,7)--(39.5,9), dashed); draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed); draw((37,9)--(38,11), dashed); draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed); draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed); draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed); draw((60,0)--(60,5)--(61,7), dashed); draw((65,0)--(65,5)--(66,7), dashed); draw((70,0)--(70,5)--(71,7), dashed); draw((50,5)--(51,7)--(56,7), dashed); draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed); draw((58.5,7)--(59.5,9), dashed); draw((63.5,7)--(64.5,9), dashed); draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed); draw((62,9)--(63,11), dashed); draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed); draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed); draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed); draw((85,0)--(85,5)--(86,7), dashed); draw((90,0)--(90,5)--(91,7), dashed); draw((95,0)--(95,5)--(96,7), dashed); draw((75,5)--(76,7)--(81,7), dashed); draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed); draw((83.5,7)--(84.5,9), dashed); draw((88.5,7)--(89.5,9), dashed); draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed); draw((87,9)--(88,11), dashed); draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed); draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1)); draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1)); draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1)); draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1)); draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1)); draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed); draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed); draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed); draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed); draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed); draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed); draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed); draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed); draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed); draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed); draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1)); draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1)); draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1)); draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed); draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed); draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed); draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));[/asy]\nTwenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.\n$\\text{(A) } 55\\quad \\text{(B) } 83\\quad \\text{(C) } 114\\quad \\text{(D) } 137\\quad \\text{(E) } 144$" + } + }, + { + "question": "Return your final response within \\boxed{}. If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount:\n$\\textbf{(A)}\\ {\\pm 2ax + a^2}\\qquad \\textbf{(B)}\\ {2ax \\pm a^2}\\qquad \\textbf{(C)}\\ {\\pm a^2 - 3} \\qquad \\textbf{(D)}\\ {(x + a)^2 - 3}\\qquad\\\\ \\textbf{(E)}\\ {(x - a)^2 - 3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1959", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount:\n$\\textbf{(A)}\\ {\\pm 2ax + a^2}\\qquad \\textbf{(B)}\\ {2ax \\pm a^2}\\qquad \\textbf{(C)}\\ {\\pm a^2 - 3} \\qquad \\textbf{(D)}\\ {(x + a)^2 - 3}\\qquad\\\\ \\textbf{(E)}\\ {(x - a)^2 - 3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning.\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 1\\frac{1}{2} \\qquad \\text{(C)}\\ 2 \\qquad \\text{(D)}\\ 2\\frac{1}{2} \\qquad \\text{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1960", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning.\n$\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 1\\frac{1}{2} \\qquad \\text{(C)}\\ 2 \\qquad \\text{(D)}\\ 2\\frac{1}{2} \\qquad \\text{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a parallelogram and let $\\overrightarrow{AA^\\prime}$, $\\overrightarrow{BB^\\prime}$, $\\overrightarrow{CC^\\prime}$, and $\\overrightarrow{DD^\\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA^{\\prime} = 10$, $BB^{\\prime}= 8$, $CC^\\prime = 18$, and $DD^\\prime = 22$ and $M$ and $N$ are the midpoints of $A^{\\prime} C^{\\prime}$ and $B^{\\prime}D^{\\prime}$, respectively, then $MN =$\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1961", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a parallelogram and let $\\overrightarrow{AA^\\prime}$, $\\overrightarrow{BB^\\prime}$, $\\overrightarrow{CC^\\prime}$, and $\\overrightarrow{DD^\\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA^{\\prime} = 10$, $BB^{\\prime}= 8$, $CC^\\prime = 18$, and $DD^\\prime = 22$ and $M$ and $N$ are the midpoints of $A^{\\prime} C^{\\prime}$ and $B^{\\prime}D^{\\prime}$, respectively, then $MN =$\n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qquad\\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. A calculator has a key that replaces the displayed entry with its [square](https://artofproblemsolving.com/wiki/index.php/Perfect_square), and another key which replaces the displayed entry with its [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal). Let $y$ be the final result when one starts with a number $x\\not=0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then $y$ equals\n$\\mathrm{(A) \\ }x^{((-2)^n)} \\qquad \\mathrm{(B) \\ }x^{2n} \\qquad \\mathrm{(C) \\ } x^{-2n} \\qquad \\mathrm{(D) \\ }x^{-(2^n)} \\qquad \\mathrm{(E) \\ } x^{((-1)^n2n)}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1962", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A calculator has a key that replaces the displayed entry with its [square](https://artofproblemsolving.com/wiki/index.php/Perfect_square), and another key which replaces the displayed entry with its [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal). Let $y$ be the final result when one starts with a number $x\\not=0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then $y$ equals\n$\\mathrm{(A) \\ }x^{((-2)^n)} \\qquad \\mathrm{(B) \\ }x^{2n} \\qquad \\mathrm{(C) \\ } x^{-2n} \\qquad \\mathrm{(D) \\ }x^{-(2^n)} \\qquad \\mathrm{(E) \\ } x^{((-1)^n2n)}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum $\\sqrt[3] {5+2\\sqrt{13}}+\\sqrt[3]{5-2\\sqrt{13}}$ equals \n$\\text{(A)} \\ \\frac 32 \\qquad \\text{(B)} \\ \\frac{\\sqrt[3]{65}}{4} \\qquad \\text{(C)} \\ \\frac{1+\\sqrt[6]{13}}{2} \\qquad \\text{(D)}\\ \\sqrt[3]{2}\\qquad \\text{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1963", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum $\\sqrt[3] {5+2\\sqrt{13}}+\\sqrt[3]{5-2\\sqrt{13}}$ equals \n$\\text{(A)} \\ \\frac 32 \\qquad \\text{(B)} \\ \\frac{\\sqrt[3]{65}}{4} \\qquad \\text{(C)} \\ \\frac{1+\\sqrt[6]{13}}{2} \\qquad \\text{(D)}\\ \\sqrt[3]{2}\\qquad \\text{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation\n$\\textbf{(A) }r^2+r+1=0\\qquad \\textbf{(B) }r^2-r+1=0\\qquad \\textbf{(C) }r^4+r^2-1=0\\qquad\\\\ \\textbf{(D) }(r+1)^4+r=0\\qquad \\textbf{(E) }(r-1)^4+r=0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1964", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation\n$\\textbf{(A) }r^2+r+1=0\\qquad \\textbf{(B) }r^2-r+1=0\\qquad \\textbf{(C) }r^4+r^2-1=0\\qquad\\\\ \\textbf{(D) }(r+1)^4+r=0\\qquad \\textbf{(E) }(r-1)^4+r=0$" + } + }, + { + "question": "Return your final response within \\boxed{}. When a student multiplied the number $66$ by the repeating decimal, \n\\[\\underline{1}.\\underline{a} \\ \\underline{b} \\ \\underline{a} \\ \\underline{b}\\ldots=\\underline{1}.\\overline{\\underline{a} \\ \\underline{b}},\\] \nwhere $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\\underline{1}.\\underline{a} \\ \\underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $\\underline{a} \\ \\underline{b}?$\n$\\textbf{(A) }15 \\qquad \\textbf{(B) }30 \\qquad \\textbf{(C) }45 \\qquad \\textbf{(D) }60 \\qquad \\textbf{(E) }75$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1965", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When a student multiplied the number $66$ by the repeating decimal, \n\\[\\underline{1}.\\underline{a} \\ \\underline{b} \\ \\underline{a} \\ \\underline{b}\\ldots=\\underline{1}.\\overline{\\underline{a} \\ \\underline{b}},\\] \nwhere $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\\underline{1}.\\underline{a} \\ \\underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $\\underline{a} \\ \\underline{b}?$\n$\\textbf{(A) }15 \\qquad \\textbf{(B) }30 \\qquad \\textbf{(C) }45 \\qquad \\textbf{(D) }60 \\qquad \\textbf{(E) }75$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1966", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Bricklayer Brenda takes $9$ hours to build a chimney alone, and bricklayer Brandon takes $10$ hours to build it alone. When they work together, they talk a lot, and their combined output decreases by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chimney?\n$\\mathrm{(A)}\\ 500\\qquad\\mathrm{(B)}\\ 900\\qquad\\mathrm{(C)}\\ 950\\qquad\\mathrm{(D)}\\ 1000\\qquad\\mathrm{(E)}\\ 1900$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1967", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Bricklayer Brenda takes $9$ hours to build a chimney alone, and bricklayer Brandon takes $10$ hours to build it alone. When they work together, they talk a lot, and their combined output decreases by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chimney?\n$\\mathrm{(A)}\\ 500\\qquad\\mathrm{(B)}\\ 900\\qquad\\mathrm{(C)}\\ 950\\qquad\\mathrm{(D)}\\ 1000\\qquad\\mathrm{(E)}\\ 1900$" + } + }, + { + "question": "Return your final response within \\boxed{}. An $8$ by $2\\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is\n$\\textbf{(A)}\\ 2\\pi \\qquad\\textbf{(B)}\\ 2\\pi+2 \\qquad\\textbf{(C)}\\ 4\\pi-4 \\qquad\\textbf{(D)}\\ 2\\pi+4 \\qquad\\textbf{(E)}\\ 4\\pi-2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1968", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An $8$ by $2\\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is\n$\\textbf{(A)}\\ 2\\pi \\qquad\\textbf{(B)}\\ 2\\pi+2 \\qquad\\textbf{(C)}\\ 4\\pi-4 \\qquad\\textbf{(D)}\\ 2\\pi+4 \\qquad\\textbf{(E)}\\ 4\\pi-2$" + } + }, + { + "question": "Return your final response within \\boxed{}. Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black?\n[asy] real d=320; pair O=origin; pair P=O+8*dir(d); pair A0 = origin; pair A1 = O+1*dir(d); pair A2 = O+2*dir(d); pair A3 = O+3*dir(d); pair A4 = O+4*dir(d); pair A5 = O+5*dir(d); filldraw(Circle(A0, 6), white, black); filldraw(circle(A1, 5), black, black); filldraw(circle(A2, 4), white, black); filldraw(circle(A3, 3), black, black); filldraw(circle(A4, 2), white, black); filldraw(circle(A5, 1), black, black); [/asy]\n$\\textbf{(A)}\\ 42\\qquad \\textbf{(B)}\\ 44\\qquad \\textbf{(C)}\\ 45\\qquad \\textbf{(D)}\\ 46\\qquad \\textbf{(E)}\\ 48\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1969", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black?\n[asy] real d=320; pair O=origin; pair P=O+8*dir(d); pair A0 = origin; pair A1 = O+1*dir(d); pair A2 = O+2*dir(d); pair A3 = O+3*dir(d); pair A4 = O+4*dir(d); pair A5 = O+5*dir(d); filldraw(Circle(A0, 6), white, black); filldraw(circle(A1, 5), black, black); filldraw(circle(A2, 4), white, black); filldraw(circle(A3, 3), black, black); filldraw(circle(A4, 2), white, black); filldraw(circle(A5, 1), black, black); [/asy]\n$\\textbf{(A)}\\ 42\\qquad \\textbf{(B)}\\ 44\\qquad \\textbf{(C)}\\ 45\\qquad \\textbf{(D)}\\ 46\\qquad \\textbf{(E)}\\ 48\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. What number is directly above $142$ in this array of numbers?\n\\[\\begin{array}{cccccc}& & & 1 & &\\\\ & & 2 & 3 & 4 &\\\\ & 5 & 6 & 7 & 8 & 9\\\\ 10 & 11 & 12 &\\cdots & &\\\\ \\end{array}\\]\n$\\text{(A)}\\ 99 \\qquad \\text{(B)}\\ 119 \\qquad \\text{(C)}\\ 120 \\qquad \\text{(D)}\\ 121 \\qquad \\text{(E)}\\ 122$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1970", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What number is directly above $142$ in this array of numbers?\n\\[\\begin{array}{cccccc}& & & 1 & &\\\\ & & 2 & 3 & 4 &\\\\ & 5 & 6 & 7 & 8 & 9\\\\ 10 & 11 & 12 &\\cdots & &\\\\ \\end{array}\\]\n$\\text{(A)}\\ 99 \\qquad \\text{(B)}\\ 119 \\qquad \\text{(C)}\\ 120 \\qquad \\text{(D)}\\ 121 \\qquad \\text{(E)}\\ 122$" + } + }, + { + "question": "Return your final response within \\boxed{}. For every integer $n\\ge2$, let $\\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\\text{pow}(144)=\\text{pow}(2^4\\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides\n\n\n$\\prod_{n=2}^{5300}\\text{pow}(n)$?\n\n\n\n$\\textbf{(A)}\\ 74 \\qquad \\textbf{(B)}\\ 75 \\qquad \\textbf{(C)}\\ 76 \\qquad \\textbf{(D)}\\ 77 \\qquad \\textbf{(E)}\\ 78$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1971", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For every integer $n\\ge2$, let $\\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\\text{pow}(144)=\\text{pow}(2^4\\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides\n\n\n$\\prod_{n=2}^{5300}\\text{pow}(n)$?\n\n\n\n$\\textbf{(A)}\\ 74 \\qquad \\textbf{(B)}\\ 75 \\qquad \\textbf{(C)}\\ 76 \\qquad \\textbf{(D)}\\ 77 \\qquad \\textbf{(E)}\\ 78$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for\n$\\begin{tabular}{ccc} X & X & X \\\\ & Y & X \\\\ + & & X \\\\ \\hline \\end{tabular}$\nhas the form\n$\\text{(A)}\\ XXY \\qquad \\text{(B)}\\ XYZ \\qquad \\text{(C)}\\ YYX \\qquad \\text{(D)}\\ YYZ \\qquad \\text{(E)}\\ ZZY$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1972", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for\n$\\begin{tabular}{ccc} X & X & X \\\\ & Y & X \\\\ + & & X \\\\ \\hline \\end{tabular}$\nhas the form\n$\\text{(A)}\\ XXY \\qquad \\text{(B)}\\ XYZ \\qquad \\text{(C)}\\ YYX \\qquad \\text{(D)}\\ YYZ \\qquad \\text{(E)}\\ ZZY$" + } + }, + { + "question": "Return your final response within \\boxed{}. With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in the passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the revote was $\\frac{12}{11}$ of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time?\n$\\text{(A) } 75\\quad \\text{(B) } 60\\quad \\text{(C) } 50\\quad \\text{(D) } 45\\quad \\text{(E) } 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1973", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in the passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the revote was $\\frac{12}{11}$ of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time?\n$\\text{(A) } 75\\quad \\text{(B) } 60\\quad \\text{(C) } 50\\quad \\text{(D) } 45\\quad \\text{(E) } 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?\n$\\text{(A) } \\dfrac32 \\quad \\text{(B) } \\dfrac{1+\\sqrt5}2 \\quad \\text{(C) } \\sqrt3 \\quad \\text{(D) } 2 \\quad \\text{(E) } \\dfrac{3+\\sqrt5}2$\n[asy] real r=(3+sqrt(5))/2; real s=sqrt(r); real Brad=r; real brad=1; real Fht = 2*s; import graph3; import solids; currentprojection=orthographic(1,0,.2); currentlight=(10,10,5); revolution sph=sphere((0,0,Fht/2),Fht/2); //draw(surface(sph),green+white+opacity(0.5)); //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} triple f(pair t) { triple v0 = Brad*(cos(t.x),sin(t.x),0); triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); return (v0 + t.y*(v1-v0)); } triple g(pair t) { return (t.y*cos(t.x),t.y*sin(t.x),0); } surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); surface base = surface(g,(0,0),(2pi,Brad),80,2); draw(sback,gray(0.3)); draw(sfront,gray(0.5)); draw(base,gray(0.9)); draw(surface(sph),gray(0.4));[/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1974", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?\n$\\text{(A) } \\dfrac32 \\quad \\text{(B) } \\dfrac{1+\\sqrt5}2 \\quad \\text{(C) } \\sqrt3 \\quad \\text{(D) } 2 \\quad \\text{(E) } \\dfrac{3+\\sqrt5}2$\n[asy] real r=(3+sqrt(5))/2; real s=sqrt(r); real Brad=r; real brad=1; real Fht = 2*s; import graph3; import solids; currentprojection=orthographic(1,0,.2); currentlight=(10,10,5); revolution sph=sphere((0,0,Fht/2),Fht/2); //draw(surface(sph),green+white+opacity(0.5)); //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} triple f(pair t) { triple v0 = Brad*(cos(t.x),sin(t.x),0); triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); return (v0 + t.y*(v1-v0)); } triple g(pair t) { return (t.y*cos(t.x),t.y*sin(t.x),0); } surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); surface base = surface(g,(0,0),(2pi,Brad),80,2); draw(sback,gray(0.3)); draw(sfront,gray(0.5)); draw(base,gray(0.9)); draw(surface(sph),gray(0.4));[/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?\n$\\textbf{(A)} \\ \\frac{553}{715} \\qquad \\textbf{(B)} \\ \\frac{443}{572} \\qquad \\textbf{(C)} \\ \\frac{111}{143} \\qquad \\textbf{(D)} \\ \\frac{81}{104} \\qquad \\textbf{(E)} \\ \\frac{223}{286}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1975", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?\n$\\textbf{(A)} \\ \\frac{553}{715} \\qquad \\textbf{(B)} \\ \\frac{443}{572} \\qquad \\textbf{(C)} \\ \\frac{111}{143} \\qquad \\textbf{(D)} \\ \\frac{81}{104} \\qquad \\textbf{(E)} \\ \\frac{223}{286}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two equal circles in the same plane cannot have the following number of common tangents.\n$\\textbf{(A) \\ }1 \\qquad \\textbf{(B) \\ }2 \\qquad \\textbf{(C) \\ }3 \\qquad \\textbf{(D) \\ }4 \\qquad \\textbf{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1976", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two equal circles in the same plane cannot have the following number of common tangents.\n$\\textbf{(A) \\ }1 \\qquad \\textbf{(B) \\ }2 \\qquad \\textbf{(C) \\ }3 \\qquad \\textbf{(D) \\ }4 \\qquad \\textbf{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock $A.$ How many hours did it take the power boat to go from $A$ to $B$?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 3.5 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 4.5 \\qquad \\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1977", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock $A.$ How many hours did it take the power boat to go from $A$ to $B$?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 3.5 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 4.5 \\qquad \\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following figures has the greatest number of lines of symmetry?\n$\\textbf{(A)}\\ \\text{equilateral triangle}$\n$\\textbf{(B)}\\ \\text{non-square rhombus}$\n$\\textbf{(C)}\\ \\text{non-square rectangle}$\n$\\textbf{(D)}\\ \\text{isosceles trapezoid}$\n$\\textbf{(E)}\\ \\text{square}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1978", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following figures has the greatest number of lines of symmetry?\n$\\textbf{(A)}\\ \\text{equilateral triangle}$\n$\\textbf{(B)}\\ \\text{non-square rhombus}$\n$\\textbf{(C)}\\ \\text{non-square rectangle}$\n$\\textbf{(D)}\\ \\text{isosceles trapezoid}$\n$\\textbf{(E)}\\ \\text{square}$" + } + }, + { + "question": "Return your final response within \\boxed{}. When $p = \\sum\\limits_{k=1}^{6} k \\text{ ln }{k}$, the number $e^p$ is an integer. What is the largest power of $2$ that is a factor of $e^p$?\n$\\textbf{(A)}\\ 2^{12}\\qquad\\textbf{(B)}\\ 2^{14}\\qquad\\textbf{(C)}\\ 2^{16}\\qquad\\textbf{(D)}\\ 2^{18}\\qquad\\textbf{(E)}\\ 2^{20}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1979", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When $p = \\sum\\limits_{k=1}^{6} k \\text{ ln }{k}$, the number $e^p$ is an integer. What is the largest power of $2$ that is a factor of $e^p$?\n$\\textbf{(A)}\\ 2^{12}\\qquad\\textbf{(B)}\\ 2^{14}\\qquad\\textbf{(C)}\\ 2^{16}\\qquad\\textbf{(D)}\\ 2^{18}\\qquad\\textbf{(E)}\\ 2^{20}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A customer who intends to purchase an appliance has three coupons, only one of which may be used:\nCoupon 1: $10\\%$ off the listed price if the listed price is at least $\\textdollar50$\nCoupon 2: $\\textdollar 20$ off the listed price if the listed price is at least $\\textdollar100$\nCoupon 3: $18\\%$ off the amount by which the listed price exceeds $\\textdollar100$\nFor which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?\n$\\textbf{(A) }\\textdollar179.95\\qquad \\textbf{(B) }\\textdollar199.95\\qquad \\textbf{(C) }\\textdollar219.95\\qquad \\textbf{(D) }\\textdollar239.95\\qquad \\textbf{(E) }\\textdollar259.95\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1980", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A customer who intends to purchase an appliance has three coupons, only one of which may be used:\nCoupon 1: $10\\%$ off the listed price if the listed price is at least $\\textdollar50$\nCoupon 2: $\\textdollar 20$ off the listed price if the listed price is at least $\\textdollar100$\nCoupon 3: $18\\%$ off the amount by which the listed price exceeds $\\textdollar100$\nFor which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?\n$\\textbf{(A) }\\textdollar179.95\\qquad \\textbf{(B) }\\textdollar199.95\\qquad \\textbf{(C) }\\textdollar219.95\\qquad \\textbf{(D) }\\textdollar239.95\\qquad \\textbf{(E) }\\textdollar259.95\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?\n$\\textbf{(A) }$ $n$ and $m$ are even $\\qquad\\textbf{(B) }$ $n$ and $m$ are odd $\\qquad\\textbf{(C) }$ $n+m$ is even $\\qquad\\textbf{(D) }$ $n+m$ is odd $\\qquad \\textbf{(E) }$ none of these are impossible", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1981", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?\n$\\textbf{(A) }$ $n$ and $m$ are even $\\qquad\\textbf{(B) }$ $n$ and $m$ are odd $\\qquad\\textbf{(C) }$ $n+m$ is even $\\qquad\\textbf{(D) }$ $n+m$ is odd $\\qquad \\textbf{(E) }$ none of these are impossible" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following rigid transformations (isometries) maps the line segment $\\overline{AB}$ onto the line segment $\\overline{A'B'}$ so that the image of $A(-2, 1)$ is $A'(2, -1)$ and the image of $B(-1, 4)$ is $B'(1, -4)$?\n$\\textbf{(A) }$ reflection in the $y$-axis\n$\\textbf{(B) }$ counterclockwise rotation around the origin by $90^{\\circ}$\n$\\textbf{(C) }$ translation by 3 units to the right and 5 units down\n$\\textbf{(D) }$ reflection in the $x$-axis\n$\\textbf{(E) }$ clockwise rotation about the origin by $180^{\\circ}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1982", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following rigid transformations (isometries) maps the line segment $\\overline{AB}$ onto the line segment $\\overline{A'B'}$ so that the image of $A(-2, 1)$ is $A'(2, -1)$ and the image of $B(-1, 4)$ is $B'(1, -4)$?\n$\\textbf{(A) }$ reflection in the $y$-axis\n$\\textbf{(B) }$ counterclockwise rotation around the origin by $90^{\\circ}$\n$\\textbf{(C) }$ translation by 3 units to the right and 5 units down\n$\\textbf{(D) }$ reflection in the $x$-axis\n$\\textbf{(E) }$ clockwise rotation about the origin by $180^{\\circ}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A square of side length $1$ and a circle of radius $\\dfrac{\\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?\n$\\textbf{(A)}\\ \\dfrac{\\pi}{3}-1 \\qquad \\textbf{(B)}\\ \\dfrac{2\\pi}{9}-\\dfrac{\\sqrt{3}}{3} \\qquad \\textbf{(C)}\\ \\dfrac{\\pi}{18} \\qquad \\textbf{(D)}\\ \\dfrac{1}{4} \\qquad \\textbf{(E)}\\ \\dfrac{2\\pi}{9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1983", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square of side length $1$ and a circle of radius $\\dfrac{\\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?\n$\\textbf{(A)}\\ \\dfrac{\\pi}{3}-1 \\qquad \\textbf{(B)}\\ \\dfrac{2\\pi}{9}-\\dfrac{\\sqrt{3}}{3} \\qquad \\textbf{(C)}\\ \\dfrac{\\pi}{18} \\qquad \\textbf{(D)}\\ \\dfrac{1}{4} \\qquad \\textbf{(E)}\\ \\dfrac{2\\pi}{9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $(6?3) + 4 - (2 - 1) = 5.$ To make this statement true, the question mark between the 6 and the 3 should be replaced by\n$\\text{(A)} \\div \\qquad \\text{(B)}\\ \\times \\qquad \\text{(C)} + \\qquad \\text{(D)}\\ - \\qquad \\text{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_1984", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $(6?3) + 4 - (2 - 1) = 5.$ To make this statement true, the question mark between the 6 and the 3 should be replaced by\n$\\text{(A)} \\div \\qquad \\text{(B)}\\ \\times \\qquad \\text{(C)} + \\qquad \\text{(D)}\\ - \\qquad \\text{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following equations does NOT have a solution?\n$\\text{(A)}\\:(x+7)^2=0$\n$\\text{(B)}\\:|-3x|+5=0$\n$\\text{(C)}\\:\\sqrt{-x}-2=0$\n$\\text{(D)}\\:\\sqrt{x}-8=0$\n$\\text{(E)}\\:|-3x|-4=0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1985", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following equations does NOT have a solution?\n$\\text{(A)}\\:(x+7)^2=0$\n$\\text{(B)}\\:|-3x|+5=0$\n$\\text{(C)}\\:\\sqrt{-x}-2=0$\n$\\text{(D)}\\:\\sqrt{x}-8=0$\n$\\text{(E)}\\:|-3x|-4=0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?\n[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1)); [/asy]\n$\\text{(A)}\\ 24 \\qquad \\text{(B)}\\ 30 \\qquad \\text{(C)}\\ 36 \\qquad \\text{(D)}\\ 42 \\qquad \\text{(E)}\\ 48$\nAssume that the planes cutting the prism do not intersect anywhere in or on the prism.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1986", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?\n[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1)); [/asy]\n$\\text{(A)}\\ 24 \\qquad \\text{(B)}\\ 30 \\qquad \\text{(C)}\\ 36 \\qquad \\text{(D)}\\ 42 \\qquad \\text{(E)}\\ 48$\nAssume that the planes cutting the prism do not intersect anywhere in or on the prism." + } + }, + { + "question": "Return your final response within \\boxed{}. Last summer $30\\%$ of the birds living on Town Lake were geese, $25\\%$ were swans, $10\\%$ were herons, and $35\\%$ were ducks. What percent of the birds that were not swans were geese?\n$\\textbf{(A)}\\ 20 \\qquad \\textbf{(B)}\\ 30 \\qquad \\textbf{(C)}\\ 40 \\qquad \\textbf{(D)}\\ 50 \\qquad \\textbf{(E)}\\ 60$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1987", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Last summer $30\\%$ of the birds living on Town Lake were geese, $25\\%$ were swans, $10\\%$ were herons, and $35\\%$ were ducks. What percent of the birds that were not swans were geese?\n$\\textbf{(A)}\\ 20 \\qquad \\textbf{(B)}\\ 30 \\qquad \\textbf{(C)}\\ 40 \\qquad \\textbf{(D)}\\ 50 \\qquad \\textbf{(E)}\\ 60$" + } + }, + { + "question": "Return your final response within \\boxed{}. The product $\\left(1-\\frac{1}{2^{2}}\\right)\\left(1-\\frac{1}{3^{2}}\\right)\\ldots\\left(1-\\frac{1}{9^{2}}\\right)\\left(1-\\frac{1}{10^{2}}\\right)$ equals\n$\\textbf{(A)}\\ \\frac{5}{12}\\qquad \\textbf{(B)}\\ \\frac{1}{2}\\qquad \\textbf{(C)}\\ \\frac{11}{20}\\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ \\frac{7}{10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1988", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The product $\\left(1-\\frac{1}{2^{2}}\\right)\\left(1-\\frac{1}{3^{2}}\\right)\\ldots\\left(1-\\frac{1}{9^{2}}\\right)\\left(1-\\frac{1}{10^{2}}\\right)$ equals\n$\\textbf{(A)}\\ \\frac{5}{12}\\qquad \\textbf{(B)}\\ \\frac{1}{2}\\qquad \\textbf{(C)}\\ \\frac{11}{20}\\qquad \\textbf{(D)}\\ \\frac{2}{3}\\qquad \\textbf{(E)}\\ \\frac{7}{10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many positive factors does $23,232$ have?\n$\\textbf{(A) }9\\qquad\\textbf{(B) }12\\qquad\\textbf{(C) }28\\qquad\\textbf{(D) }36\\qquad\\textbf{(E) }42$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1989", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many positive factors does $23,232$ have?\n$\\textbf{(A) }9\\qquad\\textbf{(B) }12\\qquad\\textbf{(C) }28\\qquad\\textbf{(D) }36\\qquad\\textbf{(E) }42$" + } + }, + { + "question": "Return your final response within \\boxed{}. A coin is biased in such a way that on each toss the probability of heads is $\\frac{2}{3}$ and the probability of tails is $\\frac{1}{3}$. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?\n$\\textbf{(A)}$ The probability of winning Game A is $\\frac{4}{81}$ less than the probability of winning Game B. \n$\\textbf{(B)}$ The probability of winning Game A is $\\frac{2}{81}$ less than the probability of winning Game B. \n$\\textbf{(C)}$ The probabilities are the same.\n$\\textbf{(D)}$ The probability of winning Game A is $\\frac{2}{81}$ greater than the probability of winning Game B.\n$\\textbf{(E)}$ The probability of winning Game A is $\\frac{4}{81}$ greater than the probability of winning Game B.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1990", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A coin is biased in such a way that on each toss the probability of heads is $\\frac{2}{3}$ and the probability of tails is $\\frac{1}{3}$. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?\n$\\textbf{(A)}$ The probability of winning Game A is $\\frac{4}{81}$ less than the probability of winning Game B. \n$\\textbf{(B)}$ The probability of winning Game A is $\\frac{2}{81}$ less than the probability of winning Game B. \n$\\textbf{(C)}$ The probabilities are the same.\n$\\textbf{(D)}$ The probability of winning Game A is $\\frac{2}{81}$ greater than the probability of winning Game B.\n$\\textbf{(E)}$ The probability of winning Game A is $\\frac{4}{81}$ greater than the probability of winning Game B." + } + }, + { + "question": "Return your final response within \\boxed{}. Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?\n$\\textbf{(A)}\\ a+3\\qquad\\textbf{(B)}\\ a+4\\qquad\\textbf{(C)}\\ a+5\\qquad\\textbf{(D)}\\ a+6\\qquad\\textbf{(E)}\\ a+7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_1991", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?\n$\\textbf{(A)}\\ a+3\\qquad\\textbf{(B)}\\ a+4\\qquad\\textbf{(C)}\\ a+5\\qquad\\textbf{(D)}\\ a+6\\qquad\\textbf{(E)}\\ a+7$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $i^2 = -1$, then the sum \\[\\cos{45^\\circ} + i\\cos{135^\\circ} + \\cdots + i^n\\cos{(45 + 90n)^\\circ} + \\cdots + i^{40}\\cos{3645^\\circ}\\]\nequals \n$\\text{(A)}\\ \\frac{\\sqrt{2}}{2} \\qquad \\text{(B)}\\ -10i\\sqrt{2} \\qquad \\text{(C)}\\ \\frac{21\\sqrt{2}}{2} \\qquad\\\\ \\text{(D)}\\ \\frac{\\sqrt{2}}{2}(21 - 20i) \\qquad \\text{(E)}\\ \\frac{\\sqrt{2}}{2}(21 + 20i)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1992", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $i^2 = -1$, then the sum \\[\\cos{45^\\circ} + i\\cos{135^\\circ} + \\cdots + i^n\\cos{(45 + 90n)^\\circ} + \\cdots + i^{40}\\cos{3645^\\circ}\\]\nequals \n$\\text{(A)}\\ \\frac{\\sqrt{2}}{2} \\qquad \\text{(B)}\\ -10i\\sqrt{2} \\qquad \\text{(C)}\\ \\frac{21\\sqrt{2}}{2} \\qquad\\\\ \\text{(D)}\\ \\frac{\\sqrt{2}}{2}(21 - 20i) \\qquad \\text{(E)}\\ \\frac{\\sqrt{2}}{2}(21 + 20i)$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many different real numbers $x$ satisfy the equation \\[(x^{2}-5)^{2}=16?\\]\n$\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\\textbf{(C) }2\\qquad\\textbf{(D) }4\\qquad\\textbf{(E) }8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_1993", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many different real numbers $x$ satisfy the equation \\[(x^{2}-5)^{2}=16?\\]\n$\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\\textbf{(C) }2\\qquad\\textbf{(D) }4\\qquad\\textbf{(E) }8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures\nFour friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.\n$\\circ$ Art's cookies are trapezoids: \n\n$\\circ$ Roger's cookies are rectangles: \n\n$\\circ$ Paul's cookies are parallelograms: \n\n$\\circ$ Trisha's cookies are triangles: \n\nEach friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?\n$\\textbf{(A)}\\ 18\\qquad\\textbf{(B)}\\ 25\\qquad\\textbf{(C)}\\ 40\\qquad\\textbf{(D)}\\ 75\\qquad\\textbf{(E)}\\ 90$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1994", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures\nFour friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.\n$\\circ$ Art's cookies are trapezoids: \n\n$\\circ$ Roger's cookies are rectangles: \n\n$\\circ$ Paul's cookies are parallelograms: \n\n$\\circ$ Trisha's cookies are triangles: \n\nEach friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?\n$\\textbf{(A)}\\ 18\\qquad\\textbf{(B)}\\ 25\\qquad\\textbf{(C)}\\ 40\\qquad\\textbf{(D)}\\ 75\\qquad\\textbf{(E)}\\ 90$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $( \\sqrt{\\pi} , a)$ and $( \\sqrt{\\pi} , b)$ are distinct points on the graph of $y^2 + x^4 = 2x^2 y + 1$. What is $|a-b|$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)} \\ \\frac{\\pi}{2} \\qquad\\textbf{(C)} \\ 2 \\qquad\\textbf{(D)} \\ \\sqrt{1+\\pi} \\qquad\\textbf{(E)} \\ 1 + \\sqrt{\\pi}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1995", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $( \\sqrt{\\pi} , a)$ and $( \\sqrt{\\pi} , b)$ are distinct points on the graph of $y^2 + x^4 = 2x^2 y + 1$. What is $|a-b|$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)} \\ \\frac{\\pi}{2} \\qquad\\textbf{(C)} \\ 2 \\qquad\\textbf{(D)} \\ \\sqrt{1+\\pi} \\qquad\\textbf{(E)} \\ 1 + \\sqrt{\\pi}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is:\n$\\text{(A) equal to } \\frac{1}{4}\\quad\\\\ \\text{(B) equal to or greater than } \\frac{1}{8} \\text{ and less than }\\frac{1}{2}\\quad\\\\ \\text{(C) greater than } \\frac{1}{8} \\text{ and less than }\\frac{1}{2}\\quad\\\\ \\text{(D) equal to or greater than } \\frac{1}{8} \\text{ and less than }\\frac{1}{4}\\quad\\\\ \\text{(E) less than }\\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_1996", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is:\n$\\text{(A) equal to } \\frac{1}{4}\\quad\\\\ \\text{(B) equal to or greater than } \\frac{1}{8} \\text{ and less than }\\frac{1}{2}\\quad\\\\ \\text{(C) greater than } \\frac{1}{8} \\text{ and less than }\\frac{1}{2}\\quad\\\\ \\text{(D) equal to or greater than } \\frac{1}{8} \\text{ and less than }\\frac{1}{4}\\quad\\\\ \\text{(E) less than }\\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. As the number of sides of a polygon increases from $3$ to $n$, the sum of the exterior angles formed by extending each side in succession:\n$\\textbf{(A)}\\ \\text{Increases}\\qquad\\textbf{(B)}\\ \\text{Decreases}\\qquad\\textbf{(C)}\\ \\text{Remains constant}\\qquad\\textbf{(D)}\\ \\text{Cannot be predicted}\\qquad\\\\ \\textbf{(E)}\\ \\text{Becomes }(n-3)\\text{ straight angles}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1997", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. As the number of sides of a polygon increases from $3$ to $n$, the sum of the exterior angles formed by extending each side in succession:\n$\\textbf{(A)}\\ \\text{Increases}\\qquad\\textbf{(B)}\\ \\text{Decreases}\\qquad\\textbf{(C)}\\ \\text{Remains constant}\\qquad\\textbf{(D)}\\ \\text{Cannot be predicted}\\qquad\\\\ \\textbf{(E)}\\ \\text{Becomes }(n-3)\\text{ straight angles}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Assume that, for a certain school, it is true that\nI: Some students are not honest.\nII: All fraternity members are honest.\nA necessary conclusion is:\n$\\text{(A) Some students are fraternity members.} \\quad\\\\ \\text{(B) Some fraternity member are not students.} \\quad\\\\ \\text{(C) Some students are not fraternity members.} \\quad\\\\ \\text{(D) No fraternity member is a student.} \\quad\\\\ \\text{(E) No student is a fraternity member.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_1998", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Assume that, for a certain school, it is true that\nI: Some students are not honest.\nII: All fraternity members are honest.\nA necessary conclusion is:\n$\\text{(A) Some students are fraternity members.} \\quad\\\\ \\text{(B) Some fraternity member are not students.} \\quad\\\\ \\text{(C) Some students are not fraternity members.} \\quad\\\\ \\text{(D) No fraternity member is a student.} \\quad\\\\ \\text{(E) No student is a fraternity member.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given a [rational number](https://artofproblemsolving.com/wiki/index.php/Rational_number), write it as a [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) in lowest terms and calculate the product of the resulting [numerator](https://artofproblemsolving.com/wiki/index.php/Numerator) and [denominator](https://artofproblemsolving.com/wiki/index.php/Denominator). For how many rational numbers between $0$ and $1$ will $20_{}^{}!$ be the resulting [product](https://artofproblemsolving.com/wiki/index.php/Product)?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "128", + "index": "Sky-T1_10k_1999", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given a [rational number](https://artofproblemsolving.com/wiki/index.php/Rational_number), write it as a [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) in lowest terms and calculate the product of the resulting [numerator](https://artofproblemsolving.com/wiki/index.php/Numerator) and [denominator](https://artofproblemsolving.com/wiki/index.php/Denominator). For how many rational numbers between $0$ and $1$ will $20_{}^{}!$ be the resulting [product](https://artofproblemsolving.com/wiki/index.php/Product)?" + } + }, + { + "question": "Return your final response within \\boxed{}. Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?\n$\\textbf{(A) }1:4\\qquad\\textbf{(B) }1:2\\qquad\\textbf{(C) }1:1\\qquad\\textbf{(D) }2:1\\qquad\\textbf{(E) }4:1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2000", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?\n$\\textbf{(A) }1:4\\qquad\\textbf{(B) }1:2\\qquad\\textbf{(C) }1:1\\qquad\\textbf{(D) }2:1\\qquad\\textbf{(E) }4:1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?\n$\\textbf{(A) }a=\\frac14 \\qquad \\textbf{(B) }\\frac14 < a < \\frac12 \\qquad \\textbf{(C) }a>\\frac14 \\qquad \\textbf{(D) }a=\\frac12 \\qquad \\textbf{(E) }a>\\frac12 \\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2001", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?\n$\\textbf{(A) }a=\\frac14 \\qquad \\textbf{(B) }\\frac14 < a < \\frac12 \\qquad \\textbf{(C) }a>\\frac14 \\qquad \\textbf{(D) }a=\\frac12 \\qquad \\textbf{(E) }a>\\frac12 \\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is:\n$\\text{(A)}\\quad 16 \\qquad \\text{(B)}\\quad 24 \\qquad \\text{(C)}\\quad 32 \\qquad \\text{(D)}\\quad 44 \\qquad \\text{(E)} \\quad 52$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2002", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is:\n$\\text{(A)}\\quad 16 \\qquad \\text{(B)}\\quad 24 \\qquad \\text{(C)}\\quad 32 \\qquad \\text{(D)}\\quad 44 \\qquad \\text{(E)} \\quad 52$" + } + }, + { + "question": "Return your final response within \\boxed{}. A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?\n\n$\\mathrm{(A)}\\ 2\\sqrt{5}-\\sqrt{3}\\qquad\\mathrm{(B)}\\ 3\\qquad\\mathrm{(C)}\\ \\frac{3\\sqrt{7}-\\sqrt{3}}{2}\\qquad\\mathrm{(D)}\\ 2\\sqrt{3}\\qquad\\mathrm{(E)}\\ \\frac{5+2\\sqrt{3}}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2003", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?\n\n$\\mathrm{(A)}\\ 2\\sqrt{5}-\\sqrt{3}\\qquad\\mathrm{(B)}\\ 3\\qquad\\mathrm{(C)}\\ \\frac{3\\sqrt{7}-\\sqrt{3}}{2}\\qquad\\mathrm{(D)}\\ 2\\sqrt{3}\\qquad\\mathrm{(E)}\\ \\frac{5+2\\sqrt{3}}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the radius of a circle is increased $100\\%$, the area is increased:\n$\\textbf{(A)}\\ 100\\%\\qquad\\textbf{(B)}\\ 200\\%\\qquad\\textbf{(C)}\\ 300\\%\\qquad\\textbf{(D)}\\ 400\\%\\qquad\\textbf{(E)}\\ \\text{By none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2004", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the radius of a circle is increased $100\\%$, the area is increased:\n$\\textbf{(A)}\\ 100\\%\\qquad\\textbf{(B)}\\ 200\\%\\qquad\\textbf{(C)}\\ 300\\%\\qquad\\textbf{(D)}\\ 400\\%\\qquad\\textbf{(E)}\\ \\text{By none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Through a point inside a triangle, three lines are drawn from the vertices to the opposite sides forming six triangular sections. Then: \n$\\textbf{(A)}\\ \\text{the triangles are similar in opposite pairs}\\qquad\\textbf{(B)}\\ \\text{the triangles are congruent in opposite pairs}$\n$\\textbf{(C)}\\ \\text{the triangles are equal in area in opposite pairs}\\qquad\\textbf{(D)}\\ \\text{three similar quadrilaterals are formed}$\n$\\textbf{(E)}\\ \\text{none of the above relations are true}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2005", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Through a point inside a triangle, three lines are drawn from the vertices to the opposite sides forming six triangular sections. Then: \n$\\textbf{(A)}\\ \\text{the triangles are similar in opposite pairs}\\qquad\\textbf{(B)}\\ \\text{the triangles are congruent in opposite pairs}$\n$\\textbf{(C)}\\ \\text{the triangles are equal in area in opposite pairs}\\qquad\\textbf{(D)}\\ \\text{three similar quadrilaterals are formed}$\n$\\textbf{(E)}\\ \\text{none of the above relations are true}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?\n$\\textbf{(A)} \\text{ Bela will always win.} \\qquad \\textbf{(B)} \\text{ Jenn will always win.} \\qquad \\textbf{(C)} \\text{ Bela will win if and only if }n \\text{ is odd.} \\\\ \\textbf{(D)} \\text{ Jenn will win if and only if }n \\text{ is odd.} \\qquad \\textbf{(E)} \\text { Jenn will win if and only if } n>8.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2006", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?\n$\\textbf{(A)} \\text{ Bela will always win.} \\qquad \\textbf{(B)} \\text{ Jenn will always win.} \\qquad \\textbf{(C)} \\text{ Bela will win if and only if }n \\text{ is odd.} \\\\ \\textbf{(D)} \\text{ Jenn will win if and only if }n \\text{ is odd.} \\qquad \\textbf{(E)} \\text { Jenn will win if and only if } n>8.$" + } + }, + { + "question": "Return your final response within \\boxed{}. The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to:\n$\\textbf{(A)}\\ \\infty \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ \\sqrt[4]{2} \\qquad \\textbf{(D)}\\ \\sqrt{2} \\qquad \\textbf{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2007", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to:\n$\\textbf{(A)}\\ \\infty \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ \\sqrt[4]{2} \\qquad \\textbf{(D)}\\ \\sqrt{2} \\qquad \\textbf{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$\n$\\textbf{(A)}\\ 3 - \\frac{\\pi}{2} \\qquad \\textbf{(B)}\\ \\frac{\\pi}{2} \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ \\frac{3\\pi}{4} \\qquad \\textbf{(E)}\\ 1+\\frac{\\pi}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2008", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$\n$\\textbf{(A)}\\ 3 - \\frac{\\pi}{2} \\qquad \\textbf{(B)}\\ \\frac{\\pi}{2} \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ \\frac{3\\pi}{4} \\qquad \\textbf{(E)}\\ 1+\\frac{\\pi}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A [circle](https://artofproblemsolving.com/wiki/index.php/Circle) and two distinct [lines](https://artofproblemsolving.com/wiki/index.php/Line) are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?\n$\\text {(A)}\\ 2 \\qquad \\text {(B)}\\ 3 \\qquad {(C)}\\ 4 \\qquad {(D)}\\ 5 \\qquad {(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2009", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A [circle](https://artofproblemsolving.com/wiki/index.php/Circle) and two distinct [lines](https://artofproblemsolving.com/wiki/index.php/Line) are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?\n$\\text {(A)}\\ 2 \\qquad \\text {(B)}\\ 3 \\qquad {(C)}\\ 4 \\qquad {(D)}\\ 5 \\qquad {(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\\cdot5^5\\cdot7^7.$ Find the number of positive integer divisors of $n.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "270", + "index": "Sky-T1_10k_2010", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\\cdot5^5\\cdot7^7.$ Find the number of positive integer divisors of $n.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\le 4$. Prove that\n\\[\\frac{ab + 1}{(a + b)^2} + \\frac{bc + 1}{(b + c)^2} + \\frac{ca + 1}{(c + a)^2} \\ge 3.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "3", + "index": "Sky-T1_10k_2011", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\le 4$. Prove that\n\\[\\frac{ab + 1}{(a + b)^2} + \\frac{bc + 1}{(b + c)^2} + \\frac{ca + 1}{(c + a)^2} \\ge 3.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCDEF$ be a convex hexagon satisfying $\\overline{AB} \\parallel \\overline{DE}$, $\\overline{BC} \\parallel \\overline{EF}$, $\\overline{CD} \\parallel \\overline{FA}$, and\\[AB \\cdot DE = BC \\cdot EF = CD \\cdot FA.\\]Let $X$, $Y$, and $Z$ be the midpoints of $\\overline{AD}$, $\\overline{BE}$, and $\\overline{CF}$. Prove that the circumcenter of $\\triangle ACE$, the circumcenter of $\\triangle BDF$, and the orthocenter of $\\triangle XYZ$ are collinear.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The circumcenters of \\triangle ACE and \\triangle BDF and the orthocenter of \\triangle XYZ are collinear.", + "index": "Sky-T1_10k_2012", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCDEF$ be a convex hexagon satisfying $\\overline{AB} \\parallel \\overline{DE}$, $\\overline{BC} \\parallel \\overline{EF}$, $\\overline{CD} \\parallel \\overline{FA}$, and\\[AB \\cdot DE = BC \\cdot EF = CD \\cdot FA.\\]Let $X$, $Y$, and $Z$ be the midpoints of $\\overline{AD}$, $\\overline{BE}$, and $\\overline{CF}$. Prove that the circumcenter of $\\triangle ACE$, the circumcenter of $\\triangle BDF$, and the orthocenter of $\\triangle XYZ$ are collinear." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be the smallest nonprime [integer](https://artofproblemsolving.com/wiki/index.php/Integer) greater than $1$ with no [prime factor](https://artofproblemsolving.com/wiki/index.php/Prime_factorization) less than $10$. Then \n$\\mathrm{(A) \\ }100y>0$ , then $\\frac{x^y y^x}{y^y x^x}=$\n\n$\\text{(A) } (x-y)^{y/x}\\quad \\text{(B) } \\left(\\frac{x}{y}\\right)^{x-y}\\quad \\text{(C) } 1\\quad \\text{(D) } \\left(\\frac{x}{y}\\right)^{y-x}\\quad \\text{(E) } (x-y)^{x/y}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2016", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x>y>0$ , then $\\frac{x^y y^x}{y^y x^x}=$\n\n$\\text{(A) } (x-y)^{y/x}\\quad \\text{(B) } \\left(\\frac{x}{y}\\right)^{x-y}\\quad \\text{(C) } 1\\quad \\text{(D) } \\left(\\frac{x}{y}\\right)^{y-x}\\quad \\text{(E) } (x-y)^{x/y}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many [positive integers](https://artofproblemsolving.com/wiki/index.php/Positive_integer) have exactly three [proper divisors](https://artofproblemsolving.com/wiki/index.php/Proper_divisor) (positive integral [divisors](https://artofproblemsolving.com/wiki/index.php/Divisor) excluding itself), each of which is less than 50?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "109", + "index": "Sky-T1_10k_2017", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many [positive integers](https://artofproblemsolving.com/wiki/index.php/Positive_integer) have exactly three [proper divisors](https://artofproblemsolving.com/wiki/index.php/Proper_divisor) (positive integral [divisors](https://artofproblemsolving.com/wiki/index.php/Divisor) excluding itself), each of which is less than 50?" + } + }, + { + "question": "Return your final response within \\boxed{}. A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\\dfrac{x}{y}$?\n$\\textbf{(A) } \\dfrac{12}{13} \\qquad \\textbf{(B) } \\dfrac{35}{37} \\qquad \\textbf{(C) } 1 \\qquad \\textbf{(D) } \\dfrac{37}{35} \\qquad \\textbf{(E) } \\dfrac{13}{12}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2018", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\\dfrac{x}{y}$?\n$\\textbf{(A) } \\dfrac{12}{13} \\qquad \\textbf{(B) } \\dfrac{35}{37} \\qquad \\textbf{(C) } 1 \\qquad \\textbf{(D) } \\dfrac{37}{35} \\qquad \\textbf{(E) } \\dfrac{13}{12}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy]\nHow many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?\n\nsome rotation around a point of line $\\ell$\nsome translation in the direction parallel to line $\\ell$\nthe reflection across line $\\ell$\nsome reflection across a line perpendicular to line $\\ell$\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2019", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy]\nHow many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?\n\nsome rotation around a point of line $\\ell$\nsome translation in the direction parallel to line $\\ell$\nthe reflection across line $\\ell$\nsome reflection across a line perpendicular to line $\\ell$\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$?\n$\\textbf{(A) }15\\qquad\\textbf{(B) }16\\qquad\\textbf{(C) }17\\qquad\\textbf{(D) }18\\qquad\\textbf{(E) }20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2020", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$?\n$\\textbf{(A) }15\\qquad\\textbf{(B) }16\\qquad\\textbf{(C) }17\\qquad\\textbf{(D) }18\\qquad\\textbf{(E) }20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Azar and Carl play a game of tic-tac-toe. Azar places an in $X$ one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O$. How many ways can the board look after the game is over?\n$\\textbf{(A) } 36 \\qquad\\textbf{(B) } 112 \\qquad\\textbf{(C) } 120 \\qquad\\textbf{(D) } 148 \\qquad\\textbf{(E) } 160$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2021", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Azar and Carl play a game of tic-tac-toe. Azar places an in $X$ one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O$. How many ways can the board look after the game is over?\n$\\textbf{(A) } 36 \\qquad\\textbf{(B) } 112 \\qquad\\textbf{(C) } 120 \\qquad\\textbf{(D) } 148 \\qquad\\textbf{(E) } 160$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $p$ and $q$ are positive numbers for which \\[\\operatorname{log}_{9}(p) = \\operatorname{log}_{12}(q) = \\operatorname{log}_{16}(p+q).\\] What is the value of $\\frac{q}{p}$?\n$\\textbf{(A)}\\ \\frac{4}{3}\\qquad \\textbf{(B)}\\ \\frac{1+\\sqrt{3}}{2}\\qquad \\textbf{(C)}\\ \\frac{8}{5}\\qquad \\textbf{(D)}\\ \\frac{1+\\sqrt{5}}{2}\\qquad \\textbf{(E)}\\ \\frac{16}{9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2022", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $p$ and $q$ are positive numbers for which \\[\\operatorname{log}_{9}(p) = \\operatorname{log}_{12}(q) = \\operatorname{log}_{16}(p+q).\\] What is the value of $\\frac{q}{p}$?\n$\\textbf{(A)}\\ \\frac{4}{3}\\qquad \\textbf{(B)}\\ \\frac{1+\\sqrt{3}}{2}\\qquad \\textbf{(C)}\\ \\frac{8}{5}\\qquad \\textbf{(D)}\\ \\frac{1+\\sqrt{5}}{2}\\qquad \\textbf{(E)}\\ \\frac{16}{9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Here is a list of the numbers of fish that Tyler caught in nine outings last summer: \\[2,0,1,3,0,3,3,1,2.\\] Which statement about the mean, median, and mode is true?\n$\\textbf{(A)}\\ \\text{median} < \\text{mean} < \\text{mode} \\qquad \\textbf{(B)}\\ \\text{mean} < \\text{mode} < \\text{median} \\\\ \\\\ \\textbf{(C)}\\ \\text{mean} < \\text{median} < \\text{mode} \\qquad \\textbf{(D)}\\ \\text{median} < \\text{mode} < \\text{mean} \\\\ \\\\ \\textbf{(E)}\\ \\text{mode} < \\text{median} < \\text{mean}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2023", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Here is a list of the numbers of fish that Tyler caught in nine outings last summer: \\[2,0,1,3,0,3,3,1,2.\\] Which statement about the mean, median, and mode is true?\n$\\textbf{(A)}\\ \\text{median} < \\text{mean} < \\text{mode} \\qquad \\textbf{(B)}\\ \\text{mean} < \\text{mode} < \\text{median} \\\\ \\\\ \\textbf{(C)}\\ \\text{mean} < \\text{median} < \\text{mode} \\qquad \\textbf{(D)}\\ \\text{median} < \\text{mode} < \\text{mean} \\\\ \\\\ \\textbf{(E)}\\ \\text{mode} < \\text{median} < \\text{mean}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $x = .123456789101112....998999$, where the digits are obtained by writing the integers $1$ through $999$ in order. \nThe $1983$rd digit to the right of the decimal point is\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 3\\qquad \\textbf{(C)}\\ 5\\qquad \\textbf{(D)}\\ 7\\qquad \\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2024", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $x = .123456789101112....998999$, where the digits are obtained by writing the integers $1$ through $999$ in order. \nThe $1983$rd digit to the right of the decimal point is\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 3\\qquad \\textbf{(C)}\\ 5\\qquad \\textbf{(D)}\\ 7\\qquad \\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. For each real number $a$ with $0 \\leq a \\leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that\n\\[\\sin^2{(\\pi x)} + \\sin^2{(\\pi y)} > 1\\]\nWhat is the maximum value of $P(a)?$\n$\\textbf{(A)}\\ \\frac{7}{12} \\qquad\\textbf{(B)}\\ 2 - \\sqrt{2} \\qquad\\textbf{(C)}\\ \\frac{1+\\sqrt{2}}{4} \\qquad\\textbf{(D)}\\ \\frac{\\sqrt{5}-1}{2} \\qquad\\textbf{(E)}\\ \\frac{5}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2025", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For each real number $a$ with $0 \\leq a \\leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that\n\\[\\sin^2{(\\pi x)} + \\sin^2{(\\pi y)} > 1\\]\nWhat is the maximum value of $P(a)?$\n$\\textbf{(A)}\\ \\frac{7}{12} \\qquad\\textbf{(B)}\\ 2 - \\sqrt{2} \\qquad\\textbf{(C)}\\ \\frac{1+\\sqrt{2}}{4} \\qquad\\textbf{(D)}\\ \\frac{\\sqrt{5}-1}{2} \\qquad\\textbf{(E)}\\ \\frac{5}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The fraction $\\frac{2(\\sqrt2+\\sqrt6)}{3\\sqrt{2+\\sqrt3}}$ is equal to \n$\\textbf{(A)}\\ \\frac{2\\sqrt2}{3} \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ \\frac{2\\sqrt3}3 \\qquad \\textbf{(D)}\\ \\frac43 \\qquad \\textbf{(E)}\\ \\frac{16}{9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2026", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The fraction $\\frac{2(\\sqrt2+\\sqrt6)}{3\\sqrt{2+\\sqrt3}}$ is equal to \n$\\textbf{(A)}\\ \\frac{2\\sqrt2}{3} \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ \\frac{2\\sqrt3}3 \\qquad \\textbf{(D)}\\ \\frac43 \\qquad \\textbf{(E)}\\ \\frac{16}{9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f$ be a function for which $f\\left(\\dfrac{x}{3}\\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$.\n\\[\\text {(A)}\\ -1/3 \\qquad \\text {(B)}\\ -1/9 \\qquad \\text {(C)}\\ 0 \\qquad \\text {(D)}\\ 5/9 \\qquad \\text {(E)}\\ 5/3\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2027", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f$ be a function for which $f\\left(\\dfrac{x}{3}\\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$.\n\\[\\text {(A)}\\ -1/3 \\qquad \\text {(B)}\\ -1/9 \\qquad \\text {(C)}\\ 0 \\qquad \\text {(D)}\\ 5/9 \\qquad \\text {(E)}\\ 5/3\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ has $AB=40,AC=31,$ and $\\sin{A}=\\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "744", + "index": "Sky-T1_10k_2028", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ has $AB=40,AC=31,$ and $\\sin{A}=\\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$." + } + }, + { + "question": "Return your final response within \\boxed{}. Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\\tfrac{m}{n}?$\n$\\textbf{(A) } \\frac{\\sqrt{2}}{4} \\qquad \\textbf{(B) } \\frac{\\sqrt{2}}{2} \\qquad \\textbf{(C) } \\frac{3}{4} \\qquad \\textbf{(D) } \\frac{3\\sqrt{2}}{5} \\qquad \\textbf{(E) } \\frac{2\\sqrt{2}}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2029", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\\tfrac{m}{n}?$\n$\\textbf{(A) } \\frac{\\sqrt{2}}{4} \\qquad \\textbf{(B) } \\frac{\\sqrt{2}}{2} \\qquad \\textbf{(C) } \\frac{3}{4} \\qquad \\textbf{(D) } \\frac{3\\sqrt{2}}{5} \\qquad \\textbf{(E) } \\frac{2\\sqrt{2}}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A positive number $x$ has the property that $x\\%$ of $x$ is $4$. What is $x$?\n$\\textbf{(A) }\\ 2 \\qquad \\textbf{(B) }\\ 4 \\qquad \\textbf{(C) }\\ 10 \\qquad \\textbf{(D) }\\ 20 \\qquad \\textbf{(E) }\\ 40$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2030", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A positive number $x$ has the property that $x\\%$ of $x$ is $4$. What is $x$?\n$\\textbf{(A) }\\ 2 \\qquad \\textbf{(B) }\\ 4 \\qquad \\textbf{(C) }\\ 10 \\qquad \\textbf{(D) }\\ 20 \\qquad \\textbf{(E) }\\ 40$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many three-digit numbers are divisible by 13?\n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 67\\qquad\\textbf{(C)}\\ 69\\qquad\\textbf{(D)}\\ 76\\qquad\\textbf{(E)}\\ 77$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2031", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many three-digit numbers are divisible by 13?\n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 67\\qquad\\textbf{(C)}\\ 69\\qquad\\textbf{(D)}\\ 76\\qquad\\textbf{(E)}\\ 77$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the accompanying figure, the outer square $S$ has side length $40$. A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two closest vertices of $S'$. The result is a four-pointed starlike figure inscribed in $S$. The star figure is cut out and then folded to form a pyramid with base $S'$. Find the volume of this pyramid.\n\n[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "750", + "index": "Sky-T1_10k_2032", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the accompanying figure, the outer square $S$ has side length $40$. A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two closest vertices of $S'$. The result is a four-pointed starlike figure inscribed in $S$. The star figure is cut out and then folded to form a pyramid with base $S'$. Find the volume of this pyramid.\n\n[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. The sides of a triangle are in the ratio $6:8:9$. Then:\n$\\textbf{(A) \\ }\\text{the triangle is obtuse}$\n$\\textbf{(B) \\ }\\text{the angles are in the ratio }6:8:9$\n$\\textbf{(C) \\ }\\text{the triangle is acute}$\n$\\textbf{(D) \\ }\\text{the angle opposite the largest side is double the angle opposite the smallest side}$\n$\\textbf{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2033", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sides of a triangle are in the ratio $6:8:9$. Then:\n$\\textbf{(A) \\ }\\text{the triangle is obtuse}$\n$\\textbf{(B) \\ }\\text{the angles are in the ratio }6:8:9$\n$\\textbf{(C) \\ }\\text{the triangle is acute}$\n$\\textbf{(D) \\ }\\text{the angle opposite the largest side is double the angle opposite the smallest side}$\n$\\textbf{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish?\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2034", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish?\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 5 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 7 \\qquad \\text{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A triangle has vertices $(0,0)$, $(1,1)$, and $(6m,0)$, and the line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$?\n$\\textbf{A} - \\!\\frac {1}{3} \\qquad \\textbf{(B)} - \\!\\frac {1}{6} \\qquad \\textbf{(C)}\\ \\frac {1}{6} \\qquad \\textbf{(D)}\\ \\frac {1}{3} \\qquad \\textbf{(E)}\\ \\frac {1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2035", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A triangle has vertices $(0,0)$, $(1,1)$, and $(6m,0)$, and the line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$?\n$\\textbf{A} - \\!\\frac {1}{3} \\qquad \\textbf{(B)} - \\!\\frac {1}{6} \\qquad \\textbf{(C)}\\ \\frac {1}{6} \\qquad \\textbf{(D)}\\ \\frac {1}{3} \\qquad \\textbf{(E)}\\ \\frac {1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded? \n\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 18\\qquad\\textbf{(C)}\\ 20\\qquad\\textbf{(D)}\\ 24\\qquad\\textbf{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2036", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded? \n\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 18\\qquad\\textbf{(C)}\\ 20\\qquad\\textbf{(D)}\\ 24\\qquad\\textbf{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is equal to $1 + \\frac {1}{1 + \\frac {1}{1 + 1}}$?\n$\\textbf{(A)}\\ \\frac {5}{4} \\qquad \\textbf{(B)}\\ \\frac {3}{2} \\qquad \\textbf{(C)}\\ \\frac {5}{3} \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2037", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is equal to $1 + \\frac {1}{1 + \\frac {1}{1 + 1}}$?\n$\\textbf{(A)}\\ \\frac {5}{4} \\qquad \\textbf{(B)}\\ \\frac {3}{2} \\qquad \\textbf{(C)}\\ \\frac {5}{3} \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. The square of an integer is called a perfect square. If $x$ is a perfect square, the next larger perfect square is\n$\\textbf{(A) }x+1\\qquad \\textbf{(B) }x^2+1\\qquad \\textbf{(C) }x^2+2x+1\\qquad \\textbf{(D) }x^2+x\\qquad \\textbf{(E) }x+2\\sqrt{x}+1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2038", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The square of an integer is called a perfect square. If $x$ is a perfect square, the next larger perfect square is\n$\\textbf{(A) }x+1\\qquad \\textbf{(B) }x^2+1\\qquad \\textbf{(C) }x^2+2x+1\\qquad \\textbf{(D) }x^2+x\\qquad \\textbf{(E) }x+2\\sqrt{x}+1$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\\overline{AC}$, in inches, is:\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ \\sqrt{50}\\qquad \\textbf{(C)}\\ 10\\qquad \\textbf{(D)}\\ \\sqrt{200}\\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2039", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\\overline{AC}$, in inches, is:\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ \\sqrt{50}\\qquad \\textbf{(C)}\\ 10\\qquad \\textbf{(D)}\\ \\sqrt{200}\\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$, then:\n$\\text{(A) } r=1 \\text{ sometimes, } r>1 \\text{ sometimes}\\quad\\\\ \\text{(B) } r=1 \\text{ sometimes, } r<1 \\text{ sometimes}\\quad\\\\ \\text{(C) } 01\\quad \\text{(E) } r=1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2040", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$, then:\n$\\text{(A) } r=1 \\text{ sometimes, } r>1 \\text{ sometimes}\\quad\\\\ \\text{(B) } r=1 \\text{ sometimes, } r<1 \\text{ sometimes}\\quad\\\\ \\text{(C) } 01\\quad \\text{(E) } r=1$" + } + }, + { + "question": "Return your final response within \\boxed{}. For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 54$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2041", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 54$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?\n$\\textbf{(A)} \\text{ 60} \\qquad \\textbf{(B)} \\text{ 72} \\qquad \\textbf{(C)} \\text{ 92} \\qquad \\textbf{(D)} \\text{ 96} \\qquad \\textbf{(E)} \\text{ 120}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2042", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?\n$\\textbf{(A)} \\text{ 60} \\qquad \\textbf{(B)} \\text{ 72} \\qquad \\textbf{(C)} \\text{ 92} \\qquad \\textbf{(D)} \\text{ 96} \\qquad \\textbf{(E)} \\text{ 120}$" + } + }, + { + "question": "Return your final response within \\boxed{}. When $x$ is added to both the numerator and denominator of the fraction \n$\\frac{a}{b},a \\ne b,b \\ne 0$, the value of the fraction is changed to $\\frac{c}{d}$. \nThen $x$ equals:\n$\\text{(A) } \\frac{1}{c-d}\\quad \\text{(B) } \\frac{ad-bc}{c-d}\\quad \\text{(C) } \\frac{ad-bc}{c+d}\\quad \\text{(D) }\\frac{bc-ad}{c-d} \\quad \\text{(E) } \\frac{bc+ad}{c-d}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2043", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When $x$ is added to both the numerator and denominator of the fraction \n$\\frac{a}{b},a \\ne b,b \\ne 0$, the value of the fraction is changed to $\\frac{c}{d}$. \nThen $x$ equals:\n$\\text{(A) } \\frac{1}{c-d}\\quad \\text{(B) } \\frac{ad-bc}{c-d}\\quad \\text{(C) } \\frac{ad-bc}{c+d}\\quad \\text{(D) }\\frac{bc-ad}{c-d} \\quad \\text{(E) } \\frac{bc+ad}{c-d}$" + } + }, + { + "question": "Return your final response within \\boxed{}. George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\\frac{1}{2}$ mile in order to arrive just as school begins today?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }6\\qquad\\textbf{(C) }8\\qquad\\textbf{(D) }10\\qquad \\textbf{(E) }12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2044", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\\frac{1}{2}$ mile in order to arrive just as school begins today?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }6\\qquad\\textbf{(C) }8\\qquad\\textbf{(D) }10\\qquad \\textbf{(E) }12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed\ncircle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?\n$\\textbf{(A)} \\: 9\\pi \\qquad\\textbf{(B)} \\: 12\\pi \\qquad\\textbf{(C)} \\: 18\\pi \\qquad\\textbf{(D)} \\: 24\\pi \\qquad\\textbf{(E)} \\: 27\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2045", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed\ncircle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?\n$\\textbf{(A)} \\: 9\\pi \\qquad\\textbf{(B)} \\: 12\\pi \\qquad\\textbf{(C)} \\: 18\\pi \\qquad\\textbf{(D)} \\: 24\\pi \\qquad\\textbf{(E)} \\: 27\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?\n$\\textbf{(A) }\\dfrac{\\sqrt3}4\\qquad \\textbf{(B) }\\dfrac{\\sqrt3}3\\qquad \\textbf{(C) }\\dfrac23\\qquad \\textbf{(D) }\\dfrac{\\sqrt2}2\\qquad \\textbf{(E) }\\dfrac{\\sqrt3}2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2046", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?\n$\\textbf{(A) }\\dfrac{\\sqrt3}4\\qquad \\textbf{(B) }\\dfrac{\\sqrt3}3\\qquad \\textbf{(C) }\\dfrac23\\qquad \\textbf{(D) }\\dfrac{\\sqrt2}2\\qquad \\textbf{(E) }\\dfrac{\\sqrt3}2$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals:\n$\\text{(A) } -3\\quad \\text{(B) } 0\\quad \\text{(C) } 3\\quad \\text{(D) } \\sqrt{ac}\\quad \\text{(E) } \\frac{a+c}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2047", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals:\n$\\text{(A) } -3\\quad \\text{(B) } 0\\quad \\text{(C) } 3\\quad \\text{(D) } \\sqrt{ac}\\quad \\text{(E) } \\frac{a+c}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. When $x^5, x+\\frac{1}{x}$ and $1+\\frac{2}{x} + \\frac{3}{x^2}$ are multiplied, the product is a polynomial of degree.\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 3\\qquad \\textbf{(C)}\\ 6\\qquad \\textbf{(D)}\\ 7\\qquad \\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2048", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When $x^5, x+\\frac{1}{x}$ and $1+\\frac{2}{x} + \\frac{3}{x^2}$ are multiplied, the product is a polynomial of degree.\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 3\\qquad \\textbf{(C)}\\ 6\\qquad \\textbf{(D)}\\ 7\\qquad \\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain?\n$\\text{(A)}\\ 6 \\qquad \\text{(B)}\\ 7 \\qquad \\text{(C)}\\ 8 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2049", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain?\n$\\text{(A)}\\ 6 \\qquad \\text{(B)}\\ 7 \\qquad \\text{(C)}\\ 8 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. A merchant bought some goods at a discount of $20\\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\\%$ of the marked price and still make a profit of $20\\%$ of the selling price. The per cent of the list price at which he should mark them is:\n$\\textbf{(A) \\ }20 \\qquad \\textbf{(B) \\ }100 \\qquad \\textbf{(C) \\ }125 \\qquad \\textbf{(D) \\ }80 \\qquad \\textbf{(E) \\ }120$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2050", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A merchant bought some goods at a discount of $20\\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\\%$ of the marked price and still make a profit of $20\\%$ of the selling price. The per cent of the list price at which he should mark them is:\n$\\textbf{(A) \\ }20 \\qquad \\textbf{(B) \\ }100 \\qquad \\textbf{(C) \\ }125 \\qquad \\textbf{(D) \\ }80 \\qquad \\textbf{(E) \\ }120$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?\n\n$\\textbf{(A)}\\ 5\\qquad\\textbf{(B)}\\ 5.5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 6.5\\qquad\\textbf{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2051", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?\n\n$\\textbf{(A)}\\ 5\\qquad\\textbf{(B)}\\ 5.5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 6.5\\qquad\\textbf{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always\n$\\text{(A) } 6W\\quad \\text{(B) } 6[W]\\quad \\text{(C) } 6([W]-1)\\quad \\text{(D) } 6([W]+1)\\quad \\text{(E) } -6[-W]$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2052", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always\n$\\text{(A) } 6W\\quad \\text{(B) } 6[W]\\quad \\text{(C) } 6([W]-1)\\quad \\text{(D) } 6([W]+1)\\quad \\text{(E) } -6[-W]$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $m=\\frac{cab}{a-b}$, then $b$ equals:\n$\\textbf{(A) \\ }\\frac{m(a-b)}{ca} \\qquad \\textbf{(B) \\ }\\frac{cab-ma}{-m} \\qquad \\textbf{(C) \\ }\\frac{1}{1+c} \\qquad \\textbf{(D) \\ }\\frac{ma}{m+ca} \\qquad \\textbf{(E) \\ }\\frac{m+ca}{ma}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2053", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $m=\\frac{cab}{a-b}$, then $b$ equals:\n$\\textbf{(A) \\ }\\frac{m(a-b)}{ca} \\qquad \\textbf{(B) \\ }\\frac{cab-ma}{-m} \\qquad \\textbf{(C) \\ }\\frac{1}{1+c} \\qquad \\textbf{(D) \\ }\\frac{ma}{m+ca} \\qquad \\textbf{(E) \\ }\\frac{m+ca}{ma}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\\overline{AQ}$, and $\\measuredangle QPC = 60^\\circ$, then the length of $PQ$ divided by the length of $AQ$ is \n\n$\\text{(A)} \\ \\frac{\\sqrt{3}}{2} \\qquad \\text{(B)} \\ \\frac{\\sqrt{3}}{3} \\qquad \\text{(C)} \\ \\frac{\\sqrt{2}}{2} \\qquad \\text{(D)} \\ \\frac12 \\qquad \\text{(E)} \\ \\frac23$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2054", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\\overline{AQ}$, and $\\measuredangle QPC = 60^\\circ$, then the length of $PQ$ divided by the length of $AQ$ is \n\n$\\text{(A)} \\ \\frac{\\sqrt{3}}{2} \\qquad \\text{(B)} \\ \\frac{\\sqrt{3}}{3} \\qquad \\text{(C)} \\ \\frac{\\sqrt{2}}{2} \\qquad \\text{(D)} \\ \\frac12 \\qquad \\text{(E)} \\ \\frac23$" + } + }, + { + "question": "Return your final response within \\boxed{}. A game uses a deck of $n$ different cards, where $n$ is an integer and $n \\geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "13", + "index": "Sky-T1_10k_2055", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A game uses a deck of $n$ different cards, where $n$ is an integer and $n \\geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the number of five-digit positive integers, $n$, that satisfy the following conditions:\n\n\n(a) the number $n$ is divisible by $5,$\n\n\n(b) the first and last digits of $n$ are equal, and\n\n\n(c) the sum of the digits of $n$ is divisible by $5.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "200", + "index": "Sky-T1_10k_2056", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the number of five-digit positive integers, $n$, that satisfy the following conditions:\n\n\n(a) the number $n$ is divisible by $5,$\n\n\n(b) the first and last digits of $n$ are equal, and\n\n\n(c) the sum of the digits of $n$ is divisible by $5.$" + } + }, + { + "question": "Return your final response within \\boxed{}. A \"stair-step\" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37\\text{th}$ row is\n[asy] draw((0,0)--(7,0)--(7,1)--(0,1)--cycle); draw((1,0)--(6,0)--(6,2)--(1,2)--cycle); draw((2,0)--(5,0)--(5,3)--(2,3)--cycle); draw((3,0)--(4,0)--(4,4)--(3,4)--cycle); fill((1,0)--(2,0)--(2,1)--(1,1)--cycle,black); fill((3,0)--(4,0)--(4,1)--(3,1)--cycle,black); fill((5,0)--(6,0)--(6,1)--(5,1)--cycle,black); fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,black); fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,black); fill((3,2)--(4,2)--(4,3)--(3,3)--cycle,black); [/asy]\n$\\text{(A)}\\ 34 \\qquad \\text{(B)}\\ 35 \\qquad \\text{(C)}\\ 36 \\qquad \\text{(D)}\\ 37 \\qquad \\text{(E)}\\ 38$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2057", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A \"stair-step\" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37\\text{th}$ row is\n[asy] draw((0,0)--(7,0)--(7,1)--(0,1)--cycle); draw((1,0)--(6,0)--(6,2)--(1,2)--cycle); draw((2,0)--(5,0)--(5,3)--(2,3)--cycle); draw((3,0)--(4,0)--(4,4)--(3,4)--cycle); fill((1,0)--(2,0)--(2,1)--(1,1)--cycle,black); fill((3,0)--(4,0)--(4,1)--(3,1)--cycle,black); fill((5,0)--(6,0)--(6,1)--(5,1)--cycle,black); fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,black); fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,black); fill((3,2)--(4,2)--(4,3)--(3,3)--cycle,black); [/asy]\n$\\text{(A)}\\ 34 \\qquad \\text{(B)}\\ 35 \\qquad \\text{(C)}\\ 36 \\qquad \\text{(D)}\\ 37 \\qquad \\text{(E)}\\ 38$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a,b,c,d$ be real numbers such that $b-d \\ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "16", + "index": "Sky-T1_10k_2058", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a,b,c,d$ be real numbers such that $b-d \\ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take." + } + }, + { + "question": "Return your final response within \\boxed{}. The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The tetrahedron is regular.", + "index": "Sky-T1_10k_2059", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is: \n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B) }\\ 3 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 1 \\qquad \\textbf{(E) }\\ 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2060", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is: \n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B) }\\ 3 \\qquad \\textbf{(C) }\\ 2 \\qquad \\textbf{(D) }\\ 1 \\qquad \\textbf{(E) }\\ 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. [asy] draw((-10,-10)--(-10,10)--(10,10)--(10,-10)--cycle,dashed+linewidth(.75)); draw((-7,-7)--(-7,7)--(7,7)--(7,-7)--cycle,dashed+linewidth(.75)); draw((-10,-10)--(10,10),dashed+linewidth(.75)); draw((-10,10)--(10,-10),dashed+linewidth(.75)); fill((10,10)--(10,9)--(9,9)--(9,10)--cycle,black); fill((9,9)--(9,8)--(8,8)--(8,9)--cycle,black); fill((8,8)--(8,7)--(7,7)--(7,8)--cycle,black); fill((-10,-10)--(-10,-9)--(-9,-9)--(-9,-10)--cycle,black); fill((-9,-9)--(-9,-8)--(-8,-8)--(-8,-9)--cycle,black); fill((-8,-8)--(-8,-7)--(-7,-7)--(-7,-8)--cycle,black); fill((10,-10)--(10,-9)--(9,-9)--(9,-10)--cycle,black); fill((9,-9)--(9,-8)--(8,-8)--(8,-9)--cycle,black); fill((8,-8)--(8,-7)--(7,-7)--(7,-8)--cycle,black); fill((-10,10)--(-10,9)--(-9,9)--(-9,10)--cycle,black); fill((-9,9)--(-9,8)--(-8,8)--(-8,9)--cycle,black); fill((-8,8)--(-8,7)--(-7,7)--(-7,8)--cycle,black); [/asy]\nA square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is\n$\\text{(A) } 121\\quad \\text{(B) } 625\\quad \\text{(C) } 676\\quad \\text{(D) } 2500\\quad \\text{(E) } 2601$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2061", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [asy] draw((-10,-10)--(-10,10)--(10,10)--(10,-10)--cycle,dashed+linewidth(.75)); draw((-7,-7)--(-7,7)--(7,7)--(7,-7)--cycle,dashed+linewidth(.75)); draw((-10,-10)--(10,10),dashed+linewidth(.75)); draw((-10,10)--(10,-10),dashed+linewidth(.75)); fill((10,10)--(10,9)--(9,9)--(9,10)--cycle,black); fill((9,9)--(9,8)--(8,8)--(8,9)--cycle,black); fill((8,8)--(8,7)--(7,7)--(7,8)--cycle,black); fill((-10,-10)--(-10,-9)--(-9,-9)--(-9,-10)--cycle,black); fill((-9,-9)--(-9,-8)--(-8,-8)--(-8,-9)--cycle,black); fill((-8,-8)--(-8,-7)--(-7,-7)--(-7,-8)--cycle,black); fill((10,-10)--(10,-9)--(9,-9)--(9,-10)--cycle,black); fill((9,-9)--(9,-8)--(8,-8)--(8,-9)--cycle,black); fill((8,-8)--(8,-7)--(7,-7)--(7,-8)--cycle,black); fill((-10,10)--(-10,9)--(-9,9)--(-9,10)--cycle,black); fill((-9,9)--(-9,8)--(-8,8)--(-8,9)--cycle,black); fill((-8,8)--(-8,7)--(-7,7)--(-7,8)--cycle,black); [/asy]\nA square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is\n$\\text{(A) } 121\\quad \\text{(B) } 625\\quad \\text{(C) } 676\\quad \\text{(D) } 2500\\quad \\text{(E) } 2601$" + } + }, + { + "question": "Return your final response within \\boxed{}. A fair die is rolled six times. The probability of rolling at least a five at least five times is \n$\\mathrm{(A)\\ } \\frac{13}{729} \\qquad \\mathrm{(B) \\ }\\frac{12}{729} \\qquad \\mathrm{(C) \\ } \\frac{2}{729} \\qquad \\mathrm{(D) \\ } \\frac{3}{729} \\qquad \\mathrm{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2062", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A fair die is rolled six times. The probability of rolling at least a five at least five times is \n$\\mathrm{(A)\\ } \\frac{13}{729} \\qquad \\mathrm{(B) \\ }\\frac{12}{729} \\qquad \\mathrm{(C) \\ } \\frac{2}{729} \\qquad \\mathrm{(D) \\ } \\frac{3}{729} \\qquad \\mathrm{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{1-\\frac{1}{3}}{1-\\frac{1}{2}} =$\n$\\text{(A)}\\ \\frac{1}{3} \\qquad \\text{(B)}\\ \\frac{2}{3} \\qquad \\text{(C)}\\ \\frac{3}{4} \\qquad \\text{(D)}\\ \\frac{3}{2} \\qquad \\text{(E)}\\ \\frac{4}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2063", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{1-\\frac{1}{3}}{1-\\frac{1}{2}} =$\n$\\text{(A)}\\ \\frac{1}{3} \\qquad \\text{(B)}\\ \\frac{2}{3} \\qquad \\text{(C)}\\ \\frac{3}{4} \\qquad \\text{(D)}\\ \\frac{3}{2} \\qquad \\text{(E)}\\ \\frac{4}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. When simplified, $(-\\frac{1}{125})^{-2/3}$ becomes:\n$\\textbf{(A)}\\ \\frac{1}{25} \\qquad \\textbf{(B)}\\ -\\frac{1}{25} \\qquad \\textbf{(C)}\\ 25\\qquad \\textbf{(D)}\\ -25\\qquad \\textbf{(E)}\\ 25\\sqrt{-1}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2064", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When simplified, $(-\\frac{1}{125})^{-2/3}$ becomes:\n$\\textbf{(A)}\\ \\frac{1}{25} \\qquad \\textbf{(B)}\\ -\\frac{1}{25} \\qquad \\textbf{(C)}\\ 25\\qquad \\textbf{(D)}\\ -25\\qquad \\textbf{(E)}\\ 25\\sqrt{-1}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many of the base-ten numerals for the positive integers less than or equal to $2017$ contain the digit $0$?\n$\\textbf{(A)}\\ 469\\qquad\\textbf{(B)}\\ 471\\qquad\\textbf{(C)}\\ 475\\qquad\\textbf{(D)}\\ 478\\qquad\\textbf{(E)}\\ 481$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2065", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many of the base-ten numerals for the positive integers less than or equal to $2017$ contain the digit $0$?\n$\\textbf{(A)}\\ 469\\qquad\\textbf{(B)}\\ 471\\qquad\\textbf{(C)}\\ 475\\qquad\\textbf{(D)}\\ 478\\qquad\\textbf{(E)}\\ 481$" + } + }, + { + "question": "Return your final response within \\boxed{}. Joyce made $12$ of her first $30$ shots in the first three games of this basketball game, so her seasonal shooting [average](https://artofproblemsolving.com/wiki/index.php/Average) was $40\\%$. In her next game, she took $10$ shots and raised her seasonal shooting average to $50\\%$. How many of these $10$ shots did she make?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2066", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Joyce made $12$ of her first $30$ shots in the first three games of this basketball game, so her seasonal shooting [average](https://artofproblemsolving.com/wiki/index.php/Average) was $40\\%$. In her next game, she took $10$ shots and raised her seasonal shooting average to $50\\%$. How many of these $10$ shots did she make?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of values of $x$ satisfying the [equation](https://artofproblemsolving.com/wiki/index.php/Equation)\n\\[\\frac {2x^2 - 10x}{x^2 - 5x} = x - 3\\]\nis:\n$\\text{(A)} \\ \\text{zero} \\qquad \\text{(B)} \\ \\text{one} \\qquad \\text{(C)} \\ \\text{two} \\qquad \\text{(D)} \\ \\text{three} \\qquad \\text{(E)} \\ \\text{an integer greater than 3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2067", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of values of $x$ satisfying the [equation](https://artofproblemsolving.com/wiki/index.php/Equation)\n\\[\\frac {2x^2 - 10x}{x^2 - 5x} = x - 3\\]\nis:\n$\\text{(A)} \\ \\text{zero} \\qquad \\text{(B)} \\ \\text{one} \\qquad \\text{(C)} \\ \\text{two} \\qquad \\text{(D)} \\ \\text{three} \\qquad \\text{(E)} \\ \\text{an integer greater than 3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The roots of $(x^{2}-3x+2)(x)(x-4)=0$ are:\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 0\\text{ and }4\\qquad\\textbf{(C)}\\ 1\\text{ and }2\\qquad\\textbf{(D)}\\ 0,1,2\\text{ and }4\\qquad\\textbf{(E)}\\ 1,2\\text{ and }4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2068", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The roots of $(x^{2}-3x+2)(x)(x-4)=0$ are:\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 0\\text{ and }4\\qquad\\textbf{(C)}\\ 1\\text{ and }2\\qquad\\textbf{(D)}\\ 0,1,2\\text{ and }4\\qquad\\textbf{(E)}\\ 1,2\\text{ and }4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? \n$\\mathrm{(A)}\\ 12 \\qquad\\mathrm{(B)}\\ 16 \\qquad\\mathrm{(C)}\\ 20 \\qquad\\mathrm{(D)}\\ 22 \\qquad\\mathrm{(E)}\\ 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2069", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? \n$\\mathrm{(A)}\\ 12 \\qquad\\mathrm{(B)}\\ 16 \\qquad\\mathrm{(C)}\\ 20 \\qquad\\mathrm{(D)}\\ 22 \\qquad\\mathrm{(E)}\\ 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $c$ be a constant. The simultaneous equations\n\\begin{align*}x-y = &\\ 2 \\\\cx+y = &\\ 3 \\\\\\end{align*}\nhave a solution $(x, y)$ inside Quadrant I if and only if\n$\\textbf{(A)}\\ c=-1 \\qquad \\textbf{(B)}\\ c>-1 \\qquad \\textbf{(C)}\\ c<\\frac{3}{2} \\qquad \\textbf{(D)}\\ 0-1 \\qquad \\textbf{(C)}\\ c<\\frac{3}{2} \\qquad \\textbf{(D)}\\ 0 ax > a^2 \\qquad \\textbf{(C)}\\ x^2 < a^2 < 0 \\\\ \\textbf{(D)}\\ x^2 > ax\\text{ but }ax < 0 \\qquad \\textbf{(E)}\\ x^2 > a^2\\text{ but }a^2 < 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2085", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x < a < 0$ means that $x$ and $a$ are numbers such that $x$ is less than $a$ and $a$ is less than zero, then: \n$\\textbf{(A)}\\ x^2 < ax < 0 \\qquad \\textbf{(B)}\\ x^2 > ax > a^2 \\qquad \\textbf{(C)}\\ x^2 < a^2 < 0 \\\\ \\textbf{(D)}\\ x^2 > ax\\text{ but }ax < 0 \\qquad \\textbf{(E)}\\ x^2 > a^2\\text{ but }a^2 < 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\\gcd(a+i, b+j)>1$ for all $i, j\\in\\{0, 1, \\ldots n\\}$, then\\[\\min\\{a, b\\}>c^n\\cdot n^{\\frac{n}{2}}.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\min\\{a, b\\} > c^n \\cdot n^{n/2}", + "index": "Sky-T1_10k_2086", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\\gcd(a+i, b+j)>1$ for all $i, j\\in\\{0, 1, \\ldots n\\}$, then\\[\\min\\{a, b\\}>c^n\\cdot n^{\\frac{n}{2}}.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is:\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ \\text{about }\\frac{\\sqrt{5}}{2} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{5}-1}{2}\\qquad \\textbf{(D)}\\ \\frac{1-\\sqrt{5}}{2}\\qquad \\textbf{(E)}\\ \\frac{2}{\\sqrt{5}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2087", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is:\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ \\text{about }\\frac{\\sqrt{5}}{2} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{5}-1}{2}\\qquad \\textbf{(D)}\\ \\frac{1-\\sqrt{5}}{2}\\qquad \\textbf{(E)}\\ \\frac{2}{\\sqrt{5}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The product of all real [roots](https://artofproblemsolving.com/wiki/index.php/Root_(polynomial)) of the equation $x^{\\log_{10}{x}}=10$ is\n$\\mathrm{(A) \\ }1 \\qquad \\mathrm{(B) \\ }-1 \\qquad \\mathrm{(C) \\ } 10 \\qquad \\mathrm{(D) \\ }10^{-1} \\qquad \\mathrm{(E) \\ } \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2088", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The product of all real [roots](https://artofproblemsolving.com/wiki/index.php/Root_(polynomial)) of the equation $x^{\\log_{10}{x}}=10$ is\n$\\mathrm{(A) \\ }1 \\qquad \\mathrm{(B) \\ }-1 \\qquad \\mathrm{(C) \\ } 10 \\qquad \\mathrm{(D) \\ }10^{-1} \\qquad \\mathrm{(E) \\ } \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two integers have a sum of $26$. when two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2089", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two integers have a sum of $26$. when two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?\n$\\textbf{(A)}\\ 18 \\qquad \\textbf{(B)}\\ 21 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 30$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2090", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?\n$\\textbf{(A)}\\ 18 \\qquad \\textbf{(B)}\\ 21 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 30$" + } + }, + { + "question": "Return your final response within \\boxed{}. The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. \nFind the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.\n\n\n\n$\\textbf{(A) }\\frac{1}{2}\\qquad \\textbf{(B) }\\frac{3}{4}\\qquad \\textbf{(C) }\\frac{\\sqrt{2}}{2}\\qquad \\textbf{(D) }\\frac{\\sqrt{3}}{2}\\qquad \\textbf{(E) }\\frac{\\sqrt{3}}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2091", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. \nFind the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.\n\n\n\n$\\textbf{(A) }\\frac{1}{2}\\qquad \\textbf{(B) }\\frac{3}{4}\\qquad \\textbf{(C) }\\frac{\\sqrt{2}}{2}\\qquad \\textbf{(D) }\\frac{\\sqrt{3}}{2}\\qquad \\textbf{(E) }\\frac{\\sqrt{3}}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \\sqrt{n}$ and $P(n+48) = \\sqrt{n+48}$?\n$\\textbf{(A) } 0\\qquad \\textbf{(B) } 1\\qquad \\textbf{(C) } 3\\qquad \\textbf{(D) } 4\\qquad \\textbf{(E) } 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2092", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \\sqrt{n}$ and $P(n+48) = \\sqrt{n+48}$?\n$\\textbf{(A) } 0\\qquad \\textbf{(B) } 1\\qquad \\textbf{(C) } 3\\qquad \\textbf{(D) } 4\\qquad \\textbf{(E) } 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The yearly changes in the population census of a town for four consecutive years are, \nrespectively, 25% increase, 25% increase, 25% decrease, 25% decrease. \nThe net change over the four years, to the nearest percent, is:\n$\\textbf{(A)}\\ -12 \\qquad \\textbf{(B)}\\ -1 \\qquad \\textbf{(C)}\\ 0 \\qquad \\textbf{(D)}\\ 1\\qquad \\textbf{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2093", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The yearly changes in the population census of a town for four consecutive years are, \nrespectively, 25% increase, 25% increase, 25% decrease, 25% decrease. \nThe net change over the four years, to the nearest percent, is:\n$\\textbf{(A)}\\ -12 \\qquad \\textbf{(B)}\\ -1 \\qquad \\textbf{(C)}\\ 0 \\qquad \\textbf{(D)}\\ 1\\qquad \\textbf{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Determine all the [roots](https://artofproblemsolving.com/wiki/index.php/Root), [real](https://artofproblemsolving.com/wiki/index.php/Real) or [complex](https://artofproblemsolving.com/wiki/index.php/Complex), of the system of simultaneous [equations](https://artofproblemsolving.com/wiki/index.php/Equation)\n\n$x+y+z=3$,\n$x^2+y^2+z^2=3$,\n\n$x^3+y^3+z^3=3$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "(1, 1, 1)", + "index": "Sky-T1_10k_2094", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Determine all the [roots](https://artofproblemsolving.com/wiki/index.php/Root), [real](https://artofproblemsolving.com/wiki/index.php/Real) or [complex](https://artofproblemsolving.com/wiki/index.php/Complex), of the system of simultaneous [equations](https://artofproblemsolving.com/wiki/index.php/Equation)\n\n$x+y+z=3$,\n$x^2+y^2+z^2=3$,\n\n$x^3+y^3+z^3=3$." + } + }, + { + "question": "Return your final response within \\boxed{}. A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is\n$\\textbf{(A) }\\frac{1}{2}\\qquad \\textbf{(B) }\\frac{2}{x}\\qquad \\textbf{(C) }\\frac{1}{x+1}\\qquad \\textbf{(D) }\\frac{1}{x}\\qquad \\textbf{(E) }\\frac{2}{x+1}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2095", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is\n$\\textbf{(A) }\\frac{1}{2}\\qquad \\textbf{(B) }\\frac{2}{x}\\qquad \\textbf{(C) }\\frac{1}{x+1}\\qquad \\textbf{(D) }\\frac{1}{x}\\qquad \\textbf{(E) }\\frac{2}{x+1}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\\angle{BAC}$?\n$\\textbf{(A)}\\ 24 \\qquad \\textbf{(B)}\\ 30 \\qquad \\textbf{(C)}\\ 36 \\qquad \\textbf{(D)}\\ 48 \\qquad \\textbf{(E)}\\ 60$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2096", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\\angle{BAC}$?\n$\\textbf{(A)}\\ 24 \\qquad \\textbf{(B)}\\ 30 \\qquad \\textbf{(C)}\\ 36 \\qquad \\textbf{(D)}\\ 48 \\qquad \\textbf{(E)}\\ 60$" + } + }, + { + "question": "Return your final response within \\boxed{}. The lengths of the sides of a triangle are integers, and its area is also an integer. \nOne side is $21$ and the perimeter is $48$. The shortest side is:\n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 10\\qquad \\textbf{(C)}\\ 12 \\qquad \\textbf{(D)}\\ 14 \\qquad \\textbf{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2097", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The lengths of the sides of a triangle are integers, and its area is also an integer. \nOne side is $21$ and the perimeter is $48$. The shortest side is:\n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 10\\qquad \\textbf{(C)}\\ 12 \\qquad \\textbf{(D)}\\ 14 \\qquad \\textbf{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2098", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) of a [cube](https://artofproblemsolving.com/wiki/index.php/Cube) whose [surface area](https://artofproblemsolving.com/wiki/index.php/Surface_area) is twice that of a cube with volume 1? \n$\\mathrm{(A)}\\ \\sqrt{2}\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 2\\sqrt{2}\\qquad\\mathrm{(D)}\\ 4\\qquad\\mathrm{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2099", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) of a [cube](https://artofproblemsolving.com/wiki/index.php/Cube) whose [surface area](https://artofproblemsolving.com/wiki/index.php/Surface_area) is twice that of a cube with volume 1? \n$\\mathrm{(A)}\\ \\sqrt{2}\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 2\\sqrt{2}\\qquad\\mathrm{(D)}\\ 4\\qquad\\mathrm{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, \nthen the measure of $\\measuredangle A_{44}A_{45}A_{43}$ equals\n$\\textbf{(A) }30^\\circ\\qquad \\textbf{(B) }45^\\circ\\qquad \\textbf{(C) }60^\\circ\\qquad \\textbf{(D) }90^\\circ\\qquad \\textbf{(E) }120^\\circ$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2100", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, \nthen the measure of $\\measuredangle A_{44}A_{45}A_{43}$ equals\n$\\textbf{(A) }30^\\circ\\qquad \\textbf{(B) }45^\\circ\\qquad \\textbf{(C) }60^\\circ\\qquad \\textbf{(D) }90^\\circ\\qquad \\textbf{(E) }120^\\circ$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x = (\\log_82)^{(\\log_28)}$, then $\\log_3x$ equals:\n$\\text{(A)} \\ - 3 \\qquad \\text{(B)} \\ - \\frac13 \\qquad \\text{(C)} \\ \\frac13 \\qquad \\text{(D)} \\ 3 \\qquad \\text{(E)} \\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2101", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x = (\\log_82)^{(\\log_28)}$, then $\\log_3x$ equals:\n$\\text{(A)} \\ - 3 \\qquad \\text{(B)} \\ - \\frac13 \\qquad \\text{(C)} \\ \\frac13 \\qquad \\text{(D)} \\ 3 \\qquad \\text{(E)} \\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is:\n$\\text{(A) } xy\\quad \\text{(B) } \\frac{y}{x+y}\\quad \\text{(C) } \\frac{xy}{x-1}\\quad \\text{(D) } \\frac{x+y}{x+1}\\quad \\text{(E) } \\frac{x+y}{x-1}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2102", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is:\n$\\text{(A) } xy\\quad \\text{(B) } \\frac{y}{x+y}\\quad \\text{(C) } \\frac{xy}{x-1}\\quad \\text{(D) } \\frac{x+y}{x+1}\\quad \\text{(E) } \\frac{x+y}{x-1}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?\n\n$\\textbf{(A)}\\ \\frac{1}3\\qquad\\textbf{(B)}\\ \\frac{4}9\\qquad\\textbf{(C)}\\ \\frac{1}2\\qquad\\textbf{(D)}\\ \\frac{5}9\\qquad\\textbf{(E)}\\ \\frac{5}8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2103", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?\n\n$\\textbf{(A)}\\ \\frac{1}3\\qquad\\textbf{(B)}\\ \\frac{4}9\\qquad\\textbf{(C)}\\ \\frac{1}2\\qquad\\textbf{(D)}\\ \\frac{5}9\\qquad\\textbf{(E)}\\ \\frac{5}8$" + } + }, + { + "question": "Return your final response within \\boxed{}. The two roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are $1$ and: \n$\\textbf{(A)}\\ \\frac{b(c-a)}{a(b-c)}\\qquad\\textbf{(B)}\\ \\frac{a(b-c)}{c(a-b)}\\qquad\\textbf{(C)}\\ \\frac{a(b-c)}{b(c-a)}\\qquad\\textbf{(D)}\\ \\frac{c(a-b)}{a(b-c)}\\qquad\\textbf{(E)}\\ \\frac{c(a-b)}{b(c-a)}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2104", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The two roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are $1$ and: \n$\\textbf{(A)}\\ \\frac{b(c-a)}{a(b-c)}\\qquad\\textbf{(B)}\\ \\frac{a(b-c)}{c(a-b)}\\qquad\\textbf{(C)}\\ \\frac{a(b-c)}{b(c-a)}\\qquad\\textbf{(D)}\\ \\frac{c(a-b)}{a(b-c)}\\qquad\\textbf{(E)}\\ \\frac{c(a-b)}{b(c-a)}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of $\\angle GDA$ is\n$\\text{(A)} \\ 90^\\circ \\qquad \\text{(B)} \\ 105^\\circ \\qquad \\text{(C)} \\ 120^\\circ \\qquad \\text{(D)} \\ 135^\\circ \\qquad \\text{(E)} \\ 150^\\circ$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2105", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of $\\angle GDA$ is\n$\\text{(A)} \\ 90^\\circ \\qquad \\text{(B)} \\ 105^\\circ \\qquad \\text{(C)} \\ 120^\\circ \\qquad \\text{(D)} \\ 135^\\circ \\qquad \\text{(E)} \\ 150^\\circ$" + } + }, + { + "question": "Return your final response within \\boxed{}. What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?\n\n$\\textbf{(A) }$ It is a negative number ending with a 1. \n$\\textbf{(B) }$ It is a positive number ending with a 1. \n$\\textbf{(C) }$ It is a negative number ending with a 5. \n$\\textbf{(D) }$ It is a positive number ending with a 5. \n$\\textbf{(E) }$ It is a negative number ending with a 0.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2106", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?\n\n$\\textbf{(A) }$ It is a negative number ending with a 1. \n$\\textbf{(B) }$ It is a positive number ending with a 1. \n$\\textbf{(C) }$ It is a negative number ending with a 5. \n$\\textbf{(D) }$ It is a positive number ending with a 5. \n$\\textbf{(E) }$ It is a negative number ending with a 0." + } + }, + { + "question": "Return your final response within \\boxed{}. Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }5\\qquad\\textbf{(C) }6\\qquad\\textbf{(D) }8\\qquad \\textbf{(E) }10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2107", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }5\\qquad\\textbf{(C) }6\\qquad\\textbf{(D) }8\\qquad \\textbf{(E) }10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?\n$\\textbf{(A) }105\\qquad\\textbf{(B) }114\\qquad\\textbf{(C) }190\\qquad\\textbf{(D) }210\\qquad\\textbf{(E) }380$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2108", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?\n$\\textbf{(A) }105\\qquad\\textbf{(B) }114\\qquad\\textbf{(C) }190\\qquad\\textbf{(D) }210\\qquad\\textbf{(E) }380$" + } + }, + { + "question": "Return your final response within \\boxed{}. The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$?\n$\\textbf{(A)} ~\\frac{14-8}{k-12} \\qquad\\textbf{(B)} ~\\frac{8k-168}{k-12} \\qquad\\textbf{(C)} ~\\frac{14}{12} - \\frac{8}{k} \\qquad\\textbf{(D)} ~\\frac{14(k-12)}{k^2} \\qquad\\textbf{(E)} ~\\frac{14(k-12)}{8k}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2109", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$?\n$\\textbf{(A)} ~\\frac{14-8}{k-12} \\qquad\\textbf{(B)} ~\\frac{8k-168}{k-12} \\qquad\\textbf{(C)} ~\\frac{14}{12} - \\frac{8}{k} \\qquad\\textbf{(D)} ~\\frac{14(k-12)}{k^2} \\qquad\\textbf{(E)} ~\\frac{14(k-12)}{8k}$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square?\n\n$\\textbf{(A) }\\frac{1}{5}\\qquad\\textbf{(B) }\\frac{1}{4} \\qquad\\textbf{(C) }\\frac{2}{5} \\qquad\\textbf{(D) }\\frac{9}{20} \\qquad\\textbf{(E) }\\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2110", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square?\n\n$\\textbf{(A) }\\frac{1}{5}\\qquad\\textbf{(B) }\\frac{1}{4} \\qquad\\textbf{(C) }\\frac{2}{5} \\qquad\\textbf{(D) }\\frac{9}{20} \\qquad\\textbf{(E) }\\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be\n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 11 \\qquad \\text{(C)}\\ 19 \\qquad \\text{(D)}\\ 20 \\qquad \\text{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2111", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be\n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 11 \\qquad \\text{(C)}\\ 19 \\qquad \\text{(D)}\\ 20 \\qquad \\text{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. While eating out, Mike and Joe each tipped their server $2$ dollars. Mike tipped $10\\%$ of his bill and Joe tipped $20\\%$ of his bill. What was the difference, in dollars between their bills? \n$\\textbf{(A) } 2\\qquad \\textbf{(B) } 4\\qquad \\textbf{(C) } 5\\qquad \\textbf{(D) } 10\\qquad \\textbf{(E) } 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2112", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. While eating out, Mike and Joe each tipped their server $2$ dollars. Mike tipped $10\\%$ of his bill and Joe tipped $20\\%$ of his bill. What was the difference, in dollars between their bills? \n$\\textbf{(A) } 2\\qquad \\textbf{(B) } 4\\qquad \\textbf{(C) } 5\\qquad \\textbf{(D) } 10\\qquad \\textbf{(E) } 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. A positive integer $N$ with three digits in its base ten representation is chosen at random, \nwith each three digit number having an equal chance of being chosen. The probability that $\\log_2 N$ is an integer is \n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 3/899 \\qquad \\textbf{(C)}\\ 1/225 \\qquad \\textbf{(D)}\\ 1/300 \\qquad \\textbf{(E)}\\ 1/450$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2113", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A positive integer $N$ with three digits in its base ten representation is chosen at random, \nwith each three digit number having an equal chance of being chosen. The probability that $\\log_2 N$ is an integer is \n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 3/899 \\qquad \\textbf{(C)}\\ 1/225 \\qquad \\textbf{(D)}\\ 1/300 \\qquad \\textbf{(E)}\\ 1/450$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\\overline{BC}$. Points $I$ and $E$ lie on sides $\\overline{AC}$ and $\\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\\frac{a-\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?\n$\\textbf{(A) }9 \\qquad \\textbf{(B) }10 \\qquad \\textbf{(C) }11 \\qquad \\textbf{(D) }12 \\qquad \\textbf{(E) }13 \\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2114", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\\overline{BC}$. Points $I$ and $E$ lie on sides $\\overline{AC}$ and $\\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\\frac{a-\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?\n$\\textbf{(A) }9 \\qquad \\textbf{(B) }10 \\qquad \\textbf{(C) }11 \\qquad \\textbf{(D) }12 \\qquad \\textbf{(E) }13 \\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Melanie computes the mean $\\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\\text{s}$, $12$ $2\\text{s}$, . . . , $12$ $28\\text{s}$, $11$ $29\\text{s}$, $11$ $30\\text{s}$, and $7$ $31\\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true?\n$\\textbf{(A) } \\mu < d < M \\qquad\\textbf{(B) } M < d < \\mu \\qquad\\textbf{(C) } d = M =\\mu \\qquad\\textbf{(D) } d < M < \\mu \\qquad\\textbf{(E) } d < \\mu < M$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2115", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Melanie computes the mean $\\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\\text{s}$, $12$ $2\\text{s}$, . . . , $12$ $28\\text{s}$, $11$ $29\\text{s}$, $11$ $30\\text{s}$, and $7$ $31\\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true?\n$\\textbf{(A) } \\mu < d < M \\qquad\\textbf{(B) } M < d < \\mu \\qquad\\textbf{(C) } d = M =\\mu \\qquad\\textbf{(D) } d < M < \\mu \\qquad\\textbf{(E) } d < \\mu < M$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$. What is the distance between two adjacent parallel lines?\n$\\textbf{(A) }5\\frac12 \\qquad \\textbf{(B) }6 \\qquad \\textbf{(C) }6\\frac12 \\qquad \\textbf{(D) }7 \\qquad \\textbf{(E) }7\\frac12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2116", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$. What is the distance between two adjacent parallel lines?\n$\\textbf{(A) }5\\frac12 \\qquad \\textbf{(B) }6 \\qquad \\textbf{(C) }6\\frac12 \\qquad \\textbf{(D) }7 \\qquad \\textbf{(E) }7\\frac12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?\n$\\mathrm{(A)}\\ 240\\qquad\\mathrm{(B)}\\ 360\\qquad\\mathrm{(C)}\\ 480\\qquad\\mathrm{(D)}\\ 540\\qquad\\mathrm{(E)}\\ 720$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2117", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?\n$\\mathrm{(A)}\\ 240\\qquad\\mathrm{(B)}\\ 360\\qquad\\mathrm{(C)}\\ 480\\qquad\\mathrm{(D)}\\ 540\\qquad\\mathrm{(E)}\\ 720$" + } + }, + { + "question": "Return your final response within \\boxed{}. Assume every 7-digit [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) is a possible telephone number except those that begin with $0$ or $1$. What [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) of telephone numbers begin with $9$ and end with $0$?\n$\\text{(A)}\\ \\frac{1}{63} \\qquad \\text{(B)}\\ \\frac{1}{80} \\qquad \\text{(C)}\\ \\frac{1}{81} \\qquad \\text{(D)}\\ \\frac{1}{90} \\qquad \\text{(E)}\\ \\frac{1}{100}$\nNote: All telephone numbers are 7-digit whole numbers.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2118", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Assume every 7-digit [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) is a possible telephone number except those that begin with $0$ or $1$. What [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) of telephone numbers begin with $9$ and end with $0$?\n$\\text{(A)}\\ \\frac{1}{63} \\qquad \\text{(B)}\\ \\frac{1}{80} \\qquad \\text{(C)}\\ \\frac{1}{81} \\qquad \\text{(D)}\\ \\frac{1}{90} \\qquad \\text{(E)}\\ \\frac{1}{100}$\nNote: All telephone numbers are 7-digit whole numbers." + } + }, + { + "question": "Return your final response within \\boxed{}. If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that:\n$\\textbf{(A)}\\ |r_1+r_2|>4\\sqrt{2}\\qquad \\textbf{(B)}\\ |r_1|>3 \\; \\text{or} \\; |r_2| >3 \\\\ \\textbf{(C)}\\ |r_1|>2 \\; \\text{and} \\; |r_2|>2\\qquad \\textbf{(D)}\\ r_1<0 \\; \\text{and} \\; r_2<0\\qquad \\textbf{(E)}\\ |r_1+r_2|<4\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2119", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that:\n$\\textbf{(A)}\\ |r_1+r_2|>4\\sqrt{2}\\qquad \\textbf{(B)}\\ |r_1|>3 \\; \\text{or} \\; |r_2| >3 \\\\ \\textbf{(C)}\\ |r_1|>2 \\; \\text{and} \\; |r_2|>2\\qquad \\textbf{(D)}\\ r_1<0 \\; \\text{and} \\; r_2<0\\qquad \\textbf{(E)}\\ |r_1+r_2|<4\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$?\n$\\textbf{(A) } \\frac17\\qquad \\textbf{(B) } \\frac27\\qquad \\textbf{(C) } 1\\qquad \\textbf{(D) } 7\\qquad \\textbf{(E) } 14$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2120", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$?\n$\\textbf{(A) } \\frac17\\qquad \\textbf{(B) } \\frac27\\qquad \\textbf{(C) } 1\\qquad \\textbf{(D) } 7\\qquad \\textbf{(E) } 14$" + } + }, + { + "question": "Return your final response within \\boxed{}. The limiting sum of the infinite series, $\\frac{1}{10} + \\frac{2}{10^2} + \\frac{3}{10^3} + \\dots$ whose $n$th term is $\\frac{n}{10^n}$ is: \n$\\textbf{(A)}\\ \\frac{1}9\\qquad\\textbf{(B)}\\ \\frac{10}{81}\\qquad\\textbf{(C)}\\ \\frac{1}8\\qquad\\textbf{(D)}\\ \\frac{17}{72}\\qquad\\textbf{(E)}\\ \\text{larger than any finite quantity}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2121", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The limiting sum of the infinite series, $\\frac{1}{10} + \\frac{2}{10^2} + \\frac{3}{10^3} + \\dots$ whose $n$th term is $\\frac{n}{10^n}$ is: \n$\\textbf{(A)}\\ \\frac{1}9\\qquad\\textbf{(B)}\\ \\frac{10}{81}\\qquad\\textbf{(C)}\\ \\frac{1}8\\qquad\\textbf{(D)}\\ \\frac{17}{72}\\qquad\\textbf{(E)}\\ \\text{larger than any finite quantity}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years $1994$, $1995$, and $1997$ were $39$, $60$, and $123$, respectively, then the population in $1996$ was\n$\\textbf{(A)}\\ 81\\qquad\\textbf{(B)}\\ 84\\qquad\\textbf{(C)}\\ 87\\qquad\\textbf{(D)}\\ 90\\qquad\\textbf{(E)}\\ 102$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2122", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years $1994$, $1995$, and $1997$ were $39$, $60$, and $123$, respectively, then the population in $1996$ was\n$\\textbf{(A)}\\ 81\\qquad\\textbf{(B)}\\ 84\\qquad\\textbf{(C)}\\ 87\\qquad\\textbf{(D)}\\ 90\\qquad\\textbf{(E)}\\ 102$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\dfrac{1}{10}+\\dfrac{2}{10}+\\dfrac{3}{10}+\\dfrac{4}{10}+\\dfrac{5}{10}+\\dfrac{6}{10}+\\dfrac{7}{10}+\\dfrac{8}{10}+\\dfrac{9}{10}+\\dfrac{55}{10}=$\n$\\text{(A)}\\ 4\\dfrac{1}{2} \\qquad \\text{(B)}\\ 6.4 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 10 \\qquad \\text{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2123", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\dfrac{1}{10}+\\dfrac{2}{10}+\\dfrac{3}{10}+\\dfrac{4}{10}+\\dfrac{5}{10}+\\dfrac{6}{10}+\\dfrac{7}{10}+\\dfrac{8}{10}+\\dfrac{9}{10}+\\dfrac{55}{10}=$\n$\\text{(A)}\\ 4\\dfrac{1}{2} \\qquad \\text{(B)}\\ 6.4 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 10 \\qquad \\text{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to\n$\\textbf{(A) }3x+2\\qquad \\textbf{(B) }3x+1\\qquad \\textbf{(C) }2x+3\\qquad \\textbf{(D) }2x+2\\qquad \\textbf{(E) }2x+1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2124", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to\n$\\textbf{(A) }3x+2\\qquad \\textbf{(B) }3x+1\\qquad \\textbf{(C) }2x+3\\qquad \\textbf{(D) }2x+2\\qquad \\textbf{(E) }2x+1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Nine copies of a certain pamphlet cost less than $$ 10.00$ while ten copies of the same pamphlet (at the same price) cost more than $$ 11.00$. How much does one copy of this pamphlet cost?\n$\\text{(A)}$ $$1.07$\n$\\text{(B)}$ $$1.08$\n$\\text{(C)}$ $$1.09$\n$\\text{(D)}$ $$1.10$\n$\\text{(E)}$ $$1.11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2125", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Nine copies of a certain pamphlet cost less than $$ 10.00$ while ten copies of the same pamphlet (at the same price) cost more than $$ 11.00$. How much does one copy of this pamphlet cost?\n$\\text{(A)}$ $$1.07$\n$\\text{(B)}$ $$1.08$\n$\\text{(C)}$ $$1.09$\n$\\text{(D)}$ $$1.10$\n$\\text{(E)}$ $$1.11$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the area of the region defined by the [inequality](https://artofproblemsolving.com/wiki/index.php/Inequality) $|3x-18|+|2y+7|\\le3$?\n$\\mathrm{(A)}\\ 3\\qquad\\mathrm{(B)}\\ \\frac{7}{2}\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ \\frac{9}{2}\\qquad\\mathrm{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2126", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the area of the region defined by the [inequality](https://artofproblemsolving.com/wiki/index.php/Inequality) $|3x-18|+|2y+7|\\le3$?\n$\\mathrm{(A)}\\ 3\\qquad\\mathrm{(B)}\\ \\frac{7}{2}\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ \\frac{9}{2}\\qquad\\mathrm{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be a [parallelogram](https://artofproblemsolving.com/wiki/index.php/Parallelogram). Extend $\\overline{DA}$ through $A$ to a point $P,$ and let $\\overline{PC}$ meet $\\overline{AB}$ at $Q$ and $\\overline{DB}$ at $R.$ Given that $PQ = 735$ and $QR = 112,$ find $RC.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "308", + "index": "Sky-T1_10k_2127", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $ABCD$ be a [parallelogram](https://artofproblemsolving.com/wiki/index.php/Parallelogram). Extend $\\overline{DA}$ through $A$ to a point $P,$ and let $\\overline{PC}$ meet $\\overline{AB}$ at $Q$ and $\\overline{DB}$ at $R.$ Given that $PQ = 735$ and $QR = 112,$ find $RC.$" + } + }, + { + "question": "Return your final response within \\boxed{}. $2000(2000^{2000}) = ?$\n$\\textbf{(A)} \\ 2000^{2001} \\qquad \\textbf{(B)} \\ 4000^{2000} \\qquad \\textbf{(C)} \\ 2000^{4000} \\qquad \\textbf{(D)} \\ 4,000,000^{2000} \\qquad \\textbf{(E)} \\ 2000^{4,000,000}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2128", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $2000(2000^{2000}) = ?$\n$\\textbf{(A)} \\ 2000^{2001} \\qquad \\textbf{(B)} \\ 4000^{2000} \\qquad \\textbf{(C)} \\ 2000^{4000} \\qquad \\textbf{(D)} \\ 4,000,000^{2000} \\qquad \\textbf{(E)} \\ 2000^{4,000,000}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In trapezoid $ABCD$, $\\overline{AD}$ is perpendicular to $\\overline{DC}$,\n$AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\\overline{DC}$, and $\\overline{BE}$ is parallel to $\\overline{AD}$. Find the area of $\\triangle BEC$.\n\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4.5 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 9 \\qquad \\textbf{(E)}\\ 18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2129", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In trapezoid $ABCD$, $\\overline{AD}$ is perpendicular to $\\overline{DC}$,\n$AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\\overline{DC}$, and $\\overline{BE}$ is parallel to $\\overline{AD}$. Find the area of $\\triangle BEC$.\n\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4.5 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 9 \\qquad \\textbf{(E)}\\ 18$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?\n$\\textbf{(A)}\\ \\frac{2}{7}\\qquad\\textbf{(B)}\\ \\frac{3}{8}\\qquad\\textbf{(C)}\\ \\frac{1}{2}\\qquad\\textbf{(D)}\\ \\frac{4}{7}\\qquad\\textbf{(E)}\\ \\frac{5}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2130", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?\n$\\textbf{(A)}\\ \\frac{2}{7}\\qquad\\textbf{(B)}\\ \\frac{3}{8}\\qquad\\textbf{(C)}\\ \\frac{1}{2}\\qquad\\textbf{(D)}\\ \\frac{4}{7}\\qquad\\textbf{(E)}\\ \\frac{5}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is\n$\\mathrm{(A) \\ 16 } \\qquad \\mathrm{(B) \\ 17 } \\qquad \\mathrm{(C) \\ 18 } \\qquad \\mathrm{(D) \\ 19 } \\qquad \\mathrm{(E) \\ 20 }$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2131", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is\n$\\mathrm{(A) \\ 16 } \\qquad \\mathrm{(B) \\ 17 } \\qquad \\mathrm{(C) \\ 18 } \\qquad \\mathrm{(D) \\ 19 } \\qquad \\mathrm{(E) \\ 20 }$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$, $b$, $c$ be positive real numbers. Prove that\n\\[\\frac{a^3 + 3b^3}{5a + b} + \\frac{b^3 + 3c^3}{5b + c} + \\frac{c^3 + 3a^3}{5c + a} \\ge \\frac{2}{3} (a^2 + b^2 + c^2).\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\frac{2}{3} (a^2 + b^2 + c^2)", + "index": "Sky-T1_10k_2132", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$, $b$, $c$ be positive real numbers. Prove that\n\\[\\frac{a^3 + 3b^3}{5a + b} + \\frac{b^3 + 3c^3}{5b + c} + \\frac{c^3 + 3a^3}{5c + a} \\ge \\frac{2}{3} (a^2 + b^2 + c^2).\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. In the set of equations $z^x = y^{2x},\\quad 2^z = 2\\cdot4^x, \\quad x + y + z = 16$, the integral roots in the order $x,y,z$ are: \n$\\textbf{(A) } 3,4,9 \\qquad \\textbf{(B) } 9,-5,-12 \\qquad \\textbf{(C) } 12,-5,9 \\qquad \\textbf{(D) } 4,3,9 \\qquad \\textbf{(E) } 4,9,3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2133", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the set of equations $z^x = y^{2x},\\quad 2^z = 2\\cdot4^x, \\quad x + y + z = 16$, the integral roots in the order $x,y,z$ are: \n$\\textbf{(A) } 3,4,9 \\qquad \\textbf{(B) } 9,-5,-12 \\qquad \\textbf{(C) } 12,-5,9 \\qquad \\textbf{(D) } 4,3,9 \\qquad \\textbf{(E) } 4,9,3$" + } + }, + { + "question": "Return your final response within \\boxed{}. Label one disk \"$1$\", two disks \"$2$\", three disks \"$3$\"$, ...,$ fifty disks \"$50$\". Put these $1+2+3+ \\cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label is\n$\\textbf{(A)}\\ 10 \\qquad\\textbf{(B)}\\ 51 \\qquad\\textbf{(C)}\\ 415 \\qquad\\textbf{(D)}\\ 451 \\qquad\\textbf{(E)}\\ 501$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2134", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Label one disk \"$1$\", two disks \"$2$\", three disks \"$3$\"$, ...,$ fifty disks \"$50$\". Put these $1+2+3+ \\cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label is\n$\\textbf{(A)}\\ 10 \\qquad\\textbf{(B)}\\ 51 \\qquad\\textbf{(C)}\\ 415 \\qquad\\textbf{(D)}\\ 451 \\qquad\\textbf{(E)}\\ 501$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many pairs of parallel edges, such as $\\overline{AB}$ and $\\overline{GH}$ or $\\overline{EH}$ and $\\overline{FG}$, does a cube have?\n\n$\\text{(A) }6 \\quad\\text{(B) }12 \\quad\\text{(C) } 18 \\quad\\text{(D) } 24 \\quad \\text{(E) } 36$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2135", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many pairs of parallel edges, such as $\\overline{AB}$ and $\\overline{GH}$ or $\\overline{EH}$ and $\\overline{FG}$, does a cube have?\n\n$\\text{(A) }6 \\quad\\text{(B) }12 \\quad\\text{(C) } 18 \\quad\\text{(D) } 24 \\quad \\text{(E) } 36$" + } + }, + { + "question": "Return your final response within \\boxed{}. About the equation $ax^2 - 2x\\sqrt {2} + c = 0$, with $a$ and $c$ real constants, \nwe are told that the discriminant is zero. The roots are necessarily: \n$\\textbf{(A)}\\ \\text{equal and integral}\\qquad \\textbf{(B)}\\ \\text{equal and rational}\\qquad \\textbf{(C)}\\ \\text{equal and real} \\\\ \\textbf{(D)}\\ \\text{equal and irrational} \\qquad \\textbf{(E)}\\ \\text{equal and imaginary}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2136", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. About the equation $ax^2 - 2x\\sqrt {2} + c = 0$, with $a$ and $c$ real constants, \nwe are told that the discriminant is zero. The roots are necessarily: \n$\\textbf{(A)}\\ \\text{equal and integral}\\qquad \\textbf{(B)}\\ \\text{equal and rational}\\qquad \\textbf{(C)}\\ \\text{equal and real} \\\\ \\textbf{(D)}\\ \\text{equal and irrational} \\qquad \\textbf{(E)}\\ \\text{equal and imaginary}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?\n$\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2137", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?\n$\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?\n$\\textbf{(A) }\\ 100 \\qquad \\textbf{(B) }\\ 200 \\qquad \\textbf{(C) }\\ 300 \\qquad \\textbf{(D) }\\ 400 \\qquad \\textbf{(E) }\\ 500$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2138", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?\n$\\textbf{(A) }\\ 100 \\qquad \\textbf{(B) }\\ 200 \\qquad \\textbf{(C) }\\ 300 \\qquad \\textbf{(D) }\\ 400 \\qquad \\textbf{(E) }\\ 500$" + } + }, + { + "question": "Return your final response within \\boxed{}. A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. What is the maximum possiblle salary, in dollars, for a single player?\n$\\textbf{(A)}\\ 270,000 \\qquad \\textbf{(B)}\\ 385,000 \\qquad \\textbf{(C)}\\ 400,000 \\qquad \\textbf{(D)}\\ 430,000 \\qquad \\textbf{(E)}\\ 700,000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2139", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. What is the maximum possiblle salary, in dollars, for a single player?\n$\\textbf{(A)}\\ 270,000 \\qquad \\textbf{(B)}\\ 385,000 \\qquad \\textbf{(C)}\\ 400,000 \\qquad \\textbf{(D)}\\ 430,000 \\qquad \\textbf{(E)}\\ 700,000$" + } + }, + { + "question": "Return your final response within \\boxed{}. The polygon(s) formed by $y=3x+2, y=-3x+2$, and $y=-2$, is (are):\n$\\textbf{(A) }\\text{An equilateral triangle}\\qquad\\textbf{(B) }\\text{an isosceles triangle} \\qquad\\textbf{(C) }\\text{a right triangle} \\qquad \\\\ \\textbf{(D) }\\text{a triangle and a trapezoid}\\qquad\\textbf{(E) }\\text{a quadrilateral}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2140", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The polygon(s) formed by $y=3x+2, y=-3x+2$, and $y=-2$, is (are):\n$\\textbf{(A) }\\text{An equilateral triangle}\\qquad\\textbf{(B) }\\text{an isosceles triangle} \\qquad\\textbf{(C) }\\text{a right triangle} \\qquad \\\\ \\textbf{(D) }\\text{a triangle and a trapezoid}\\qquad\\textbf{(E) }\\text{a quadrilateral}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\\triangle AED$ is twice the perimeter of $\\triangle BEC$. Find $AB$.\n$\\text{(A)}\\ 15/2 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 17/2 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 19/2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2141", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\\triangle AED$ is twice the perimeter of $\\triangle BEC$. Find $AB$.\n$\\text{(A)}\\ 15/2 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 17/2 \\qquad \\text{(D)}\\ 9 \\qquad \\text{(E)}\\ 19/2$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the margin made on an article costing $C$ dollars and selling for $S$ dollars is $M=\\frac{1}{n}C$, then the margin is given by: \n$\\textbf{(A)}\\ M=\\frac{1}{n-1}S\\qquad\\textbf{(B)}\\ M=\\frac{1}{n}S\\qquad\\textbf{(C)}\\ M=\\frac{n}{n+1}S\\\\ \\textbf{(D)}\\ M=\\frac{1}{n+1}S\\qquad\\textbf{(E)}\\ M=\\frac{n}{n-1}S$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2142", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the margin made on an article costing $C$ dollars and selling for $S$ dollars is $M=\\frac{1}{n}C$, then the margin is given by: \n$\\textbf{(A)}\\ M=\\frac{1}{n-1}S\\qquad\\textbf{(B)}\\ M=\\frac{1}{n}S\\qquad\\textbf{(C)}\\ M=\\frac{n}{n+1}S\\\\ \\textbf{(D)}\\ M=\\frac{1}{n+1}S\\qquad\\textbf{(E)}\\ M=\\frac{n}{n-1}S$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $p \\geq 5$ is a prime number, then $24$ divides $p^2 - 1$ without remainder \n$\\textbf{(A)}\\ \\text{never} \\qquad \\textbf{(B)}\\ \\text{sometimes only} \\qquad \\textbf{(C)}\\ \\text{always} \\qquad$\n$\\textbf{(D)}\\ \\text{only if } p =5 \\qquad \\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2143", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $p \\geq 5$ is a prime number, then $24$ divides $p^2 - 1$ without remainder \n$\\textbf{(A)}\\ \\text{never} \\qquad \\textbf{(B)}\\ \\text{sometimes only} \\qquad \\textbf{(C)}\\ \\text{always} \\qquad$\n$\\textbf{(D)}\\ \\text{only if } p =5 \\qquad \\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. On Monday, Millie puts a quart of seeds, $25\\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?\n$\\textbf{(A)}\\ \\text{Tuesday}\\qquad \\textbf{(B)}\\ \\text{Wednesday}\\qquad \\textbf{(C)}\\ \\text{Thursday}\\qquad \\textbf{(D)}\\ \\text{Friday}\\qquad \\textbf{(E)}\\ \\text{Saturday}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2144", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On Monday, Millie puts a quart of seeds, $25\\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?\n$\\textbf{(A)}\\ \\text{Tuesday}\\qquad \\textbf{(B)}\\ \\text{Wednesday}\\qquad \\textbf{(C)}\\ \\text{Thursday}\\qquad \\textbf{(D)}\\ \\text{Friday}\\qquad \\textbf{(E)}\\ \\text{Saturday}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?\n$\\textbf{(A)}\\ 62 \\qquad \\textbf{(B)}\\ 82 \\qquad \\textbf{(C)}\\ 83 \\qquad \\textbf{(D)}\\ 102 \\qquad \\textbf{(E)}\\ 103$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2145", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?\n$\\textbf{(A)}\\ 62 \\qquad \\textbf{(B)}\\ 82 \\qquad \\textbf{(C)}\\ 83 \\qquad \\textbf{(D)}\\ 102 \\qquad \\textbf{(E)}\\ 103$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of the expression $\\frac{1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 6 \\cdot 7 \\cdot 8}{1+2+3+4+5+6+7+8}$?\n$\\textbf{(A) }1020\\qquad\\textbf{(B) }1120\\qquad\\textbf{(C) }1220\\qquad\\textbf{(D) }2240\\qquad\\textbf{(E) }3360$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2146", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of the expression $\\frac{1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 6 \\cdot 7 \\cdot 8}{1+2+3+4+5+6+7+8}$?\n$\\textbf{(A) }1020\\qquad\\textbf{(B) }1120\\qquad\\textbf{(C) }1220\\qquad\\textbf{(D) }2240\\qquad\\textbf{(E) }3360$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the [obtuse triangle](https://artofproblemsolving.com/wiki/index.php/Obtuse_triangle) $ABC$ with $\\angle C>90^\\circ$, $AM=MB$, $MD\\perp BC$, and $EC\\perp BC$ ($D$ is on $BC$, $E$ is on $AB$, and $M$ is on $EB$). If the [area](https://artofproblemsolving.com/wiki/index.php/Area) of $\\triangle ABC$ is $24$, then the area of $\\triangle BED$ is\n$\\mathrm{(A) \\ }9 \\qquad \\mathrm{(B) \\ }12 \\qquad \\mathrm{(C) \\ } 15 \\qquad \\mathrm{(D) \\ }18 \\qquad \\mathrm{(E) \\ } \\text{Not uniquely determined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2147", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the [obtuse triangle](https://artofproblemsolving.com/wiki/index.php/Obtuse_triangle) $ABC$ with $\\angle C>90^\\circ$, $AM=MB$, $MD\\perp BC$, and $EC\\perp BC$ ($D$ is on $BC$, $E$ is on $AB$, and $M$ is on $EB$). If the [area](https://artofproblemsolving.com/wiki/index.php/Area) of $\\triangle ABC$ is $24$, then the area of $\\triangle BED$ is\n$\\mathrm{(A) \\ }9 \\qquad \\mathrm{(B) \\ }12 \\qquad \\mathrm{(C) \\ } 15 \\qquad \\mathrm{(D) \\ }18 \\qquad \\mathrm{(E) \\ } \\text{Not uniquely determined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$?\n$\\textbf{(A)}\\ 7 \\qquad\\textbf{(B)}\\ 9 \\qquad\\textbf{(C)}\\ 10 \\qquad\\textbf{(D)}\\ 11 \\qquad\\textbf{(E)}\\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2148", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$?\n$\\textbf{(A)}\\ 7 \\qquad\\textbf{(B)}\\ 9 \\qquad\\textbf{(C)}\\ 10 \\qquad\\textbf{(D)}\\ 11 \\qquad\\textbf{(E)}\\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given rectangle $R_1$ with one side $2$ inches and area $12$ square inches. Rectangle $R_2$ with diagonal $15$ inches is similar to $R_1$. Expressed in square inches the area of $R_2$ is: \n$\\textbf{(A)}\\ \\frac{9}2\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ \\frac{135}{2}\\qquad\\textbf{(D)}\\ 9\\sqrt{10}\\qquad\\textbf{(E)}\\ \\frac{27\\sqrt{10}}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2149", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given rectangle $R_1$ with one side $2$ inches and area $12$ square inches. Rectangle $R_2$ with diagonal $15$ inches is similar to $R_1$. Expressed in square inches the area of $R_2$ is: \n$\\textbf{(A)}\\ \\frac{9}2\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ \\frac{135}{2}\\qquad\\textbf{(D)}\\ 9\\sqrt{10}\\qquad\\textbf{(E)}\\ \\frac{27\\sqrt{10}}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are unique integers $a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$ such that\n\\[\\frac {5}{7} = \\frac {a_{2}}{2!} + \\frac {a_{3}}{3!} + \\frac {a_{4}}{4!} + \\frac {a_{5}}{5!} + \\frac {a_{6}}{6!} + \\frac {a_{7}}{7!}\\]\nwhere $0\\leq a_{i} < i$ for $i = 2,3,\\ldots,7$. Find $a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}$.\n$\\textrm{(A)} \\ 8 \\qquad \\textrm{(B)} \\ 9 \\qquad \\textrm{(C)} \\ 10 \\qquad \\textrm{(D)} \\ 11 \\qquad \\textrm{(E)} \\ 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2150", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are unique integers $a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$ such that\n\\[\\frac {5}{7} = \\frac {a_{2}}{2!} + \\frac {a_{3}}{3!} + \\frac {a_{4}}{4!} + \\frac {a_{5}}{5!} + \\frac {a_{6}}{6!} + \\frac {a_{7}}{7!}\\]\nwhere $0\\leq a_{i} < i$ for $i = 2,3,\\ldots,7$. Find $a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}$.\n$\\textrm{(A)} \\ 8 \\qquad \\textrm{(B)} \\ 9 \\qquad \\textrm{(C)} \\ 10 \\qquad \\textrm{(D)} \\ 11 \\qquad \\textrm{(E)} \\ 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "540", + "index": "Sky-T1_10k_2151", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$." + } + }, + { + "question": "Return your final response within \\boxed{}. One can holds $12$ ounces of soda, what is the minimum number of cans needed to provide a gallon ($128$ ounces) of soda?\n$\\textbf{(A)}\\ 7\\qquad \\textbf{(B)}\\ 8\\qquad \\textbf{(C)}\\ 9\\qquad \\textbf{(D)}\\ 10\\qquad \\textbf{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2152", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. One can holds $12$ ounces of soda, what is the minimum number of cans needed to provide a gallon ($128$ ounces) of soda?\n$\\textbf{(A)}\\ 7\\qquad \\textbf{(B)}\\ 8\\qquad \\textbf{(C)}\\ 9\\qquad \\textbf{(D)}\\ 10\\qquad \\textbf{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?\n$\\textbf{(A) }55\\qquad\\textbf{(B) }60\\qquad\\textbf{(C) }65\\qquad\\textbf{(D) }70\\qquad\\textbf{(E) }75$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2153", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?\n$\\textbf{(A) }55\\qquad\\textbf{(B) }60\\qquad\\textbf{(C) }65\\qquad\\textbf{(D) }70\\qquad\\textbf{(E) }75$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "There exists a point common to at least half of the sets.", + "index": "Sky-T1_10k_2154", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family." + } + }, + { + "question": "Return your final response within \\boxed{}. If the base $8$ representation of a perfect square is $ab3c$, where $a\\ne 0$, then $c$ equals \n$\\text{(A)} 0\\qquad \\text{(B)}1 \\qquad \\text{(C)} 3\\qquad \\text{(D)} 4\\qquad \\text{(E)} \\text{not uniquely determined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2155", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the base $8$ representation of a perfect square is $ab3c$, where $a\\ne 0$, then $c$ equals \n$\\text{(A)} 0\\qquad \\text{(B)}1 \\qquad \\text{(C)} 3\\qquad \\text{(D)} 4\\qquad \\text{(E)} \\text{not uniquely determined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $p_1,p_2,p_3,...$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between $0$ and $1$. For positive integer $k$, define\n$x_{k}=\\begin{cases}0&\\text{ if }x_{k-1}=0\\\\ \\left\\{\\frac{p_{k}}{x_{k-1}}\\right\\}&\\text{ if }x_{k-1}\\ne0\\end{cases}$\nwhere $\\{x\\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x-\\lfloor{x}\\rfloor$ where $\\lfloor{x}\\rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 0$?\n$\\textbf{(A)} ~\\frac{8a^2}{(a+1)^2}\\qquad\\textbf{(B)} ~\\frac{4a}{a+1}\\qquad\\textbf{(C)} ~\\frac{8a}{a+1}\\qquad\\textbf{(D)} ~\\frac{8a^2}{a^2+1}\\qquad\\textbf{(E)} ~\\frac{8a}{a^2+1}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2176", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?\n$\\textbf{(A)} ~\\frac{8a^2}{(a+1)^2}\\qquad\\textbf{(B)} ~\\frac{4a}{a+1}\\qquad\\textbf{(C)} ~\\frac{8a}{a+1}\\qquad\\textbf{(D)} ~\\frac{8a^2}{a^2+1}\\qquad\\textbf{(E)} ~\\frac{8a}{a^2+1}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\\%$ of the juniors and $10\\%$ of the seniors are on the debate team. How many juniors are in the program?\n$\\textbf{(A)} ~5 \\qquad\\textbf{(B)} ~6 \\qquad\\textbf{(C)} ~8 \\qquad\\textbf{(D)} ~11 \\qquad\\textbf{(E)} ~20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2177", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\\%$ of the juniors and $10\\%$ of the seniors are on the debate team. How many juniors are in the program?\n$\\textbf{(A)} ~5 \\qquad\\textbf{(B)} ~6 \\qquad\\textbf{(C)} ~8 \\qquad\\textbf{(D)} ~11 \\qquad\\textbf{(E)} ~20$" + } + }, + { + "question": "Return your final response within \\boxed{}. A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?\n[asy] path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(3,0)*p); draw(shift(3,1)*p); draw(shift(4,0)*p); draw(shift(4,1)*p); draw(shift(7,0)*p); draw(shift(7,1)*p); draw(shift(7,2)*p); draw(shift(8,0)*p); draw(shift(8,1)*p); draw(shift(8,2)*p); draw(shift(9,0)*p); draw(shift(9,1)*p); draw(shift(9,2)*p);[/asy]\n$\\text{(A)}\\ 11 \\qquad \\text{(B)}\\ 12 \\qquad \\text{(C)}\\ 13 \\qquad \\text{(D)}\\ 14 \\qquad \\text{(E)}\\ 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2178", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?\n[asy] path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(3,0)*p); draw(shift(3,1)*p); draw(shift(4,0)*p); draw(shift(4,1)*p); draw(shift(7,0)*p); draw(shift(7,1)*p); draw(shift(7,2)*p); draw(shift(8,0)*p); draw(shift(8,1)*p); draw(shift(8,2)*p); draw(shift(9,0)*p); draw(shift(9,1)*p); draw(shift(9,2)*p);[/asy]\n$\\text{(A)}\\ 11 \\qquad \\text{(B)}\\ 12 \\qquad \\text{(C)}\\ 13 \\qquad \\text{(D)}\\ 14 \\qquad \\text{(E)}\\ 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \\le x \\le 2$, $-2 \\le y \\le 2$ at each step?\n$\\textbf{(A)}\\ 92 \\qquad \\textbf{(B)}\\ 144 \\qquad \\textbf{(C)}\\ 1568 \\qquad \\textbf{(D)}\\ 1698 \\qquad \\textbf{(E)}\\ 12,800$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2179", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \\le x \\le 2$, $-2 \\le y \\le 2$ at each step?\n$\\textbf{(A)}\\ 92 \\qquad \\textbf{(B)}\\ 144 \\qquad \\textbf{(C)}\\ 1568 \\qquad \\textbf{(D)}\\ 1698 \\qquad \\textbf{(E)}\\ 12,800$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set ${0,1,2,3}.$ For how many such quadruples is it true that $a\\cdot d-b\\cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0\\cdot 1-3\\cdot 1 = -3$ is odd.)\n$\\textbf{(A) } 48 \\qquad \\textbf{(B) } 64 \\qquad \\textbf{(C) } 96 \\qquad \\textbf{(D) } 128 \\qquad \\textbf{(E) } 192$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2180", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set ${0,1,2,3}.$ For how many such quadruples is it true that $a\\cdot d-b\\cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0\\cdot 1-3\\cdot 1 = -3$ is odd.)\n$\\textbf{(A) } 48 \\qquad \\textbf{(B) } 64 \\qquad \\textbf{(C) } 96 \\qquad \\textbf{(D) } 128 \\qquad \\textbf{(E) } 192$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2181", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is:\n$\\textbf{(A)}\\ 6\\pi \\qquad \\textbf{(B)}\\ 6 \\qquad \\textbf{(C)}\\ 12 \\qquad \\textbf{(D)}\\ 36 \\qquad \\textbf{(E)}\\ 36\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2182", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is:\n$\\textbf{(A)}\\ 6\\pi \\qquad \\textbf{(B)}\\ 6 \\qquad \\textbf{(C)}\\ 12 \\qquad \\textbf{(D)}\\ 36 \\qquad \\textbf{(E)}\\ 36\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\\left(\\tfrac{1+\\sqrt{3}i}{2}\\right)=2015+2019\\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "53", + "index": "Sky-T1_10k_2183", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\\left(\\tfrac{1+\\sqrt{3}i}{2}\\right)=2015+2019\\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$." + } + }, + { + "question": "Return your final response within \\boxed{}. Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?\n$\\textbf{(A)}\\ 60 \\qquad \\textbf{(B)}\\ 80 \\qquad \\textbf{(C)}\\ 100 \\qquad \\textbf{(D)}\\ 120 \\qquad \\textbf{(E)}\\ 140$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2184", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?\n$\\textbf{(A)}\\ 60 \\qquad \\textbf{(B)}\\ 80 \\qquad \\textbf{(C)}\\ 100 \\qquad \\textbf{(D)}\\ 120 \\qquad \\textbf{(E)}\\ 140$" + } + }, + { + "question": "Return your final response within \\boxed{}. A refrigerator is offered at sale at $250.00 less successive discounts of 20% and 15%. The sale price of the refrigerator is:\n$\\textbf{(A) } \\text{35\\% less than 250.00} \\qquad \\textbf{(B) } \\text{65\\% of 250.00} \\qquad \\textbf{(C) } \\text{77\\% of 250.00} \\qquad \\textbf{(D) } \\text{68\\% of 250.00} \\qquad \\textbf{(E) } \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2185", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A refrigerator is offered at sale at $250.00 less successive discounts of 20% and 15%. The sale price of the refrigerator is:\n$\\textbf{(A) } \\text{35\\% less than 250.00} \\qquad \\textbf{(B) } \\text{65\\% of 250.00} \\qquad \\textbf{(C) } \\text{77\\% of 250.00} \\qquad \\textbf{(D) } \\text{68\\% of 250.00} \\qquad \\textbf{(E) } \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{2^1+2^0+2^{-1}}{2^{-2}+2^{-3}+2^{-4}}$ equals \n$\\text{(A)} \\ 6 \\qquad \\text{(B)} \\ 8 \\qquad \\text{(C)} \\ \\frac{31}{2} \\qquad \\text{(D)} \\ 24 \\qquad \\text{(E)} \\ 512$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2186", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{2^1+2^0+2^{-1}}{2^{-2}+2^{-3}+2^{-4}}$ equals \n$\\text{(A)} \\ 6 \\qquad \\text{(B)} \\ 8 \\qquad \\text{(C)} \\ \\frac{31}{2} \\qquad \\text{(D)} \\ 24 \\qquad \\text{(E)} \\ 512$" + } + }, + { + "question": "Return your final response within \\boxed{}. A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?\n\n$\\textbf{(A)}\\ 238\\pi \\qquad \\textbf{(B)}\\ 240\\pi \\qquad \\textbf{(C)}\\ 260\\pi \\qquad \\textbf{(D)}\\ 280\\pi \\qquad \\textbf{(E)}\\ 500\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2187", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?\n\n$\\textbf{(A)}\\ 238\\pi \\qquad \\textbf{(B)}\\ 240\\pi \\qquad \\textbf{(C)}\\ 260\\pi \\qquad \\textbf{(D)}\\ 280\\pi \\qquad \\textbf{(E)}\\ 500\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of solutions to $\\{1,~2\\}\\subseteq~X~\\subseteq~\\{1,~2,~3,~4,~5\\}$, where $X$ is a subset of $\\{1,~2,~3,~4,~5\\}$ is\n$\\textbf{(A) }2\\qquad \\textbf{(B) }4\\qquad \\textbf{(C) }6\\qquad \\textbf{(D) }8\\qquad \\textbf{(E) }\\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2188", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of solutions to $\\{1,~2\\}\\subseteq~X~\\subseteq~\\{1,~2,~3,~4,~5\\}$, where $X$ is a subset of $\\{1,~2,~3,~4,~5\\}$ is\n$\\textbf{(A) }2\\qquad \\textbf{(B) }4\\qquad \\textbf{(C) }6\\qquad \\textbf{(D) }8\\qquad \\textbf{(E) }\\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $b>1$, $\\sin x>0$, $\\cos x>0$, and $\\log_b \\sin x = a$, then $\\log_b \\cos x$ equals\n$\\text{(A)} \\ 2\\log_b(1-b^{a/2}) ~~\\text{(B)} \\ \\sqrt{1-a^2} ~~\\text{(C)} \\ b^{a^2} ~~\\text{(D)} \\ \\frac 12 \\log_b(1-b^{2a}) ~~\\text{(E)} \\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2189", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $b>1$, $\\sin x>0$, $\\cos x>0$, and $\\log_b \\sin x = a$, then $\\log_b \\cos x$ equals\n$\\text{(A)} \\ 2\\log_b(1-b^{a/2}) ~~\\text{(B)} \\ \\sqrt{1-a^2} ~~\\text{(C)} \\ b^{a^2} ~~\\text{(D)} \\ \\frac 12 \\log_b(1-b^{2a}) ~~\\text{(E)} \\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For any real value of $x$ the maximum value of $8x - 3x^2$ is: \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ \\frac{8}3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\frac{16}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2190", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For any real value of $x$ the maximum value of $8x - 3x^2$ is: \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ \\frac{8}3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\frac{16}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Steve's empty swimming pool will hold $24,000$ gallons of water when full. It will be filled by $4$ hoses, each of which supplies $2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool?\n$\\textbf{(A)}\\ 40 \\qquad \\textbf{(B)}\\ 42 \\qquad \\textbf{(C)}\\ 44 \\qquad \\textbf{(D)}\\ 46 \\qquad \\textbf{(E)}\\ 48$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2191", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Steve's empty swimming pool will hold $24,000$ gallons of water when full. It will be filled by $4$ hoses, each of which supplies $2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool?\n$\\textbf{(A)}\\ 40 \\qquad \\textbf{(B)}\\ 42 \\qquad \\textbf{(C)}\\ 44 \\qquad \\textbf{(D)}\\ 46 \\qquad \\textbf{(E)}\\ 48$" + } + }, + { + "question": "Return your final response within \\boxed{}. The taxi fare in Gotham City is $2.40 for the first $\\frac12$ mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?\n$\\textbf{(A) }3.0\\qquad\\textbf{(B) }3.25\\qquad\\textbf{(C) }3.3\\qquad\\textbf{(D) }3.5\\qquad\\textbf{(E) }3.75$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2192", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The taxi fare in Gotham City is $2.40 for the first $\\frac12$ mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?\n$\\textbf{(A) }3.0\\qquad\\textbf{(B) }3.25\\qquad\\textbf{(C) }3.3\\qquad\\textbf{(D) }3.5\\qquad\\textbf{(E) }3.75$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$ and $CA$ are, respectively, $x+75^{\\circ} , 2x+25^{\\circ},3x-22^{\\circ}$. Then one interior angle of the triangle is:\n$\\text{(A) } 57\\tfrac{1}{2}^{\\circ}\\quad \\text{(B) } 59^{\\circ}\\quad \\text{(C) } 60^{\\circ}\\quad \\text{(D) } 61^{\\circ}\\quad \\text{(E) } 122^{\\circ}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2193", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$ and $CA$ are, respectively, $x+75^{\\circ} , 2x+25^{\\circ},3x-22^{\\circ}$. Then one interior angle of the triangle is:\n$\\text{(A) } 57\\tfrac{1}{2}^{\\circ}\\quad \\text{(B) } 59^{\\circ}\\quad \\text{(C) } 60^{\\circ}\\quad \\text{(D) } 61^{\\circ}\\quad \\text{(E) } 122^{\\circ}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other?\n$\\textbf{(A) 1:24}\\qquad \\textbf{(B) 1:28}\\qquad \\textbf{(C) 1:36}\\qquad \\textbf{(D) 1:40}\\qquad \\textbf{(E) 1:48}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2194", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other?\n$\\textbf{(A) 1:24}\\qquad \\textbf{(B) 1:28}\\qquad \\textbf{(C) 1:36}\\qquad \\textbf{(D) 1:40}\\qquad \\textbf{(E) 1:48}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The smallest [sum](https://artofproblemsolving.com/wiki/index.php/Sum) one could get by adding three different numbers from the [set](https://artofproblemsolving.com/wiki/index.php/Set) $\\{ 7,25,-1,12,-3 \\}$ is\n$\\text{(A)}\\ -3 \\qquad \\text{(B)}\\ -1 \\qquad \\text{(C)}\\ 3 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 21$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2195", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The smallest [sum](https://artofproblemsolving.com/wiki/index.php/Sum) one could get by adding three different numbers from the [set](https://artofproblemsolving.com/wiki/index.php/Set) $\\{ 7,25,-1,12,-3 \\}$ is\n$\\text{(A)}\\ -3 \\qquad \\text{(B)}\\ -1 \\qquad \\text{(C)}\\ 3 \\qquad \\text{(D)}\\ 5 \\qquad \\text{(E)}\\ 21$" + } + }, + { + "question": "Return your final response within \\boxed{}. In year $N$, the $300^{\\text{th}}$ day of the year is a Tuesday. In year $N+1$, the $200^{\\text{th}}$ day is also a Tuesday. On what day of the week did the $100$th day of year $N-1$ occur?\n$\\text {(A)}\\ \\text{Thursday} \\qquad \\text {(B)}\\ \\text{Friday}\\qquad \\text {(C)}\\ \\text{Saturday}\\qquad \\text {(D)}\\ \\text{Sunday}\\qquad \\text {(E)}\\ \\text{Monday}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2196", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In year $N$, the $300^{\\text{th}}$ day of the year is a Tuesday. In year $N+1$, the $200^{\\text{th}}$ day is also a Tuesday. On what day of the week did the $100$th day of year $N-1$ occur?\n$\\text {(A)}\\ \\text{Thursday} \\qquad \\text {(B)}\\ \\text{Friday}\\qquad \\text {(C)}\\ \\text{Saturday}\\qquad \\text {(D)}\\ \\text{Sunday}\\qquad \\text {(E)}\\ \\text{Monday}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $xy \\equal{} b$ (Error compiling LaTeX. Unknown error_msg) and $\\frac{1}{x^2} \\plus{} \\frac{1}{y^2} \\equal{} a$ (Error compiling LaTeX. Unknown error_msg), then $(x \\plus{} y)^2$ (Error compiling LaTeX. Unknown error_msg) equals:\n$\\textbf{(A)}\\ (a \\plus{} 2b)^2\\qquad \n\\textbf{(B)}\\ a^2 \\plus{} b^2\\qquad \n\\textbf{(C)}\\ b(ab \\plus{} 2)\\qquad \n\\textbf{(D)}\\ ab(b \\plus{} 2)\\qquad \n\\textbf{(E)}\\ \\frac{1}{a} \\plus{} 2b$ (Error compiling LaTeX. Unknown error_msg)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2197", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $xy \\equal{} b$ (Error compiling LaTeX. Unknown error_msg) and $\\frac{1}{x^2} \\plus{} \\frac{1}{y^2} \\equal{} a$ (Error compiling LaTeX. Unknown error_msg), then $(x \\plus{} y)^2$ (Error compiling LaTeX. Unknown error_msg) equals:\n$\\textbf{(A)}\\ (a \\plus{} 2b)^2\\qquad \n\\textbf{(B)}\\ a^2 \\plus{} b^2\\qquad \n\\textbf{(C)}\\ b(ab \\plus{} 2)\\qquad \n\\textbf{(D)}\\ ab(b \\plus{} 2)\\qquad \n\\textbf{(E)}\\ \\frac{1}{a} \\plus{} 2b$ (Error compiling LaTeX. Unknown error_msg)" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $m$ be a positive integer and let the lines $13x+11y=700$ and $y=mx-1$ intersect in a point whose coordinates are integers. Then m can be:\n$\\text{(A) 4 only} \\quad \\text{(B) 5 only} \\quad \\text{(C) 6 only} \\quad \\text{(D) 7 only} \\\\ \\text{(E) one of the integers 4,5,6,7 and one other positive integer}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2198", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $m$ be a positive integer and let the lines $13x+11y=700$ and $y=mx-1$ intersect in a point whose coordinates are integers. Then m can be:\n$\\text{(A) 4 only} \\quad \\text{(B) 5 only} \\quad \\text{(C) 6 only} \\quad \\text{(D) 7 only} \\\\ \\text{(E) one of the integers 4,5,6,7 and one other positive integer}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\\tan \\theta$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "47", + "index": "Sky-T1_10k_2199", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\\tan \\theta$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. Find the positive integer $n\\,$ for which\n\\[\\lfloor\\log_2{1}\\rfloor+\\lfloor\\log_2{2}\\rfloor+\\lfloor\\log_2{3}\\rfloor+\\cdots+\\lfloor\\log_2{n}\\rfloor=1994\\]\n(For real $x\\,$, $\\lfloor x\\rfloor\\,$ is the greatest integer $\\le x.\\,$)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "312", + "index": "Sky-T1_10k_2200", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the positive integer $n\\,$ for which\n\\[\\lfloor\\log_2{1}\\rfloor+\\lfloor\\log_2{2}\\rfloor+\\lfloor\\log_2{3}\\rfloor+\\cdots+\\lfloor\\log_2{n}\\rfloor=1994\\]\n(For real $x\\,$, $\\lfloor x\\rfloor\\,$ is the greatest integer $\\le x.\\,$)" + } + }, + { + "question": "Return your final response within \\boxed{}. It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many seconds would it take Clea to ride the escalator down when she is not walking?\n$\\textbf{(A)}\\ 36\\qquad\\textbf{(B)}\\ 40\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 52$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2201", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many seconds would it take Clea to ride the escalator down when she is not walking?\n$\\textbf{(A)}\\ 36\\qquad\\textbf{(B)}\\ 40\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 52$" + } + }, + { + "question": "Return your final response within \\boxed{}. On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?\n$\\textbf{(A) }50\\qquad\\textbf{(B) }70\\qquad\\textbf{(C) }80\\qquad\\textbf{(D) }90\\qquad \\textbf{(E) }100$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2202", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?\n$\\textbf{(A) }50\\qquad\\textbf{(B) }70\\qquad\\textbf{(C) }80\\qquad\\textbf{(D) }90\\qquad \\textbf{(E) }100$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\triangle BAD$ is right-angled at $B$. On $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC$. The magnitude of $\\angle DAB$ is:\n$\\textbf{(A)}\\ 67\\tfrac{1}{2}^{\\circ}\\qquad \\textbf{(B)}\\ 60^{\\circ}\\qquad \\textbf{(C)}\\ 45^{\\circ}\\qquad \\textbf{(D)}\\ 30^{\\circ}\\qquad \\textbf{(E)}\\ 22\\tfrac{1}{2}^{\\circ}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2203", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\triangle BAD$ is right-angled at $B$. On $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC$. The magnitude of $\\angle DAB$ is:\n$\\textbf{(A)}\\ 67\\tfrac{1}{2}^{\\circ}\\qquad \\textbf{(B)}\\ 60^{\\circ}\\qquad \\textbf{(C)}\\ 45^{\\circ}\\qquad \\textbf{(D)}\\ 30^{\\circ}\\qquad \\textbf{(E)}\\ 22\\tfrac{1}{2}^{\\circ}$" + } + }, + { + "question": "Return your final response within \\boxed{}. [asy] void fillsq(int x, int y){ fill((x,y)--(x+1,y)--(x+1,y+1)--(x,y+1)--cycle, mediumgray); } int i; fillsq(1,0);fillsq(4,0);fillsq(6,0); fillsq(0,1);fillsq(1,1);fillsq(2,1);fillsq(4,1);fillsq(5,1); fillsq(0,2);fillsq(2,2);fillsq(4,2); fillsq(0,3);fillsq(1,3);fillsq(4,3);fillsq(5,3); for(i=0; i<=7; ++i){draw((i,0)--(i,4),black+0.5);} for(i=0; i<=4; ++i){draw((0,i)--(7,i),black+0.5);} draw((3,1)--(3,3)--(7,3)--(7,1)--cycle,black+1); [/asy]\n\n(a) Suppose that each square of a $4\\times 7$ chessboard, as shown above, is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as the one outlined in the figure) whose four distinct unit corner squares are all of the same color.\n(b) Exhibit a black-white coloring of a $4\\times 6$ board in which the four corner squares of every rectangle, as described above, are not all of the same color.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "Such a rectangle must exist in any 4×7 coloring.", + "index": "Sky-T1_10k_2204", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [asy] void fillsq(int x, int y){ fill((x,y)--(x+1,y)--(x+1,y+1)--(x,y+1)--cycle, mediumgray); } int i; fillsq(1,0);fillsq(4,0);fillsq(6,0); fillsq(0,1);fillsq(1,1);fillsq(2,1);fillsq(4,1);fillsq(5,1); fillsq(0,2);fillsq(2,2);fillsq(4,2); fillsq(0,3);fillsq(1,3);fillsq(4,3);fillsq(5,3); for(i=0; i<=7; ++i){draw((i,0)--(i,4),black+0.5);} for(i=0; i<=4; ++i){draw((0,i)--(7,i),black+0.5);} draw((3,1)--(3,3)--(7,3)--(7,1)--cycle,black+1); [/asy]\n\n(a) Suppose that each square of a $4\\times 7$ chessboard, as shown above, is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as the one outlined in the figure) whose four distinct unit corner squares are all of the same color.\n(b) Exhibit a black-white coloring of a $4\\times 6$ board in which the four corner squares of every rectangle, as described above, are not all of the same color." + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of the product \n\\[\\left(\\frac{1\\cdot3}{2\\cdot2}\\right)\\left(\\frac{2\\cdot4}{3\\cdot3}\\right)\\left(\\frac{3\\cdot5}{4\\cdot4}\\right)\\cdots\\left(\\frac{97\\cdot99}{98\\cdot98}\\right)\\left(\\frac{98\\cdot100}{99\\cdot99}\\right)?\\]\n$\\textbf{(A) }\\frac{1}{2}\\qquad\\textbf{(B) }\\frac{50}{99}\\qquad\\textbf{(C) }\\frac{9800}{9801}\\qquad\\textbf{(D) }\\frac{100}{99}\\qquad\\textbf{(E) }50$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2205", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of the product \n\\[\\left(\\frac{1\\cdot3}{2\\cdot2}\\right)\\left(\\frac{2\\cdot4}{3\\cdot3}\\right)\\left(\\frac{3\\cdot5}{4\\cdot4}\\right)\\cdots\\left(\\frac{97\\cdot99}{98\\cdot98}\\right)\\left(\\frac{98\\cdot100}{99\\cdot99}\\right)?\\]\n$\\textbf{(A) }\\frac{1}{2}\\qquad\\textbf{(B) }\\frac{50}{99}\\qquad\\textbf{(C) }\\frac{9800}{9801}\\qquad\\textbf{(D) }\\frac{100}{99}\\qquad\\textbf{(E) }50$" + } + }, + { + "question": "Return your final response within \\boxed{}. Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\\overline{AD}$, point $N$ is the trisection point of $\\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\\overline{CM}$ and $\\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\\overline{BP}$ bisects the area of $BCON$. Find the area of $\\triangle CDP$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "546", + "index": "Sky-T1_10k_2206", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\\overline{AD}$, point $N$ is the trisection point of $\\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\\overline{CM}$ and $\\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\\overline{BP}$ bisects the area of $BCON$. Find the area of $\\triangle CDP$." + } + }, + { + "question": "Return your final response within \\boxed{}. A town's population increased by $1,200$ people, and then this new population decreased by $11\\%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population?\n$\\mathrm{(A)\\ } 1,200 \\qquad \\mathrm{(B) \\ }11,200 \\qquad \\mathrm{(C) \\ } 9,968 \\qquad \\mathrm{(D) \\ } 10,000 \\qquad \\mathrm{(E) \\ }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2207", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A town's population increased by $1,200$ people, and then this new population decreased by $11\\%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population?\n$\\mathrm{(A)\\ } 1,200 \\qquad \\mathrm{(B) \\ }11,200 \\qquad \\mathrm{(C) \\ } 9,968 \\qquad \\mathrm{(D) \\ } 10,000 \\qquad \\mathrm{(E) \\ }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \\ldots\\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). What is the [remainder](https://artofproblemsolving.com/wiki/index.php/Remainder) when the 1994th term of the sequence is divided by 1000?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "063", + "index": "Sky-T1_10k_2208", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \\ldots\\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). What is the [remainder](https://artofproblemsolving.com/wiki/index.php/Remainder) when the 1994th term of the sequence is divided by 1000?" + } + }, + { + "question": "Return your final response within \\boxed{}. The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants:\n$\\textbf{(A)}\\ \\text{I and II}\\qquad \\textbf{(B)}\\ \\text{II and III}\\qquad \\textbf{(C)}\\ \\text{I and III}\\qquad \\textbf{(D)}\\ \\text{III and IV}\\qquad \\textbf{(E)}\\ \\text{I and IV}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2209", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants:\n$\\textbf{(A)}\\ \\text{I and II}\\qquad \\textbf{(B)}\\ \\text{II and III}\\qquad \\textbf{(C)}\\ \\text{I and III}\\qquad \\textbf{(D)}\\ \\text{III and IV}\\qquad \\textbf{(E)}\\ \\text{I and IV}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "195", + "index": "Sky-T1_10k_2210", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer." + } + }, + { + "question": "Return your final response within \\boxed{}. The [sum](https://artofproblemsolving.com/wiki/index.php/Sum) $2\\frac17+3\\frac12+5\\frac{1}{19}$ is between\n$\\text{(A)}\\ 10\\text{ and }10\\frac12 \\qquad \\text{(B)}\\ 10\\frac12 \\text{ and } 11 \\qquad \\text{(C)}\\ 11\\text{ and }11\\frac12 \\qquad \\text{(D)}\\ 11\\frac12 \\text{ and }12 \\qquad \\text{(E)}\\ 12\\text{ and }12\\frac12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2211", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The [sum](https://artofproblemsolving.com/wiki/index.php/Sum) $2\\frac17+3\\frac12+5\\frac{1}{19}$ is between\n$\\text{(A)}\\ 10\\text{ and }10\\frac12 \\qquad \\text{(B)}\\ 10\\frac12 \\text{ and } 11 \\qquad \\text{(C)}\\ 11\\text{ and }11\\frac12 \\qquad \\text{(D)}\\ 11\\frac12 \\text{ and }12 \\qquad \\text{(E)}\\ 12\\text{ and }12\\frac12$" + } + }, + { + "question": "Return your final response within \\boxed{}. Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is\n\n$\\textrm{(A)}\\ 30\\qquad\\textrm{(B)}\\ 200\\qquad\\textrm{(C)}\\ 410\\qquad\\textrm{(D)}\\ 420\\qquad\\textrm{(E)}\\ 430$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2212", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is\n\n$\\textrm{(A)}\\ 30\\qquad\\textrm{(B)}\\ 200\\qquad\\textrm{(C)}\\ 410\\qquad\\textrm{(D)}\\ 420\\qquad\\textrm{(E)}\\ 430$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P_1, \\ldots, P_{2n}$ be $2n$ distinct points on the unit circle $x^2 + y^2 = 1$ other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ of them red and exactly $n$ of them blue. Let $R_1, \\ldots, R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ traveling counterclockwise around the circle from $R_2$, and so on, until we have labeled all the blue points $B_1, \\ldots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \\rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \\ldots, R_n$ of the red points.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The number of such arcs is independent of the ordering of the red points.", + "index": "Sky-T1_10k_2213", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P_1, \\ldots, P_{2n}$ be $2n$ distinct points on the unit circle $x^2 + y^2 = 1$ other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ of them red and exactly $n$ of them blue. Let $R_1, \\ldots, R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ traveling counterclockwise around the circle from $R_2$, and so on, until we have labeled all the blue points $B_1, \\ldots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \\rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \\ldots, R_n$ of the red points." + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be\n$\\text{(A) } 200\\quad\\text{(B) } 201\\quad\\text{(C) } 202\\quad\\text{(D) } 203\\quad\\text{(E) } 204$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2214", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be\n$\\text{(A) } 200\\quad\\text{(B) } 201\\quad\\text{(C) } 202\\quad\\text{(D) } 203\\quad\\text{(E) } 204$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is\n\\[\\sum^{100}_{i=1} \\sum^{100}_{j=1} (i+j) ?\\]\n$\\textbf{(A) }100{,}100 \\qquad \\textbf{(B) }500{,}500\\qquad \\textbf{(C) }505{,}000 \\qquad \\textbf{(D) }1{,}001{,}000 \\qquad \\textbf{(E) }1{,}010{,}000 \\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2215", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is\n\\[\\sum^{100}_{i=1} \\sum^{100}_{j=1} (i+j) ?\\]\n$\\textbf{(A) }100{,}100 \\qquad \\textbf{(B) }500{,}500\\qquad \\textbf{(C) }505{,}000 \\qquad \\textbf{(D) }1{,}001{,}000 \\qquad \\textbf{(E) }1{,}010{,}000 \\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of $2-(-2)^{-2}$?\n$\\textbf{(A) } -2\\qquad\\textbf{(B) } \\dfrac{1}{16}\\qquad\\textbf{(C) } \\dfrac{7}{4}\\qquad\\textbf{(D) } \\dfrac{9}{4}\\qquad\\textbf{(E) } 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2216", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of $2-(-2)^{-2}$?\n$\\textbf{(A) } -2\\qquad\\textbf{(B) } \\dfrac{1}{16}\\qquad\\textbf{(C) } \\dfrac{7}{4}\\qquad\\textbf{(D) } \\dfrac{9}{4}\\qquad\\textbf{(E) } 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f(x) = 4x - x^{2}$. Give $x_{0}$, consider the sequence defined by $x_{n} = f(x_{n-1})$ for all $n \\ge 1$. \nFor how many real numbers $x_{0}$ will the sequence $x_{0}, x_{1}, x_{2}, \\ldots$ take on only a finite number of different values?\n$\\textbf{(A)}\\ \\text{0}\\qquad \\textbf{(B)}\\ \\text{1 or 2}\\qquad \\textbf{(C)}\\ \\text{3, 4, 5 or 6}\\qquad \\textbf{(D)}\\ \\text{more than 6 but finitely many}\\qquad \\textbf{(E) }\\infty$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2217", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f(x) = 4x - x^{2}$. Give $x_{0}$, consider the sequence defined by $x_{n} = f(x_{n-1})$ for all $n \\ge 1$. \nFor how many real numbers $x_{0}$ will the sequence $x_{0}, x_{1}, x_{2}, \\ldots$ take on only a finite number of different values?\n$\\textbf{(A)}\\ \\text{0}\\qquad \\textbf{(B)}\\ \\text{1 or 2}\\qquad \\textbf{(C)}\\ \\text{3, 4, 5 or 6}\\qquad \\textbf{(D)}\\ \\text{more than 6 but finitely many}\\qquad \\textbf{(E) }\\infty$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \\dots$ are divisible by $101$?\n$\\textbf{(A) }253 \\qquad \\textbf{(B) }504 \\qquad \\textbf{(C) }505 \\qquad \\textbf{(D) }506 \\qquad \\textbf{(E) }1009 \\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2218", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \\dots$ are divisible by $101$?\n$\\textbf{(A) }253 \\qquad \\textbf{(B) }504 \\qquad \\textbf{(C) }505 \\qquad \\textbf{(D) }506 \\qquad \\textbf{(E) }1009 \\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\log p+\\log q=\\log(p+q)$ only if:\n$\\textbf{(A) \\ }p=q=\\text{zero} \\qquad \\textbf{(B) \\ }p=\\frac{q^2}{1-q} \\qquad \\textbf{(C) \\ }p=q=1 \\qquad$\n$\\textbf{(D) \\ }p=\\frac{q}{q-1} \\qquad \\textbf{(E) \\ }p=\\frac{q}{q+1}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2219", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\log p+\\log q=\\log(p+q)$ only if:\n$\\textbf{(A) \\ }p=q=\\text{zero} \\qquad \\textbf{(B) \\ }p=\\frac{q^2}{1-q} \\qquad \\textbf{(C) \\ }p=q=1 \\qquad$\n$\\textbf{(D) \\ }p=\\frac{q}{q-1} \\qquad \\textbf{(E) \\ }p=\\frac{q}{q+1}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?\n\\[\\textrm{I. All alligators are creepy crawlers.}\\]\n\\[\\textrm{II. Some ferocious creatures are creepy crawlers.}\\]\n\\[\\textrm{III. Some alligators are not creepy crawlers.}\\]\n$\\mathrm{(A)}\\ \\text{I only} \\qquad\\mathrm{(B)}\\ \\text{II only} \\qquad\\mathrm{(C)}\\ \\text{III only} \\qquad\\mathrm{(D)}\\ \\text{II and III only} \\qquad\\mathrm{(E)}\\ \\text{None must be true}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2220", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?\n\\[\\textrm{I. All alligators are creepy crawlers.}\\]\n\\[\\textrm{II. Some ferocious creatures are creepy crawlers.}\\]\n\\[\\textrm{III. Some alligators are not creepy crawlers.}\\]\n$\\mathrm{(A)}\\ \\text{I only} \\qquad\\mathrm{(B)}\\ \\text{II only} \\qquad\\mathrm{(C)}\\ \\text{III only} \\qquad\\mathrm{(D)}\\ \\text{II and III only} \\qquad\\mathrm{(E)}\\ \\text{None must be true}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?\n$\\textbf{(A)}\\ 24.00 \\qquad \\textbf{(B)}\\ 24.50 \\qquad \\textbf{(C)}\\ 25.50 \\qquad \\textbf{(D)}\\ 28.00 \\qquad \\textbf{(E)}\\ 30.00$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2221", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?\n$\\textbf{(A)}\\ 24.00 \\qquad \\textbf{(B)}\\ 24.50 \\qquad \\textbf{(C)}\\ 25.50 \\qquad \\textbf{(D)}\\ 28.00 \\qquad \\textbf{(E)}\\ 30.00$" + } + }, + { + "question": "Return your final response within \\boxed{}. Under what conditions is $\\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?\n$\\textbf{(A) }$ It is never true.\n$\\textbf{(B) }$ It is true if and only if $ab=0$.\n$\\textbf{(C) }$ It is true if and only if $a+b\\ge 0$.\n$\\textbf{(D) }$ It is true if and only if $ab=0$ and $a+b\\ge 0$.\n$\\textbf{(E) }$ It is always true.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2222", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Under what conditions is $\\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?\n$\\textbf{(A) }$ It is never true.\n$\\textbf{(B) }$ It is true if and only if $ab=0$.\n$\\textbf{(C) }$ It is true if and only if $a+b\\ge 0$.\n$\\textbf{(D) }$ It is true if and only if $ab=0$ and $a+b\\ge 0$.\n$\\textbf{(E) }$ It is always true." + } + }, + { + "question": "Return your final response within \\boxed{}. If the ratio of the legs of a right triangle is $1: 2$, then the ratio of the corresponding segments of \nthe hypotenuse made by a perpendicular upon it from the vertex is: \n$\\textbf{(A)}\\ 1: 4\\qquad\\textbf{(B)}\\ 1:\\sqrt{2}\\qquad\\textbf{(C)}\\ 1: 2\\qquad\\textbf{(D)}\\ 1:\\sqrt{5}\\qquad\\textbf{(E)}\\ 1: 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2223", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the ratio of the legs of a right triangle is $1: 2$, then the ratio of the corresponding segments of \nthe hypotenuse made by a perpendicular upon it from the vertex is: \n$\\textbf{(A)}\\ 1: 4\\qquad\\textbf{(B)}\\ 1:\\sqrt{2}\\qquad\\textbf{(C)}\\ 1: 2\\qquad\\textbf{(D)}\\ 1:\\sqrt{5}\\qquad\\textbf{(E)}\\ 1: 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\frac{x}{y}=\\frac{3}{4}$, then the incorrect expression in the following is:\n$\\textbf{(A) \\ }\\frac{x+y}{y}=\\frac{7}{4} \\qquad \\textbf{(B) \\ }\\frac{y}{y-x}=\\frac{4}{1} \\qquad \\textbf{(C) \\ }\\frac{x+2y}{x}=\\frac{11}{3} \\qquad$\n$\\textbf{(D) \\ }\\frac{x}{2y}=\\frac{3}{8} \\qquad \\textbf{(E) \\ }\\frac{x-y}{y}=\\frac{1}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2224", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\frac{x}{y}=\\frac{3}{4}$, then the incorrect expression in the following is:\n$\\textbf{(A) \\ }\\frac{x+y}{y}=\\frac{7}{4} \\qquad \\textbf{(B) \\ }\\frac{y}{y-x}=\\frac{4}{1} \\qquad \\textbf{(C) \\ }\\frac{x+2y}{x}=\\frac{11}{3} \\qquad$\n$\\textbf{(D) \\ }\\frac{x}{2y}=\\frac{3}{8} \\qquad \\textbf{(E) \\ }\\frac{x-y}{y}=\\frac{1}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A convex polyhedron $Q$ has vertices $V_1,V_2,\\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges does $R$ have?\n$\\mathrm{(A)}\\ 200\\qquad \\mathrm{(B)}\\ 2n\\qquad \\mathrm{(C)}\\ 300\\qquad \\mathrm{(D)}\\ 400\\qquad \\mathrm{(E)}\\ 4n$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2225", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A convex polyhedron $Q$ has vertices $V_1,V_2,\\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges does $R$ have?\n$\\mathrm{(A)}\\ 200\\qquad \\mathrm{(B)}\\ 2n\\qquad \\mathrm{(C)}\\ 300\\qquad \\mathrm{(D)}\\ 400\\qquad \\mathrm{(E)}\\ 4n$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of \n\\[1-(-2)-3-(-4)-5-(-6)?\\]\n$\\textbf{(A)}\\ -20 \\qquad\\textbf{(B)}\\ -3 \\qquad\\textbf{(C)}\\ 3 \\qquad\\textbf{(D)}\\ 5 \\qquad\\textbf{(E)}\\ 21$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2226", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of \n\\[1-(-2)-3-(-4)-5-(-6)?\\]\n$\\textbf{(A)}\\ -20 \\qquad\\textbf{(B)}\\ -3 \\qquad\\textbf{(C)}\\ 3 \\qquad\\textbf{(D)}\\ 5 \\qquad\\textbf{(E)}\\ 21$" + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangle with positive integer side lengths in $\\text{cm}$ has area $A$ $\\text{cm}^2$ and perimeter $P$ $\\text{cm}$. Which of the following numbers cannot equal $A+P$?\n$\\textbf{(A) }100\\qquad\\textbf{(B) }102\\qquad\\textbf{(C) }104\\qquad\\textbf{(D) }106\\qquad\\textbf{(E) }108$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2227", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangle with positive integer side lengths in $\\text{cm}$ has area $A$ $\\text{cm}^2$ and perimeter $P$ $\\text{cm}$. Which of the following numbers cannot equal $A+P$?\n$\\textbf{(A) }100\\qquad\\textbf{(B) }102\\qquad\\textbf{(C) }104\\qquad\\textbf{(D) }106\\qquad\\textbf{(E) }108$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$, \nand $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ is:\n$\\textbf{(A)}\\ 60\\qquad \\textbf{(B)}\\ 70\\qquad \\textbf{(C)}\\ 75\\qquad \\textbf{(D)}\\ 80\\qquad \\textbf{(E)}\\ 85$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2228", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$, \nand $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ is:\n$\\textbf{(A)}\\ 60\\qquad \\textbf{(B)}\\ 70\\qquad \\textbf{(C)}\\ 75\\qquad \\textbf{(D)}\\ 80\\qquad \\textbf{(E)}\\ 85$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x=\\dfrac{1-i\\sqrt{3}}{2}$ where $i=\\sqrt{-1}$, then $\\dfrac{1}{x^2-x}$ is equal to\n$\\textbf{(A) }-2\\qquad \\textbf{(B) }-1\\qquad \\textbf{(C) }1+i\\sqrt{3}\\qquad \\textbf{(D) }1\\qquad \\textbf{(E) }2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2229", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x=\\dfrac{1-i\\sqrt{3}}{2}$ where $i=\\sqrt{-1}$, then $\\dfrac{1}{x^2-x}$ is equal to\n$\\textbf{(A) }-2\\qquad \\textbf{(B) }-1\\qquad \\textbf{(C) }1+i\\sqrt{3}\\qquad \\textbf{(D) }1\\qquad \\textbf{(E) }2$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the product of all positive odd integers less than $10000$?\n$\\text{(A)}\\ \\dfrac{10000!}{(5000!)^2}\\qquad \\text{(B)}\\ \\dfrac{10000!}{2^{5000}}\\qquad \\text{(C)}\\ \\dfrac{9999!}{2^{5000}}\\qquad \\text{(D)}\\ \\dfrac{10000!}{2^{5000} \\cdot 5000!}\\qquad \\text{(E)}\\ \\dfrac{5000!}{2^{5000}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2230", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the product of all positive odd integers less than $10000$?\n$\\text{(A)}\\ \\dfrac{10000!}{(5000!)^2}\\qquad \\text{(B)}\\ \\dfrac{10000!}{2^{5000}}\\qquad \\text{(C)}\\ \\dfrac{9999!}{2^{5000}}\\qquad \\text{(D)}\\ \\dfrac{10000!}{2^{5000} \\cdot 5000!}\\qquad \\text{(E)}\\ \\dfrac{5000!}{2^{5000}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The $7$-digit numbers $\\underline{7} \\underline{4} \\underline{A} \\underline{5} \\underline{2} \\underline{B} \\underline{1}$ and $\\underline{3} \\underline{2} \\underline{6} \\underline{A} \\underline{B} \\underline{4} \\underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$?\n$\\textbf{(A) }1\\qquad\\textbf{(B) }2\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }5\\qquad \\textbf{(E) }8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2231", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The $7$-digit numbers $\\underline{7} \\underline{4} \\underline{A} \\underline{5} \\underline{2} \\underline{B} \\underline{1}$ and $\\underline{3} \\underline{2} \\underline{6} \\underline{A} \\underline{B} \\underline{4} \\underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$?\n$\\textbf{(A) }1\\qquad\\textbf{(B) }2\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }5\\qquad \\textbf{(E) }8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "(0, 0, 0)", + "index": "Sky-T1_10k_2232", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$." + } + }, + { + "question": "Return your final response within \\boxed{}. For which non-zero real numbers $x$ is $\\frac{|x-|x|\\-|}{x}$ a positive integer? \n$\\textbf{(A)}\\ \\text{for negative } x \\text{ only} \\qquad \\\\ \\textbf{(B)}\\ \\text{for positive } x \\text{ only} \\qquad \\\\ \\textbf{(C)}\\ \\text{only for } x \\text{ an even integer} \\qquad \\\\ \\textbf{(D)}\\ \\text{for all non-zero real numbers } x \\\\ \\textbf{(E)}\\ \\text{for no non-zero real numbers } x$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2233", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For which non-zero real numbers $x$ is $\\frac{|x-|x|\\-|}{x}$ a positive integer? \n$\\textbf{(A)}\\ \\text{for negative } x \\text{ only} \\qquad \\\\ \\textbf{(B)}\\ \\text{for positive } x \\text{ only} \\qquad \\\\ \\textbf{(C)}\\ \\text{only for } x \\text{ an even integer} \\qquad \\\\ \\textbf{(D)}\\ \\text{for all non-zero real numbers } x \\\\ \\textbf{(E)}\\ \\text{for no non-zero real numbers } x$" + } + }, + { + "question": "Return your final response within \\boxed{}. Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?\n$\\textbf{(A)}\\dfrac{47}{256}\\qquad\\textbf{(B)}\\dfrac{3}{16}\\qquad\\textbf{(C) }\\dfrac{49}{256}\\qquad\\textbf{(D) }\\dfrac{25}{128}\\qquad\\textbf{(E) }\\dfrac{51}{256}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2234", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?\n$\\textbf{(A)}\\dfrac{47}{256}\\qquad\\textbf{(B)}\\dfrac{3}{16}\\qquad\\textbf{(C) }\\dfrac{49}{256}\\qquad\\textbf{(D) }\\dfrac{25}{128}\\qquad\\textbf{(E) }\\dfrac{51}{256}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations\n\\begin{align*} a + b &= -3, \\\\ ab + bc + ca &= -4, \\\\ abc + bcd + cda + dab &= 14, \\\\ abcd &= 30. \\end{align*}\nThere exist relatively prime positive integers $m$ and $n$ such that\n\\[a^2 + b^2 + c^2 + d^2 = \\frac{m}{n}.\\]Find $m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "145", + "index": "Sky-T1_10k_2235", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations\n\\begin{align*} a + b &= -3, \\\\ ab + bc + ca &= -4, \\\\ abc + bcd + cda + dab &= 14, \\\\ abcd &= 30. \\end{align*}\nThere exist relatively prime positive integers $m$ and $n$ such that\n\\[a^2 + b^2 + c^2 + d^2 = \\frac{m}{n}.\\]Find $m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. If $f(2x)=\\frac{2}{2+x}$ for all $x>0$, then $2f(x)=$\n$\\text{(A) } \\frac{2}{1+x}\\quad \\text{(B) } \\frac{2}{2+x}\\quad \\text{(C) } \\frac{4}{1+x}\\quad \\text{(D) } \\frac{4}{2+x}\\quad \\text{(E) } \\frac{8}{4+x}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2236", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $f(2x)=\\frac{2}{2+x}$ for all $x>0$, then $2f(x)=$\n$\\text{(A) } \\frac{2}{1+x}\\quad \\text{(B) } \\frac{2}{2+x}\\quad \\text{(C) } \\frac{4}{1+x}\\quad \\text{(D) } \\frac{4}{2+x}\\quad \\text{(E) } \\frac{8}{4+x}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?\n[asy] picture corner; draw(corner,(5,0)--(35,0)); draw(corner,(0,-5)--(0,-35)); for (int i=0; i<3; ++i){for (int j=0; j>-2; --j){if ((i-j)<3){add(corner,(50i,50j));}}} draw((5,-100)--(45,-100)); draw((155,0)--(185,0),dotted+linewidth(2)); draw((105,-50)--(135,-50),dotted+linewidth(2)); draw((100,-55)--(100,-85),dotted+linewidth(2)); draw((55,-100)--(85,-100),dotted+linewidth(2)); draw((50,-105)--(50,-135),dotted+linewidth(2)); draw((0,-105)--(0,-135),dotted+linewidth(2));[/asy]\n$\\textbf{(A)}\\ 1920 \\qquad \\textbf{(B)}\\ 1952 \\qquad \\textbf{(C)}\\ 1980 \\qquad \\textbf{(D)}\\ 2013 \\qquad \\textbf{(E)}\\ 3932$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2237", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?\n[asy] picture corner; draw(corner,(5,0)--(35,0)); draw(corner,(0,-5)--(0,-35)); for (int i=0; i<3; ++i){for (int j=0; j>-2; --j){if ((i-j)<3){add(corner,(50i,50j));}}} draw((5,-100)--(45,-100)); draw((155,0)--(185,0),dotted+linewidth(2)); draw((105,-50)--(135,-50),dotted+linewidth(2)); draw((100,-55)--(100,-85),dotted+linewidth(2)); draw((55,-100)--(85,-100),dotted+linewidth(2)); draw((50,-105)--(50,-135),dotted+linewidth(2)); draw((0,-105)--(0,-135),dotted+linewidth(2));[/asy]\n$\\textbf{(A)}\\ 1920 \\qquad \\textbf{(B)}\\ 1952 \\qquad \\textbf{(C)}\\ 1980 \\qquad \\textbf{(D)}\\ 2013 \\qquad \\textbf{(E)}\\ 3932$" + } + }, + { + "question": "Return your final response within \\boxed{}. A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?\n$\\textbf{(A)}\\ 30 \\qquad \\textbf{(B)}\\ 31 \\qquad \\textbf{(C)}\\ 32 \\qquad \\textbf{(D)}\\ 33 \\qquad \\textbf{(E)}\\ 34$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2238", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?\n$\\textbf{(A)}\\ 30 \\qquad \\textbf{(B)}\\ 31 \\qquad \\textbf{(C)}\\ 32 \\qquad \\textbf{(D)}\\ 33 \\qquad \\textbf{(E)}\\ 34$" + } + }, + { + "question": "Return your final response within \\boxed{}. The points of intersection of $xy = 12$ and $x^2 + y^2 = 25$ are joined in succession. The resulting figure is: \n$\\textbf{(A)}\\ \\text{a straight line}\\qquad \\textbf{(B)}\\ \\text{an equilateral triangle}\\qquad \\textbf{(C)}\\ \\text{a parallelogram} \\\\ \\textbf{(D)}\\ \\text{a rectangle} \\qquad \\textbf{(E)}\\ \\text{a square}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2239", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The points of intersection of $xy = 12$ and $x^2 + y^2 = 25$ are joined in succession. The resulting figure is: \n$\\textbf{(A)}\\ \\text{a straight line}\\qquad \\textbf{(B)}\\ \\text{an equilateral triangle}\\qquad \\textbf{(C)}\\ \\text{a parallelogram} \\\\ \\textbf{(D)}\\ \\text{a rectangle} \\qquad \\textbf{(E)}\\ \\text{a square}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. \"Look, daddy!\" she exclaims. \"That number is evenly divisible by the age of each of us kids!\" \"That's right,\" replies Mr. Jones, \"and the last two digits just happen to be my age.\" Which of the following is not the age of one of Mr. Jones's children?\n$\\mathrm{(A)}\\ 4\\qquad\\mathrm{(B)}\\ 5\\qquad\\mathrm{(C)}\\ 6\\qquad\\mathrm{(D)}\\ 7\\qquad\\mathrm{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2240", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. \"Look, daddy!\" she exclaims. \"That number is evenly divisible by the age of each of us kids!\" \"That's right,\" replies Mr. Jones, \"and the last two digits just happen to be my age.\" Which of the following is not the age of one of Mr. Jones's children?\n$\\mathrm{(A)}\\ 4\\qquad\\mathrm{(B)}\\ 5\\qquad\\mathrm{(C)}\\ 6\\qquad\\mathrm{(D)}\\ 7\\qquad\\mathrm{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the $12$-sided polygon $ABCDEFGHIJKL$, as shown. Each of its sides has length $4$, and each two consecutive sides form a right angle. Suppose that $\\overline{AG}$ and $\\overline{CH}$ meet at $M$. What is the area of quadrilateral $ABCM$?\n\n$\\text{(A)}\\ \\frac {44}{3}\\qquad \\text{(B)}\\ 16 \\qquad \\text{(C)}\\ \\frac {88}{5}\\qquad \\text{(D)}\\ 20 \\qquad \\text{(E)}\\ \\frac {62}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2241", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the $12$-sided polygon $ABCDEFGHIJKL$, as shown. Each of its sides has length $4$, and each two consecutive sides form a right angle. Suppose that $\\overline{AG}$ and $\\overline{CH}$ meet at $M$. What is the area of quadrilateral $ABCM$?\n\n$\\text{(A)}\\ \\frac {44}{3}\\qquad \\text{(B)}\\ 16 \\qquad \\text{(C)}\\ \\frac {88}{5}\\qquad \\text{(D)}\\ 20 \\qquad \\text{(E)}\\ \\frac {62}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\\frac{1}{2}$. The first term of the progression is:\n$\\textbf{(A) \\ }3 \\text{ or } 1\\frac{1}{2} \\qquad \\textbf{(B) \\ }1 \\qquad \\textbf{(C) \\ }2\\frac{1}{2} \\qquad \\textbf{(D) \\ }6 \\qquad \\textbf{(E) \\ }9\\text{ or }3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2242", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\\frac{1}{2}$. The first term of the progression is:\n$\\textbf{(A) \\ }3 \\text{ or } 1\\frac{1}{2} \\qquad \\textbf{(B) \\ }1 \\qquad \\textbf{(C) \\ }2\\frac{1}{2} \\qquad \\textbf{(D) \\ }6 \\qquad \\textbf{(E) \\ }9\\text{ or }3$" + } + }, + { + "question": "Return your final response within \\boxed{}. A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.\nI. The digit is 1.\nII. The digit is not 2.\nIII. The digit is 3.\nIV. The digit is not 4.\nWhich one of the following must necessarily be correct?\n$\\textbf{(A)}\\ \\text{I is true.} \\qquad \\textbf{(B)}\\ \\text{I is false.}\\qquad \\textbf{(C)}\\ \\text{II is true.} \\qquad \\textbf{(D)}\\ \\text{III is true.} \\qquad \\textbf{(E)}\\ \\text{IV is false.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2243", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.\nI. The digit is 1.\nII. The digit is not 2.\nIII. The digit is 3.\nIV. The digit is not 4.\nWhich one of the following must necessarily be correct?\n$\\textbf{(A)}\\ \\text{I is true.} \\qquad \\textbf{(B)}\\ \\text{I is false.}\\qquad \\textbf{(C)}\\ \\text{II is true.} \\qquad \\textbf{(D)}\\ \\text{III is true.} \\qquad \\textbf{(E)}\\ \\text{IV is false.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the largest number of solid $2\\text{-in} \\times 2\\text{-in} \\times 1\\text{-in}$ blocks that can fit in a $3\\text{-in} \\times 2\\text{-in}\\times3\\text{-in}$ box?\n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2244", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the largest number of solid $2\\text{-in} \\times 2\\text{-in} \\times 1\\text{-in}$ blocks that can fit in a $3\\text{-in} \\times 2\\text{-in}\\times3\\text{-in}$ box?\n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S$ be the set $\\{1,2,3,...,19\\}$. For $a,b \\in S$, define $a \\succ b$ to mean that either $0 < a - b \\le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \\succ y$, $y \\succ z$, and $z \\succ x$?\n$\\textbf{(A)} \\ 810 \\qquad \\textbf{(B)} \\ 855 \\qquad \\textbf{(C)} \\ 900 \\qquad \\textbf{(D)} \\ 950 \\qquad \\textbf{(E)} \\ 988$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2245", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S$ be the set $\\{1,2,3,...,19\\}$. For $a,b \\in S$, define $a \\succ b$ to mean that either $0 < a - b \\le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \\succ y$, $y \\succ z$, and $z \\succ x$?\n$\\textbf{(A)} \\ 810 \\qquad \\textbf{(B)} \\ 855 \\qquad \\textbf{(C)} \\ 900 \\qquad \\textbf{(D)} \\ 950 \\qquad \\textbf{(E)} \\ 988$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{16+8}{4-2}=$\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2246", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{16+8}{4-2}=$\n$\\text{(A)}\\ 4 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 16 \\qquad \\text{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. The expression $\\sqrt{25-t^2}+5$ equals zero for: \n$\\textbf{(A)}\\ \\text{no real or imaginary values of }t\\qquad\\textbf{(B)}\\ \\text{no real values of }t\\text{ only}\\\\ \\textbf{(C)}\\ \\text{no imaginary values of }t\\text{ only}\\qquad\\textbf{(D)}\\ t=0\\qquad\\textbf{(E)}\\ t=\\pm 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2247", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The expression $\\sqrt{25-t^2}+5$ equals zero for: \n$\\textbf{(A)}\\ \\text{no real or imaginary values of }t\\qquad\\textbf{(B)}\\ \\text{no real values of }t\\text{ only}\\\\ \\textbf{(C)}\\ \\text{no imaginary values of }t\\text{ only}\\qquad\\textbf{(D)}\\ t=0\\qquad\\textbf{(E)}\\ t=\\pm 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $11, \\Delta^k(u_n)=\\Delta^1(\\Delta^{k-1}(u_n))$. \nIf $u_n=n^3+n$, then $\\Delta^k(u_n)=0$ for all $n$\n$\\textbf{(A) }\\text{if }k=1\\qquad \\\\ \\textbf{(B) }\\text{if }k=2,\\text{ but not if }k=1\\qquad \\\\ \\textbf{(C) }\\text{if }k=3,\\text{ but not if }k=2\\qquad \\\\ \\textbf{(D) }\\text{if }k=4,\\text{ but not if }k=3\\qquad\\\\ \\textbf{(E) }\\text{for no value of }k$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2271", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a sequence $u_1,u_2\\dots$, define $\\Delta^1(u_n)=u_{n+1}-u_n$ and, for all integer \n$k>1, \\Delta^k(u_n)=\\Delta^1(\\Delta^{k-1}(u_n))$. \nIf $u_n=n^3+n$, then $\\Delta^k(u_n)=0$ for all $n$\n$\\textbf{(A) }\\text{if }k=1\\qquad \\\\ \\textbf{(B) }\\text{if }k=2,\\text{ but not if }k=1\\qquad \\\\ \\textbf{(C) }\\text{if }k=3,\\text{ but not if }k=2\\qquad \\\\ \\textbf{(D) }\\text{if }k=4,\\text{ but not if }k=3\\qquad\\\\ \\textbf{(E) }\\text{for no value of }k$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $L$ be the line with slope $\\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\\alpha,\\beta)$ in the new coordinate system. Find $\\alpha+\\beta$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "31", + "index": "Sky-T1_10k_2272", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $L$ be the line with slope $\\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\\alpha,\\beta)$ in the new coordinate system. Find $\\alpha+\\beta$." + } + }, + { + "question": "Return your final response within \\boxed{}. Quadrilateral $XABY$ is inscribed in the semicircle $\\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \\[\\dfrac{BY}{XP}+\\dfrac{CY}{XQ}=\\dfrac{AY}{AX}.\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\dfrac{BY}{XP} + \\dfrac{CY}{XQ} = \\dfrac{AY}{AX}", + "index": "Sky-T1_10k_2273", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Quadrilateral $XABY$ is inscribed in the semicircle $\\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \\[\\dfrac{BY}{XP}+\\dfrac{CY}{XQ}=\\dfrac{AY}{AX}.\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$?\n$\\textbf{(A)} ~3 \\qquad\\textbf{(B)} ~4\\qquad\\textbf{(C)} ~6\\qquad\\textbf{(D)} ~7\\qquad\\textbf{(E)} ~8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2274", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$?\n$\\textbf{(A)} ~3 \\qquad\\textbf{(B)} ~4\\qquad\\textbf{(C)} ~6\\qquad\\textbf{(D)} ~7\\qquad\\textbf{(E)} ~8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?\n$\\textbf{(A)}\\ 1,000\\qquad\\textbf{(B)}\\ 1,250\\qquad\\textbf{(C)}\\ 2,500\\qquad\\textbf{(D)}\\ 5,000\\qquad\\textbf{(E)}\\ 10,000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2275", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?\n$\\textbf{(A)}\\ 1,000\\qquad\\textbf{(B)}\\ 1,250\\qquad\\textbf{(C)}\\ 2,500\\qquad\\textbf{(D)}\\ 5,000\\qquad\\textbf{(E)}\\ 10,000$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tony works $2$ hours a day and is paid $$0.50$ per hour for each full year of his age. During a six month period Tony worked $50$ days and earned $$630$. How old was Tony at the end of the six month period?\n$\\mathrm{(A)}\\ 9 \\qquad \\mathrm{(B)}\\ 11 \\qquad \\mathrm{(C)}\\ 12 \\qquad \\mathrm{(D)}\\ 13 \\qquad \\mathrm{(E)}\\ 14$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2276", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tony works $2$ hours a day and is paid $$0.50$ per hour for each full year of his age. During a six month period Tony worked $50$ days and earned $$630$. How old was Tony at the end of the six month period?\n$\\mathrm{(A)}\\ 9 \\qquad \\mathrm{(B)}\\ 11 \\qquad \\mathrm{(C)}\\ 12 \\qquad \\mathrm{(D)}\\ 13 \\qquad \\mathrm{(E)}\\ 14$" + } + }, + { + "question": "Return your final response within \\boxed{}. The expression $21x^2 +ax +21$ is to be factored into two linear prime binomial factors with integer coefficients. This can be done if $a$ is:\n$\\textbf{(A)}\\ \\text{any odd number} \\qquad\\textbf{(B)}\\ \\text{some odd number} \\qquad\\textbf{(C)}\\ \\text{any even number}$\n$\\textbf{(D)}\\ \\text{some even number} \\qquad\\textbf{(E)}\\ \\text{zero}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2277", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The expression $21x^2 +ax +21$ is to be factored into two linear prime binomial factors with integer coefficients. This can be done if $a$ is:\n$\\textbf{(A)}\\ \\text{any odd number} \\qquad\\textbf{(B)}\\ \\text{some odd number} \\qquad\\textbf{(C)}\\ \\text{any even number}$\n$\\textbf{(D)}\\ \\text{some even number} \\qquad\\textbf{(E)}\\ \\text{zero}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?\n$\\textbf{(A) }55\\qquad\\textbf{(B) }60\\qquad\\textbf{(C) }65\\qquad\\textbf{(D) }70\\qquad\\textbf{(E) }75$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2278", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?\n$\\textbf{(A) }55\\qquad\\textbf{(B) }60\\qquad\\textbf{(C) }65\\qquad\\textbf{(D) }70\\qquad\\textbf{(E) }75$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?\n$\\textbf{(A)}\\ 15 \\qquad \\textbf{(B)}\\ 25 \\qquad \\textbf{(C)}\\ 35 \\qquad \\textbf{(D)}\\ 45 \\qquad \\textbf{(E)}\\ 55$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2279", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?\n$\\textbf{(A)}\\ 15 \\qquad \\textbf{(B)}\\ 25 \\qquad \\textbf{(C)}\\ 35 \\qquad \\textbf{(D)}\\ 45 \\qquad \\textbf{(E)}\\ 55$" + } + }, + { + "question": "Return your final response within \\boxed{}. [katex]\\dfrac{3\\times 5}{9\\times 11}\\times \\dfrac{7\\times 9\\times 11}{3\\times 5\\times 7}=[/katex]\n\n[katex]\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 0 \\qquad \\text{(C)}\\ 49 \\qquad \\text{(D)}\\ \\frac{1}{49} \\qquad \\text{(E)}\\ 50[/katex]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2280", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. [katex]\\dfrac{3\\times 5}{9\\times 11}\\times \\dfrac{7\\times 9\\times 11}{3\\times 5\\times 7}=[/katex]\n\n[katex]\\text{(A)}\\ 1 \\qquad \\text{(B)}\\ 0 \\qquad \\text{(C)}\\ 49 \\qquad \\text{(D)}\\ \\frac{1}{49} \\qquad \\text{(E)}\\ 50[/katex]" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of \\[\\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\\]\n$\\textbf{(A)}\\ -1\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ \\frac{5}{3}\\qquad\\textbf{(D)}\\ 2013\\qquad\\textbf{(E)}\\ 2^{4024}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2281", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of \\[\\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\\]\n$\\textbf{(A)}\\ -1\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ \\frac{5}{3}\\qquad\\textbf{(D)}\\ 2013\\qquad\\textbf{(E)}\\ 2^{4024}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $8\\cdot2^x = 5^{y + 8}$, then when $y = - 8,x =$\n$\\textbf{(A)}\\ - 4 \\qquad\\textbf{(B)}\\ - 3 \\qquad\\textbf{(C)}\\ 0 \\qquad\\textbf{(D)}\\ 4 \\qquad\\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2282", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $8\\cdot2^x = 5^{y + 8}$, then when $y = - 8,x =$\n$\\textbf{(A)}\\ - 4 \\qquad\\textbf{(B)}\\ - 3 \\qquad\\textbf{(C)}\\ 0 \\qquad\\textbf{(D)}\\ 4 \\qquad\\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. When $\\left ( 1 - \\frac{1}{a} \\right ) ^6$ is expanded the sum of the last three coefficients is:\n$\\textbf{(A)}\\ 22\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\qquad\\textbf{(D)}\\ -10\\qquad\\textbf{(E)}\\ -11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2283", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When $\\left ( 1 - \\frac{1}{a} \\right ) ^6$ is expanded the sum of the last three coefficients is:\n$\\textbf{(A)}\\ 22\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\qquad\\textbf{(D)}\\ -10\\qquad\\textbf{(E)}\\ -11$" + } + }, + { + "question": "Return your final response within \\boxed{}. The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when:\n$\\textbf{(A) \\ }x=-p \\qquad \\textbf{(B) \\ }x=\\frac{p}{2} \\qquad \\textbf{(C) \\ }x=-2p \\qquad \\textbf{(D) \\ }x=\\frac{p^2}{4q} \\qquad$\n$\\textbf{(E) \\ }x=\\frac{-p}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2284", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when:\n$\\textbf{(A) \\ }x=-p \\qquad \\textbf{(B) \\ }x=\\frac{p}{2} \\qquad \\textbf{(C) \\ }x=-2p \\qquad \\textbf{(D) \\ }x=\\frac{p^2}{4q} \\qquad$\n$\\textbf{(E) \\ }x=\\frac{-p}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of \\[(2^2-2)-(3^2-3)+(4^2-4)\\]\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } 2 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } 8 \\qquad\\textbf{(E) } 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2285", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of \\[(2^2-2)-(3^2-3)+(4^2-4)\\]\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } 2 \\qquad\\textbf{(C) } 5 \\qquad\\textbf{(D) } 8 \\qquad\\textbf{(E) } 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle{ABC}$, $\\angle{C} = 90^{\\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?\n$\\textbf{(A) }12+9\\sqrt{3}\\qquad\\textbf{(B) }18+6\\sqrt{3}\\qquad\\textbf{(C) }12+12\\sqrt{2}\\qquad\\textbf{(D) }30\\qquad\\textbf{(E) }32$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2286", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle{ABC}$, $\\angle{C} = 90^{\\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?\n$\\textbf{(A) }12+9\\sqrt{3}\\qquad\\textbf{(B) }18+6\\sqrt{3}\\qquad\\textbf{(C) }12+12\\sqrt{2}\\qquad\\textbf{(D) }30\\qquad\\textbf{(E) }32$" + } + }, + { + "question": "Return your final response within \\boxed{}. The school store sells 7 pencils and 8 notebooks for $\\mathdollar 4.15$. It also sells 5 pencils and 3 notebooks for $\\mathdollar 1.77$. How much do 16 pencils and 10 notebooks cost?\n$\\text{(A)}\\mathdollar 1.76 \\qquad \\text{(B)}\\mathdollar 5.84 \\qquad \\text{(C)}\\mathdollar 6.00 \\qquad \\text{(D)}\\mathdollar 6.16 \\qquad \\text{(E)}\\mathdollar 6.32$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2287", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The school store sells 7 pencils and 8 notebooks for $\\mathdollar 4.15$. It also sells 5 pencils and 3 notebooks for $\\mathdollar 1.77$. How much do 16 pencils and 10 notebooks cost?\n$\\text{(A)}\\mathdollar 1.76 \\qquad \\text{(B)}\\mathdollar 5.84 \\qquad \\text{(C)}\\mathdollar 6.00 \\qquad \\text{(D)}\\mathdollar 6.16 \\qquad \\text{(E)}\\mathdollar 6.32$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following statements is false?\n$\\mathrm{(A) \\ All\\ equilateral\\ triangles\\ are\\ congruent\\ to\\ each\\ other.}$\n$\\mathrm{(B) \\ All\\ equilateral\\ triangles\\ are\\ convex.}$\n$\\mathrm{(C) \\ All\\ equilateral\\ triangles\\ are\\ equianguilar.}$\n$\\mathrm{(D) \\ All\\ equilateral\\ triangles\\ are\\ regular\\ polygons.}$\n$\\mathrm{(E) \\ All\\ equilateral\\ triangles\\ are\\ similar\\ to\\ each\\ other.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2288", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following statements is false?\n$\\mathrm{(A) \\ All\\ equilateral\\ triangles\\ are\\ congruent\\ to\\ each\\ other.}$\n$\\mathrm{(B) \\ All\\ equilateral\\ triangles\\ are\\ convex.}$\n$\\mathrm{(C) \\ All\\ equilateral\\ triangles\\ are\\ equianguilar.}$\n$\\mathrm{(D) \\ All\\ equilateral\\ triangles\\ are\\ regular\\ polygons.}$\n$\\mathrm{(E) \\ All\\ equilateral\\ triangles\\ are\\ similar\\ to\\ each\\ other.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The smallest product one could obtain by multiplying two numbers in the set $\\{ -7,-5,-1,1,3 \\}$ is\n$\\text{(A)}\\ -35 \\qquad \\text{(B)}\\ -21 \\qquad \\text{(C)}\\ -15 \\qquad \\text{(D)}\\ -1 \\qquad \\text{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2289", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The smallest product one could obtain by multiplying two numbers in the set $\\{ -7,-5,-1,1,3 \\}$ is\n$\\text{(A)}\\ -35 \\qquad \\text{(B)}\\ -21 \\qquad \\text{(C)}\\ -15 \\qquad \\text{(D)}\\ -1 \\qquad \\text{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $P = \\frac{s}{(1 + k)^n}$ then $n$ equals:\n$\\textbf{(A)}\\ \\frac{\\log{\\left(\\frac{s}{P}\\right)}}{\\log{(1 + k)}}\\qquad \\textbf{(B)}\\ \\log{\\left(\\frac{s}{P(1 + k)}\\right)}\\qquad \\textbf{(C)}\\ \\log{\\left(\\frac{s - P}{1 + k}\\right)}\\qquad \\\\ \\textbf{(D)}\\ \\log{\\left(\\frac{s}{P}\\right)} + \\log{(1 + k)}\\qquad \\textbf{(E)}\\ \\frac{\\log{(s)}}{\\log{(P(1 + k))}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2290", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $P = \\frac{s}{(1 + k)^n}$ then $n$ equals:\n$\\textbf{(A)}\\ \\frac{\\log{\\left(\\frac{s}{P}\\right)}}{\\log{(1 + k)}}\\qquad \\textbf{(B)}\\ \\log{\\left(\\frac{s}{P(1 + k)}\\right)}\\qquad \\textbf{(C)}\\ \\log{\\left(\\frac{s - P}{1 + k}\\right)}\\qquad \\\\ \\textbf{(D)}\\ \\log{\\left(\\frac{s}{P}\\right)} + \\log{(1 + k)}\\qquad \\textbf{(E)}\\ \\frac{\\log{(s)}}{\\log{(P(1 + k))}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\\measuredangle BAD=92^\\circ$ and $\\measuredangle ADC=68^\\circ$, find $\\measuredangle EBC$.\n$\\mathrm{(A)\\ } 66^\\circ \\qquad \\mathrm{(B) \\ }68^\\circ \\qquad \\mathrm{(C) \\ } 70^\\circ \\qquad \\mathrm{(D) \\ } 88^\\circ \\qquad \\mathrm{(E) \\ }92^\\circ$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2291", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\\measuredangle BAD=92^\\circ$ and $\\measuredangle ADC=68^\\circ$, find $\\measuredangle EBC$.\n$\\mathrm{(A)\\ } 66^\\circ \\qquad \\mathrm{(B) \\ }68^\\circ \\qquad \\mathrm{(C) \\ } 70^\\circ \\qquad \\mathrm{(D) \\ } 88^\\circ \\qquad \\mathrm{(E) \\ }92^\\circ$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, \nthen $P$ in terms of $S, S'$, and $n$ is\n$\\textbf{(A) }(SS')^{\\frac{1}{2}n}\\qquad \\textbf{(B) }(S/S')^{\\frac{1}{2}n}\\qquad \\textbf{(C) }(SS')^{n-2}\\qquad \\textbf{(D) }(S/S')^n\\qquad \\textbf{(E) }(S/S')^{\\frac{1}{2}(n-1)}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2292", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, \nthen $P$ in terms of $S, S'$, and $n$ is\n$\\textbf{(A) }(SS')^{\\frac{1}{2}n}\\qquad \\textbf{(B) }(S/S')^{\\frac{1}{2}n}\\qquad \\textbf{(C) }(SS')^{n-2}\\qquad \\textbf{(D) }(S/S')^n\\qquad \\textbf{(E) }(S/S')^{\\frac{1}{2}(n-1)}$" + } + }, + { + "question": "Return your final response within \\boxed{}. When a student multiplied the number $66$ by the repeating decimal, \n\\[\\underline{1}.\\underline{a} \\ \\underline{b} \\ \\underline{a} \\ \\underline{b}\\ldots=\\underline{1}.\\overline{\\underline{a} \\ \\underline{b}},\\] \nwhere $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\\underline{1}.\\underline{a} \\ \\underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $\\underline{a} \\ \\underline{b}?$\n$\\textbf{(A) }15 \\qquad \\textbf{(B) }30 \\qquad \\textbf{(C) }45 \\qquad \\textbf{(D) }60 \\qquad \\textbf{(E) }75$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2293", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When a student multiplied the number $66$ by the repeating decimal, \n\\[\\underline{1}.\\underline{a} \\ \\underline{b} \\ \\underline{a} \\ \\underline{b}\\ldots=\\underline{1}.\\overline{\\underline{a} \\ \\underline{b}},\\] \nwhere $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\\underline{1}.\\underline{a} \\ \\underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $\\underline{a} \\ \\underline{b}?$\n$\\textbf{(A) }15 \\qquad \\textbf{(B) }30 \\qquad \\textbf{(C) }45 \\qquad \\textbf{(D) }60 \\qquad \\textbf{(E) }75$" + } + }, + { + "question": "Return your final response within \\boxed{}. A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?\n$\\textbf{(A)}\\ 4+4\\sqrt{2} \\qquad \\textbf{(B)}\\ 2+4\\sqrt{2}+2\\sqrt{3} \\qquad \\textbf{(C)}\\ 2+3\\sqrt{2}+3\\sqrt{3} \\qquad \\textbf{(D)}\\ 4\\sqrt{2}+4\\sqrt{3} \\qquad \\textbf{(E)}\\ 3\\sqrt{2}+5\\sqrt{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2294", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?\n$\\textbf{(A)}\\ 4+4\\sqrt{2} \\qquad \\textbf{(B)}\\ 2+4\\sqrt{2}+2\\sqrt{3} \\qquad \\textbf{(C)}\\ 2+3\\sqrt{2}+3\\sqrt{3} \\qquad \\textbf{(D)}\\ 4\\sqrt{2}+4\\sqrt{3} \\qquad \\textbf{(E)}\\ 3\\sqrt{2}+5\\sqrt{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A triangle is formed by joining three points whose coordinates are integers. If the $x$-coordinate and the $y$-coordinate each have a value of $1$, then the area of the triangle, in square units:\n$\\textbf{(A)}\\ \\text{must be an integer}\\qquad \\textbf{(B)}\\ \\text{may be irrational}\\qquad \\textbf{(C)}\\ \\text{must be irrational}\\qquad \\textbf{(D)}\\ \\text{must be rational}\\qquad \\\\ \\textbf{(E)}\\ \\text{will be an integer only if the triangle is equilateral.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2295", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A triangle is formed by joining three points whose coordinates are integers. If the $x$-coordinate and the $y$-coordinate each have a value of $1$, then the area of the triangle, in square units:\n$\\textbf{(A)}\\ \\text{must be an integer}\\qquad \\textbf{(B)}\\ \\text{may be irrational}\\qquad \\textbf{(C)}\\ \\text{must be irrational}\\qquad \\textbf{(D)}\\ \\text{must be rational}\\qquad \\\\ \\textbf{(E)}\\ \\text{will be an integer only if the triangle is equilateral.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first $40$ miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.02$ gallons per mile. On the whole trip he averaged $55$ miles per gallon. How long was the trip in miles?\n$\\mathrm{(A)}\\ 140 \\qquad \\mathrm{(B)}\\ 240 \\qquad \\mathrm{(C)}\\ 440 \\qquad \\mathrm{(D)}\\ 640 \\qquad \\mathrm{(E)}\\ 840$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2296", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first $40$ miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.02$ gallons per mile. On the whole trip he averaged $55$ miles per gallon. How long was the trip in miles?\n$\\mathrm{(A)}\\ 140 \\qquad \\mathrm{(B)}\\ 240 \\qquad \\mathrm{(C)}\\ 440 \\qquad \\mathrm{(D)}\\ 640 \\qquad \\mathrm{(E)}\\ 840$" + } + }, + { + "question": "Return your final response within \\boxed{}. Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,$ and $5$ are all uphill integers, but $32, 1240,$ and $466$ are not. How many uphill integers are divisible by $15$?\n$\\textbf{(A)} ~4 \\qquad\\textbf{(B)} ~5 \\qquad\\textbf{(C)} ~6 \\qquad\\textbf{(D)} ~7 \\qquad\\textbf{(E)} ~8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2297", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,$ and $5$ are all uphill integers, but $32, 1240,$ and $466$ are not. How many uphill integers are divisible by $15$?\n$\\textbf{(A)} ~4 \\qquad\\textbf{(B)} ~5 \\qquad\\textbf{(C)} ~6 \\qquad\\textbf{(D)} ~7 \\qquad\\textbf{(E)} ~8$" + } + }, + { + "question": "Return your final response within \\boxed{}. An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. \nIf the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance \nbetween the centers of the pulleys in inches is\n$\\textbf{(A) }24\\qquad \\textbf{(B) }2\\sqrt{119}\\qquad \\textbf{(C) }25\\qquad \\textbf{(D) }26\\qquad \\textbf{(E) }4\\sqrt{35}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2298", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. \nIf the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance \nbetween the centers of the pulleys in inches is\n$\\textbf{(A) }24\\qquad \\textbf{(B) }2\\sqrt{119}\\qquad \\textbf{(C) }25\\qquad \\textbf{(D) }26\\qquad \\textbf{(E) }4\\sqrt{35}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle\nin the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ b \\qquad \\textbf{(C)}\\ c \\qquad \\textbf{(D)}\\ -a \\qquad \\textbf{(E)}\\ -b$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2299", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle\nin the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ b \\qquad \\textbf{(C)}\\ c \\qquad \\textbf{(D)}\\ -a \\qquad \\textbf{(E)}\\ -b$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\\angle BAC$ is acute.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\angle BAC is acute", + "index": "Sky-T1_10k_2300", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\\angle BAC$ is acute." + } + }, + { + "question": "Return your final response within \\boxed{}. In this figure the radius of the circle is equal to the altitude of the equilateral triangle $ABC$. The circle is made to roll along the side $AB$, remaining tangent to it at a variable point $T$ and intersecting lines $AC$ and $BC$ in variable points $M$ and $N$, respectively. Let $n$ be the number of degrees in arc $MTN$. Then $n$, for all permissible positions of the circle:\n$\\textbf{(A) }\\text{varies from }30^{\\circ}\\text{ to }90^{\\circ}$\n$\\textbf{(B) }\\text{varies from }30^{\\circ}\\text{ to }60^{\\circ}$\n$\\textbf{(C) }\\text{varies from }60^{\\circ}\\text{ to }90^{\\circ}$\n$\\textbf{(D) }\\text{remains constant at }30^{\\circ}$\n$\\textbf{(E) }\\text{remains constant at }60^{\\circ}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2301", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In this figure the radius of the circle is equal to the altitude of the equilateral triangle $ABC$. The circle is made to roll along the side $AB$, remaining tangent to it at a variable point $T$ and intersecting lines $AC$ and $BC$ in variable points $M$ and $N$, respectively. Let $n$ be the number of degrees in arc $MTN$. Then $n$, for all permissible positions of the circle:\n$\\textbf{(A) }\\text{varies from }30^{\\circ}\\text{ to }90^{\\circ}$\n$\\textbf{(B) }\\text{varies from }30^{\\circ}\\text{ to }60^{\\circ}$\n$\\textbf{(C) }\\text{varies from }60^{\\circ}\\text{ to }90^{\\circ}$\n$\\textbf{(D) }\\text{remains constant at }30^{\\circ}$\n$\\textbf{(E) }\\text{remains constant at }60^{\\circ}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \\geq{1}$ and $a+b+c \\leq{7}$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2?$.\n$\\textbf{(A)}\\ 26\\qquad\\textbf{(B)}\\ 27\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 37\\qquad\\textbf{(E)}\\ 41$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2302", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \\geq{1}$ and $a+b+c \\leq{7}$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2?$.\n$\\textbf{(A)}\\ 26\\qquad\\textbf{(B)}\\ 27\\qquad\\textbf{(C)}\\ 36\\qquad\\textbf{(D)}\\ 37\\qquad\\textbf{(E)}\\ 41$" + } + }, + { + "question": "Return your final response within \\boxed{}. Bernardo randomly picks 3 distinct numbers from the set $\\{1,2,3,4,5,6,7,8,9\\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\\{1,2,3,4,5,6,7,8\\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?\n$\\textbf{(A)}\\ \\frac{47}{72} \\qquad \\textbf{(B)}\\ \\frac{37}{56} \\qquad \\textbf{(C)}\\ \\frac{2}{3} \\qquad \\textbf{(D)}\\ \\frac{49}{72} \\qquad \\textbf{(E)}\\ \\frac{39}{56}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2303", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Bernardo randomly picks 3 distinct numbers from the set $\\{1,2,3,4,5,6,7,8,9\\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\\{1,2,3,4,5,6,7,8\\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?\n$\\textbf{(A)}\\ \\frac{47}{72} \\qquad \\textbf{(B)}\\ \\frac{37}{56} \\qquad \\textbf{(C)}\\ \\frac{2}{3} \\qquad \\textbf{(D)}\\ \\frac{49}{72} \\qquad \\textbf{(E)}\\ \\frac{39}{56}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is\n$\\text{(A) }72^\\circ \\qquad \\text{(B) }75^\\circ \\qquad \\text{(C) }90^\\circ \\qquad \\text{(D) }108^\\circ \\qquad \\text{(E) }120^\\circ$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2304", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is\n$\\text{(A) }72^\\circ \\qquad \\text{(B) }75^\\circ \\qquad \\text{(C) }90^\\circ \\qquad \\text{(D) }108^\\circ \\qquad \\text{(E) }120^\\circ$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\\circ}, 180^{\\circ},$ and $270^{\\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 17 \\qquad \\textbf{(D) } 20 \\qquad \\textbf{(E) } 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2305", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\\circ}, 180^{\\circ},$ and $270^{\\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 17 \\qquad \\textbf{(D) } 20 \\qquad \\textbf{(E) } 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\Delta ABC$ is a triangle with incenter $I$. Show that the circumcenters of $\\Delta IAB$, $\\Delta IBC$, and $\\Delta ICA$ lie on a circle whose center is the circumcenter of $\\Delta ABC$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The circumcenters of \\Delta IAB, \\Delta IBC, and \\Delta ICA lie on a circle centered at the circumcenter of \\Delta ABC.", + "index": "Sky-T1_10k_2306", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\Delta ABC$ is a triangle with incenter $I$. Show that the circumcenters of $\\Delta IAB$, $\\Delta IBC$, and $\\Delta ICA$ lie on a circle whose center is the circumcenter of $\\Delta ABC$." + } + }, + { + "question": "Return your final response within \\boxed{}. $A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then the circumference of the track in yards is\n$\\text{(A) } 400\\quad \\text{(B) } 440\\quad \\text{(C) } 480\\quad \\text{(D) } 560\\quad \\text{(E) } 880$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2307", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then the circumference of the track in yards is\n$\\text{(A) } 400\\quad \\text{(B) } 440\\quad \\text{(C) } 480\\quad \\text{(D) } 560\\quad \\text{(E) } 880$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\, D \\,$ be an arbitrary point on side $\\, AB \\,$ of a given triangle $\\, ABC, \\,$ and let $\\, E \\,$ be the interior point where $\\, CD \\,$ intersects the external common tangent to the incircles of triangles $\\, ACD \\,$ and $\\, BCD$. As $\\, D \\,$ assumes all positions between $\\, A \\,$ and $\\, B \\,$, prove that the point $\\, E \\,$ traces the arc of a circle.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "The point E traces an arc of a circle.", + "index": "Sky-T1_10k_2308", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\, D \\,$ be an arbitrary point on side $\\, AB \\,$ of a given triangle $\\, ABC, \\,$ and let $\\, E \\,$ be the interior point where $\\, CD \\,$ intersects the external common tangent to the incircles of triangles $\\, ACD \\,$ and $\\, BCD$. As $\\, D \\,$ assumes all positions between $\\, A \\,$ and $\\, B \\,$, prove that the point $\\, E \\,$ traces the arc of a circle." + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is equal to $1 + \\frac {1}{1 + \\frac {1}{1 + 1}}$?\n$\\textbf{(A)}\\ \\frac {5}{4} \\qquad \\textbf{(B)}\\ \\frac {3}{2} \\qquad \\textbf{(C)}\\ \\frac {5}{3} \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2309", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is equal to $1 + \\frac {1}{1 + \\frac {1}{1 + 1}}$?\n$\\textbf{(A)}\\ \\frac {5}{4} \\qquad \\textbf{(B)}\\ \\frac {3}{2} \\qquad \\textbf{(C)}\\ \\frac {5}{3} \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of significant digits in the measurement of the side of a square whose computed area is $1.1025$ square inches to \nthe nearest ten-thousandth of a square inch is: \n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2310", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of significant digits in the measurement of the side of a square whose computed area is $1.1025$ square inches to \nthe nearest ten-thousandth of a square inch is: \n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "c exists such that cm and cn have equal non-zero digit counts", + "index": "Sky-T1_10k_2311", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten." + } + }, + { + "question": "Return your final response within \\boxed{}. A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.\n[asy] size(3cm); pair A=(0.05,0),B=(-.9,-0.6),C=(0,-0.45),D=(.9,-0.6),E=(.55,-0.85),F=(-0.55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; draw(A--B--F--E--D--A--E--A--F--A^^B--G--F--K--G--L--J--K--E--J--D--J--L--K); draw(B--C--D--C--A--C--H--I--C--H--G^^H--L--I--J^^I--D^^H--B,dashed); dot(A^^B^^C^^D^^E^^F^^G^^H^^I^^J^^K^^L); [/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "810", + "index": "Sky-T1_10k_2312", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.\n[asy] size(3cm); pair A=(0.05,0),B=(-.9,-0.6),C=(0,-0.45),D=(.9,-0.6),E=(.55,-0.85),F=(-0.55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; draw(A--B--F--E--D--A--E--A--F--A^^B--G--F--K--G--L--J--K--E--J--D--J--L--K); draw(B--C--D--C--A--C--H--I--C--H--G^^H--L--I--J^^I--D^^H--B,dashed); dot(A^^B^^C^^D^^E^^F^^G^^H^^I^^J^^K^^L); [/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. Regular polygons with $5,6,7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?\n$(\\textbf{A})\\: 52\\qquad(\\textbf{B}) \\: 56\\qquad(\\textbf{C}) \\: 60\\qquad(\\textbf{D}) \\: 64\\qquad(\\textbf{E}) \\: 68$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2313", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Regular polygons with $5,6,7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?\n$(\\textbf{A})\\: 52\\qquad(\\textbf{B}) \\: 56\\qquad(\\textbf{C}) \\: 60\\qquad(\\textbf{D}) \\: 64\\qquad(\\textbf{E}) \\: 68$" + } + }, + { + "question": "Return your final response within \\boxed{}. Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:\n\n\"Bret is next to Carl.\"\n\"Abby is between Bret and Carl.\"\n\nHowever each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?\n$\\text{(A)}\\ \\text{Abby} \\qquad \\text{(B)}\\ \\text{Bret} \\qquad \\text{(C)}\\ \\text{Carl} \\qquad \\text{(D)}\\ \\text{Dana} \\qquad \\text{(E)}\\ \\text{There is not enough information to be sure.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2314", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:\n\n\"Bret is next to Carl.\"\n\"Abby is between Bret and Carl.\"\n\nHowever each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?\n$\\text{(A)}\\ \\text{Abby} \\qquad \\text{(B)}\\ \\text{Bret} \\qquad \\text{(C)}\\ \\text{Carl} \\qquad \\text{(D)}\\ \\text{Dana} \\qquad \\text{(E)}\\ \\text{There is not enough information to be sure.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. \nHowever, the student neglected to indicate $N$, the total number of measurements. What is the smallest possible value of $N$?\n\\[\\begin{tabular}{c c}\\text{measured value}&\\text{percent frequency}\\\\ \\hline 0 & 12.5\\\\ 1 & 0\\\\ 2 & 50\\\\ 3 & 25\\\\ 4 & 12.5\\\\ \\hline\\ & 100\\\\ \\end{tabular}\\]\n$\\textbf{(A)}\\ 5 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 16 \\qquad \\textbf{(D)}\\ 25 \\qquad \\textbf{(E)}\\ 50$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2315", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. \nHowever, the student neglected to indicate $N$, the total number of measurements. What is the smallest possible value of $N$?\n\\[\\begin{tabular}{c c}\\text{measured value}&\\text{percent frequency}\\\\ \\hline 0 & 12.5\\\\ 1 & 0\\\\ 2 & 50\\\\ 3 & 25\\\\ 4 & 12.5\\\\ \\hline\\ & 100\\\\ \\end{tabular}\\]\n$\\textbf{(A)}\\ 5 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 16 \\qquad \\textbf{(D)}\\ 25 \\qquad \\textbf{(E)}\\ 50$" + } + }, + { + "question": "Return your final response within \\boxed{}. The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to\n[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy]\n$\\textbf{(A) }50 \\qquad \\textbf{(B) }52 \\qquad \\textbf{(C) }54 \\qquad \\textbf{(D) }56 \\qquad \\textbf{(E) }58$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2316", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to\n[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy]\n$\\textbf{(A) }50 \\qquad \\textbf{(B) }52 \\qquad \\textbf{(C) }54 \\qquad \\textbf{(D) }56 \\qquad \\textbf{(E) }58$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the value of \\[\\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?\\]\n$\\textbf{(A)}\\ -1 \\qquad\\textbf{(B)}\\ 1 \\qquad\\textbf{(C)}\\ \\frac{5}{3} \\qquad\\textbf{(D)}\\ 2013 \\qquad\\textbf{(E)}\\ 2^{4024}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2317", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the value of \\[\\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?\\]\n$\\textbf{(A)}\\ -1 \\qquad\\textbf{(B)}\\ 1 \\qquad\\textbf{(C)}\\ \\frac{5}{3} \\qquad\\textbf{(D)}\\ 2013 \\qquad\\textbf{(E)}\\ 2^{4024}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The set of $x$-values satisfying the inequality $2 \\leq |x-1| \\leq 5$ is: \n$\\textbf{(A)}\\ -4\\leq x\\leq-1\\text{ or }3\\leq x\\leq 6\\qquad$\n$\\textbf{(B)}\\ 3\\leq x\\leq 6\\text{ or }-6\\leq x\\leq-3\\qquad\\textbf{(C)}\\ x\\leq-1\\text{ or }x\\geq 3\\qquad$\n$\\textbf{(D)}\\ -1\\leq x\\leq 3\\qquad\\textbf{(E)}\\ -4\\leq x\\leq 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2318", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The set of $x$-values satisfying the inequality $2 \\leq |x-1| \\leq 5$ is: \n$\\textbf{(A)}\\ -4\\leq x\\leq-1\\text{ or }3\\leq x\\leq 6\\qquad$\n$\\textbf{(B)}\\ 3\\leq x\\leq 6\\text{ or }-6\\leq x\\leq-3\\qquad\\textbf{(C)}\\ x\\leq-1\\text{ or }x\\geq 3\\qquad$\n$\\textbf{(D)}\\ -1\\leq x\\leq 3\\qquad\\textbf{(E)}\\ -4\\leq x\\leq 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was\n$\\textbf{(A)}\\ 54 \\text{ or } 18 \\qquad \\textbf{(B)}\\ 60 \\text{ or } 15 \\qquad \\textbf{(C)}\\ 63 \\text{ or } 12 \\qquad \\textbf{(D)}\\ 72 \\text{ or } 36 \\qquad \\textbf{(E)}\\ 75 \\text{ or } 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2319", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was\n$\\textbf{(A)}\\ 54 \\text{ or } 18 \\qquad \\textbf{(B)}\\ 60 \\text{ or } 15 \\qquad \\textbf{(C)}\\ 63 \\text{ or } 12 \\qquad \\textbf{(D)}\\ 72 \\text{ or } 36 \\qquad \\textbf{(E)}\\ 75 \\text{ or } 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, \"At least half of you guessed too low, two of you are off by one, and my age is a prime number.\" How old is Norb?\n$\\textbf{(A) }29\\qquad\\textbf{(B) }31\\qquad\\textbf{(C) }37\\qquad\\textbf{(D) }43\\qquad\\textbf{(E) }48$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2320", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, \"At least half of you guessed too low, two of you are off by one, and my age is a prime number.\" How old is Norb?\n$\\textbf{(A) }29\\qquad\\textbf{(B) }31\\qquad\\textbf{(C) }37\\qquad\\textbf{(D) }43\\qquad\\textbf{(E) }48$" + } + }, + { + "question": "Return your final response within \\boxed{}. The [Fibonacci sequence](https://artofproblemsolving.com/wiki/index.php/Fibonacci_sequence) $1,1,2,3,5,8,13,21,\\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten [digits](https://artofproblemsolving.com/wiki/index.php/Digit) is the last to appear in the units position of a number in the Fibonacci sequence?\n$\\textbf{(A)} \\ 0 \\qquad \\textbf{(B)} \\ 4 \\qquad \\textbf{(C)} \\ 6 \\qquad \\textbf{(D)} \\ 7 \\qquad \\textbf{(E)} \\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2321", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The [Fibonacci sequence](https://artofproblemsolving.com/wiki/index.php/Fibonacci_sequence) $1,1,2,3,5,8,13,21,\\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten [digits](https://artofproblemsolving.com/wiki/index.php/Digit) is the last to appear in the units position of a number in the Fibonacci sequence?\n$\\textbf{(A)} \\ 0 \\qquad \\textbf{(B)} \\ 4 \\qquad \\textbf{(C)} \\ 6 \\qquad \\textbf{(D)} \\ 7 \\qquad \\textbf{(E)} \\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. The roots of the equation $2\\sqrt {x} + 2x^{ - \\frac {1}{2}} = 5$ can be found by solving: \n$\\textbf{(A)}\\ 16x^2-92x+1 = 0\\qquad\\textbf{(B)}\\ 4x^2-25x+4 = 0\\qquad\\textbf{(C)}\\ 4x^2-17x+4 = 0\\\\ \\textbf{(D)}\\ 2x^2-21x+2 = 0\\qquad\\textbf{(E)}\\ 4x^2-25x-4 = 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2322", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The roots of the equation $2\\sqrt {x} + 2x^{ - \\frac {1}{2}} = 5$ can be found by solving: \n$\\textbf{(A)}\\ 16x^2-92x+1 = 0\\qquad\\textbf{(B)}\\ 4x^2-25x+4 = 0\\qquad\\textbf{(C)}\\ 4x^2-17x+4 = 0\\\\ \\textbf{(D)}\\ 2x^2-21x+2 = 0\\qquad\\textbf{(E)}\\ 4x^2-25x-4 = 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by $5$ minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?\n$\\textbf{(A) }10\\qquad\\textbf{(B) }15\\qquad\\textbf{(C) }25\\qquad\\textbf{(D) }50\\qquad\\textbf{(E) }82$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2323", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by $5$ minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?\n$\\textbf{(A) }10\\qquad\\textbf{(B) }15\\qquad\\textbf{(C) }25\\qquad\\textbf{(D) }50\\qquad\\textbf{(E) }82$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four friends do yardwork for their neighbors over the weekend, earning $$15, $20, $25,$ and $$40,$ respectively. They decide to split their earnings equally among themselves. In total, how much will the friend who earned $$40$ give to the others?\n$\\textbf{(A) }$5 \\qquad \\textbf{(B) }$10 \\qquad \\textbf{(C) }$15 \\qquad \\textbf{(D) }$20 \\qquad \\textbf{(E) }$25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2324", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four friends do yardwork for their neighbors over the weekend, earning $$15, $20, $25,$ and $$40,$ respectively. They decide to split their earnings equally among themselves. In total, how much will the friend who earned $$40$ give to the others?\n$\\textbf{(A) }$5 \\qquad \\textbf{(B) }$10 \\qquad \\textbf{(C) }$15 \\qquad \\textbf{(D) }$20 \\qquad \\textbf{(E) }$25$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider these two geoboard quadrilaterals. Which of the following statements is true?\n\n$\\text{(A)}\\ \\text{The area of quadrilateral I is more than the area of quadrilateral II.}$\n$\\text{(B)}\\ \\text{The area of quadrilateral I is less than the area of quadrilateral II.}$\n$\\text{(C)}\\ \\text{The quadrilaterals have the same area and the same perimeter.}$\n$\\text{(D)}\\ \\text{The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.}$\n$\\text{(E)}\\ \\text{The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2325", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider these two geoboard quadrilaterals. Which of the following statements is true?\n\n$\\text{(A)}\\ \\text{The area of quadrilateral I is more than the area of quadrilateral II.}$\n$\\text{(B)}\\ \\text{The area of quadrilateral I is less than the area of quadrilateral II.}$\n$\\text{(C)}\\ \\text{The quadrilaterals have the same area and the same perimeter.}$\n$\\text{(D)}\\ \\text{The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.}$\n$\\text{(E)}\\ \\text{The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, then the ratio of $r$ to $s$ is:\n$\\textbf{(A)}\\ 1 : 3\\qquad \\textbf{(B)}\\ 1 : 9\\qquad \\textbf{(C)}\\ 1 : 10\\qquad \\textbf{(D)}\\ 3 : 10\\qquad \\textbf{(E)}\\ 1 : \\sqrt{10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2326", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, then the ratio of $r$ to $s$ is:\n$\\textbf{(A)}\\ 1 : 3\\qquad \\textbf{(B)}\\ 1 : 9\\qquad \\textbf{(C)}\\ 1 : 10\\qquad \\textbf{(D)}\\ 3 : 10\\qquad \\textbf{(E)}\\ 1 : \\sqrt{10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For a positive integer $n$, the factorial notation $n!$ represents the product of the integers from $n$ to $1$. What value of $N$ satisfies the following equation? \\[5!\\cdot 9!=12\\cdot N!\\]\n$\\textbf{(A) }10\\qquad\\textbf{(B) }11\\qquad\\textbf{(C) }12\\qquad\\textbf{(D) }13\\qquad\\textbf{(E) }14\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2327", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a positive integer $n$, the factorial notation $n!$ represents the product of the integers from $n$ to $1$. What value of $N$ satisfies the following equation? \\[5!\\cdot 9!=12\\cdot N!\\]\n$\\textbf{(A) }10\\qquad\\textbf{(B) }11\\qquad\\textbf{(C) }12\\qquad\\textbf{(D) }13\\qquad\\textbf{(E) }14\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?\n$\\textbf{(A)}\\hspace{.05in}76531\\qquad\\textbf{(B)}\\hspace{.05in}86724\\qquad\\textbf{(C)}\\hspace{.05in}87431\\qquad\\textbf{(D)}\\hspace{.05in}96240\\qquad\\textbf{(E)}\\hspace{.05in}97403$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2328", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?\n$\\textbf{(A)}\\hspace{.05in}76531\\qquad\\textbf{(B)}\\hspace{.05in}86724\\qquad\\textbf{(C)}\\hspace{.05in}87431\\qquad\\textbf{(D)}\\hspace{.05in}96240\\qquad\\textbf{(E)}\\hspace{.05in}97403$" + } + }, + { + "question": "Return your final response within \\boxed{}. A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? \n$\\textbf{(A)}\\ \\frac{1}{9}\\qquad\\textbf{(B)}\\ \\frac{1}{4}\\qquad\\textbf{(C)}\\ \\frac{4}{9}\\qquad\\textbf{(D)}\\ \\frac{5}{9}\\qquad\\textbf{(E)}\\ \\frac{19}{27}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2329", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? \n$\\textbf{(A)}\\ \\frac{1}{9}\\qquad\\textbf{(B)}\\ \\frac{1}{4}\\qquad\\textbf{(C)}\\ \\frac{4}{9}\\qquad\\textbf{(D)}\\ \\frac{5}{9}\\qquad\\textbf{(E)}\\ \\frac{19}{27}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 4.5 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6.2 \\qquad \\textbf{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2330", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 4.5 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6.2 \\qquad \\textbf{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0$?\n$\\textbf{(A) }\\frac{17}{32} \\qquad \\textbf{(B) }\\frac{11}{16} \\qquad \\textbf{(C) }\\frac79 \\qquad \\textbf{(D) }\\frac76\\qquad \\textbf{(E) }\\frac{25}{11}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2331", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0$?\n$\\textbf{(A) }\\frac{17}{32} \\qquad \\textbf{(B) }\\frac{11}{16} \\qquad \\textbf{(C) }\\frac79 \\qquad \\textbf{(D) }\\frac76\\qquad \\textbf{(E) }\\frac{25}{11}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values?\n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B)}\\ 5 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 7 \\qquad \\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2332", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values?\n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B)}\\ 5 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 7 \\qquad \\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $(1+x+x^2)^n=a_1x+a_2x^2+ \\cdots + a_{2n}x^{2n}$ be an identity in $x$. If we let $s=a_0+a_2+a_4+\\cdots +a_{2n}$, then $s$ equals:\n$\\text{(A) } 2^n \\quad \\text{(B) } 2^n+1 \\quad \\text{(C) } \\frac{3^n-1}{2} \\quad \\text{(D) } \\frac{3^n}{2} \\quad \\text{(E) } \\frac{3^n+1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2333", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $(1+x+x^2)^n=a_1x+a_2x^2+ \\cdots + a_{2n}x^{2n}$ be an identity in $x$. If we let $s=a_0+a_2+a_4+\\cdots +a_{2n}$, then $s$ equals:\n$\\text{(A) } 2^n \\quad \\text{(B) } 2^n+1 \\quad \\text{(C) } \\frac{3^n-1}{2} \\quad \\text{(D) } \\frac{3^n}{2} \\quad \\text{(E) } \\frac{3^n+1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.)\n$\\textbf{(A) }12\\qquad\\textbf{(B) }36\\qquad\\textbf{(C) }108\\qquad\\textbf{(D) }324\\qquad \\textbf{(E) }972$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2334", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.)\n$\\textbf{(A) }12\\qquad\\textbf{(B) }36\\qquad\\textbf{(C) }108\\qquad\\textbf{(D) }324\\qquad \\textbf{(E) }972$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following operations has the same effect on a number as multiplying by $\\dfrac{3}{4}$ and then dividing by $\\dfrac{3}{5}$?\n$\\text{(A)}\\ \\text{dividing by }\\dfrac{4}{3} \\qquad \\text{(B)}\\ \\text{dividing by }\\dfrac{9}{20} \\qquad \\text{(C)}\\ \\text{multiplying by }\\dfrac{9}{20} \\qquad \\text{(D)}\\ \\text{dividing by }\\dfrac{5}{4} \\qquad \\text{(E)}\\ \\text{multiplying by }\\dfrac{5}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2335", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following operations has the same effect on a number as multiplying by $\\dfrac{3}{4}$ and then dividing by $\\dfrac{3}{5}$?\n$\\text{(A)}\\ \\text{dividing by }\\dfrac{4}{3} \\qquad \\text{(B)}\\ \\text{dividing by }\\dfrac{9}{20} \\qquad \\text{(C)}\\ \\text{multiplying by }\\dfrac{9}{20} \\qquad \\text{(D)}\\ \\text{dividing by }\\dfrac{5}{4} \\qquad \\text{(E)}\\ \\text{multiplying by }\\dfrac{5}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of sets of two or more consecutive positive integers whose sum is 100 is\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2336", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of sets of two or more consecutive positive integers whose sum is 100 is\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The increasing [geometric sequence](https://artofproblemsolving.com/wiki/index.php/Geometric_sequence) $x_{0},x_{1},x_{2},\\ldots$ consists entirely of [integral](https://artofproblemsolving.com/wiki/index.php/Integer) powers of $3.$ Given that\n$\\sum_{n=0}^{7}\\log_{3}(x_{n}) = 308$ and $56 \\leq \\log_{3}\\left ( \\sum_{n=0}^{7}x_{n}\\right ) \\leq 57,$\nfind $\\log_{3}(x_{14}).$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "91", + "index": "Sky-T1_10k_2337", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The increasing [geometric sequence](https://artofproblemsolving.com/wiki/index.php/Geometric_sequence) $x_{0},x_{1},x_{2},\\ldots$ consists entirely of [integral](https://artofproblemsolving.com/wiki/index.php/Integer) powers of $3.$ Given that\n$\\sum_{n=0}^{7}\\log_{3}(x_{n}) = 308$ and $56 \\leq \\log_{3}\\left ( \\sum_{n=0}^{7}x_{n}\\right ) \\leq 57,$\nfind $\\log_{3}(x_{14}).$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n=5, m=7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$?\n\n$\\text{(A)}\\ 1\\qquad\\text{(B)}\\ m/n\\qquad\\text{(C)}\\ n/m\\qquad\\text{(D)}\\ 2m/n\\qquad\\text{(E)}\\ 2n/m$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2338", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n=5, m=7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$?\n\n$\\text{(A)}\\ 1\\qquad\\text{(B)}\\ m/n\\qquad\\text{(C)}\\ n/m\\qquad\\text{(D)}\\ 2m/n\\qquad\\text{(E)}\\ 2n/m$" + } + }, + { + "question": "Return your final response within \\boxed{}. Rectangle $ABCD$, pictured below, shares $50\\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\\%$ of its area with rectangle $ABCD$. What is $\\frac{AB}{AD}$?\n\n\n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B)}\\ 5 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 8 \\qquad \\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2339", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Rectangle $ABCD$, pictured below, shares $50\\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\\%$ of its area with rectangle $ABCD$. What is $\\frac{AB}{AD}$?\n\n\n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B)}\\ 5 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 8 \\qquad \\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \\times 2 \\times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)\n$\\textbf{(A)}\\ 7 \\qquad\\textbf{(B)}\\ 8 \\qquad\\textbf{(C)}\\ 9 \\qquad\\textbf{(D)}\\ 10 \\qquad\\textbf{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2340", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \\times 2 \\times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)\n$\\textbf{(A)}\\ 7 \\qquad\\textbf{(B)}\\ 8 \\qquad\\textbf{(C)}\\ 9 \\qquad\\textbf{(D)}\\ 10 \\qquad\\textbf{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively. Let $\\omega_A$, $\\omega_B$, $\\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\\omega_A$, $\\omega_B$, $\\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\dfrac{BP}{PC}", + "index": "Sky-T1_10k_2341", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively. Let $\\omega_A$, $\\omega_B$, $\\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\\omega_A$, $\\omega_B$, $\\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$." + } + }, + { + "question": "Return your final response within \\boxed{}. Four positive integers are given. Select any three of these integers, find their arithmetic average, \nand add this result to the fourth integer. Thus the numbers $29, 23, 21$, and $17$ are obtained. One of the original integers is: \n$\\textbf{(A)}\\ 19 \\qquad \\textbf{(B)}\\ 21 \\qquad \\textbf{(C)}\\ 23 \\qquad \\textbf{(D)}\\ 29 \\qquad \\textbf{(E)}\\ 17$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2342", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four positive integers are given. Select any three of these integers, find their arithmetic average, \nand add this result to the fourth integer. Thus the numbers $29, 23, 21$, and $17$ are obtained. One of the original integers is: \n$\\textbf{(A)}\\ 19 \\qquad \\textbf{(B)}\\ 21 \\qquad \\textbf{(C)}\\ 23 \\qquad \\textbf{(D)}\\ 29 \\qquad \\textbf{(E)}\\ 17$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $x=-2016$. What is the value of $\\bigg|$ $||x|-x|-|x|$ $\\bigg|$ $-x$?\n$\\textbf{(A)}\\ -2016\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2016\\qquad\\textbf{(D)}\\ 4032\\qquad\\textbf{(E)}\\ 6048$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2343", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $x=-2016$. What is the value of $\\bigg|$ $||x|-x|-|x|$ $\\bigg|$ $-x$?\n$\\textbf{(A)}\\ -2016\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2016\\qquad\\textbf{(D)}\\ 4032\\qquad\\textbf{(E)}\\ 6048$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression \nwhile the last three are in arithmetic progression. The sum of those two positive numbers is\n$\\textbf{(A) }13\\textstyle\\frac{1}{2}\\qquad \\textbf{(B) }11\\frac{1}{4}\\qquad \\textbf{(C) }10\\frac{1}{2}\\qquad \\textbf{(D) }10\\qquad \\textbf{(E) }9\\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2344", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression \nwhile the last three are in arithmetic progression. The sum of those two positive numbers is\n$\\textbf{(A) }13\\textstyle\\frac{1}{2}\\qquad \\textbf{(B) }11\\frac{1}{4}\\qquad \\textbf{(C) }10\\frac{1}{2}\\qquad \\textbf{(D) }10\\qquad \\textbf{(E) }9\\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$?\n$\\textbf{(A) } 96 \\qquad \\textbf{(B) } 97 \\qquad \\textbf{(C) } 98 \\qquad \\textbf{(D) } 102 \\qquad \\textbf{(E) } 120$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2345", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$?\n$\\textbf{(A) } 96 \\qquad \\textbf{(B) } 97 \\qquad \\textbf{(C) } 98 \\qquad \\textbf{(D) } 102 \\qquad \\textbf{(E) } 120$" + } + }, + { + "question": "Return your final response within \\boxed{}. $2(81+83+85+87+89+91+93+95+97+99)=$\n$\\text{(A)}\\ 1600 \\qquad \\text{(B)}\\ 1650 \\qquad \\text{(C)}\\ 1700 \\qquad \\text{(D)}\\ 1750 \\qquad \\text{(E)}\\ 1800$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2346", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $2(81+83+85+87+89+91+93+95+97+99)=$\n$\\text{(A)}\\ 1600 \\qquad \\text{(B)}\\ 1650 \\qquad \\text{(C)}\\ 1700 \\qquad \\text{(D)}\\ 1750 \\qquad \\text{(E)}\\ 1800$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $P$ units be the increase in circumference of a circle resulting from an increase in $\\pi$ units in the diameter. Then $P$ equals:\n$\\text{(A) } \\frac{1}{\\pi}\\quad\\text{(B) } \\pi\\quad\\text{(C) } \\frac{\\pi^2}{2}\\quad\\text{(D) } \\pi^2\\quad\\text{(E) } 2\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2347", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $P$ units be the increase in circumference of a circle resulting from an increase in $\\pi$ units in the diameter. Then $P$ equals:\n$\\text{(A) } \\frac{1}{\\pi}\\quad\\text{(B) } \\pi\\quad\\text{(C) } \\frac{\\pi^2}{2}\\quad\\text{(D) } \\pi^2\\quad\\text{(E) } 2\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?\n$\\textbf{(A) }36.2 \\qquad \\textbf{(B) }36.4 \\qquad \\textbf{(C) }36.6\\qquad \\textbf{(D) }36.8 \\qquad \\textbf{(E) }37$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2348", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?\n$\\textbf{(A) }36.2 \\qquad \\textbf{(B) }36.4 \\qquad \\textbf{(C) }36.6\\qquad \\textbf{(D) }36.8 \\qquad \\textbf{(E) }37$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two different numbers are randomly selected from the set $\\{ - 2, -1, 0, 3, 4, 5\\}$ and multiplied together. What is the probability that the product is $0$?\n$\\textbf{(A) }\\dfrac{1}{6}\\qquad\\textbf{(B) }\\dfrac{1}{5}\\qquad\\textbf{(C) }\\dfrac{1}{4}\\qquad\\textbf{(D) }\\dfrac{1}{3}\\qquad \\textbf{(E) }\\dfrac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2349", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two different numbers are randomly selected from the set $\\{ - 2, -1, 0, 3, 4, 5\\}$ and multiplied together. What is the probability that the product is $0$?\n$\\textbf{(A) }\\dfrac{1}{6}\\qquad\\textbf{(B) }\\dfrac{1}{5}\\qquad\\textbf{(C) }\\dfrac{1}{4}\\qquad\\textbf{(D) }\\dfrac{1}{3}\\qquad \\textbf{(E) }\\dfrac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is not equal to $\\dfrac{5}{4}$?\n$\\text{(A)}\\ \\dfrac{10}{8} \\qquad \\text{(B)}\\ 1\\dfrac{1}{4} \\qquad \\text{(C)}\\ 1\\dfrac{3}{12} \\qquad \\text{(D)}\\ 1\\dfrac{1}{5} \\qquad \\text{(E)}\\ 1\\dfrac{10}{40}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2350", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is not equal to $\\dfrac{5}{4}$?\n$\\text{(A)}\\ \\dfrac{10}{8} \\qquad \\text{(B)}\\ 1\\dfrac{1}{4} \\qquad \\text{(C)}\\ 1\\dfrac{3}{12} \\qquad \\text{(D)}\\ 1\\dfrac{1}{5} \\qquad \\text{(E)}\\ 1\\dfrac{10}{40}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be the number of points $P$ interior to the region bounded by a circle with radius $1$, such that the sum of squares of the distances from $P$ to the endpoints of a given diameter is $3$. Then $n$ is:\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } 4\\quad \\text{(E) } \\infty$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2351", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $n$ be the number of points $P$ interior to the region bounded by a circle with radius $1$, such that the sum of squares of the distances from $P$ to the endpoints of a given diameter is $3$. Then $n$ is:\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } 4\\quad \\text{(E) } \\infty$" + } + }, + { + "question": "Return your final response within \\boxed{}. The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?\n[asy] defaultpen(linewidth(0.6)); size(80); real r=0.5, s=1.5; path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(s,r)*p); draw(shift(s,-r)*p); draw(shift(2s,2r)*p); draw(shift(2s,0)*p); draw(shift(2s,-2r)*p); draw(shift(3s,3r)*p); draw(shift(3s,-3r)*p); draw(shift(3s,r)*p); draw(shift(3s,-r)*p); draw(shift(4s,-4r)*p); draw(shift(4s,-2r)*p); draw(shift(4s,0)*p); draw(shift(4s,2r)*p); draw(shift(4s,4r)*p); [/asy]\n[asy] size(350); defaultpen(linewidth(0.6)); path p=origin--(1,0)--(1,1)--(0,1)--cycle; pair[] a={(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (2,0), (2,1), (3,0), (3,1), (3,2), (3,3), (3,4)}; pair[] b={(5,3), (5,4), (6,2), (6,3), (6,4), (7,1), (7,2), (7,3), (7,4), (8,0), (8,1), (8,2), (9,0), (9,1), (9,2)}; pair[] c={(11,0), (11,1), (11,2), (11,3), (11,4), (12,1), (12,2), (12,3), (12,4), (13,2), (13,3), (13,4), (14,3), (14,4), (15,4)}; pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)}; pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)}; int i; for(int i=0; i<15; i=i+1) { draw(shift(a[i])*p); draw(shift(b[i])*p); draw(shift(c[i])*p); draw(shift(d[i])*p); draw(shift(e[i])*p); } [/asy]\n\\[\\textbf{(A)}\\qquad\\qquad\\qquad\\textbf{(B)}\\quad\\qquad\\qquad\\textbf{(C)}\\:\\qquad\\qquad\\qquad\\textbf{(D)}\\quad\\qquad\\qquad\\textbf{(E)}\\]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2352", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?\n[asy] defaultpen(linewidth(0.6)); size(80); real r=0.5, s=1.5; path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(s,r)*p); draw(shift(s,-r)*p); draw(shift(2s,2r)*p); draw(shift(2s,0)*p); draw(shift(2s,-2r)*p); draw(shift(3s,3r)*p); draw(shift(3s,-3r)*p); draw(shift(3s,r)*p); draw(shift(3s,-r)*p); draw(shift(4s,-4r)*p); draw(shift(4s,-2r)*p); draw(shift(4s,0)*p); draw(shift(4s,2r)*p); draw(shift(4s,4r)*p); [/asy]\n[asy] size(350); defaultpen(linewidth(0.6)); path p=origin--(1,0)--(1,1)--(0,1)--cycle; pair[] a={(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (2,0), (2,1), (3,0), (3,1), (3,2), (3,3), (3,4)}; pair[] b={(5,3), (5,4), (6,2), (6,3), (6,4), (7,1), (7,2), (7,3), (7,4), (8,0), (8,1), (8,2), (9,0), (9,1), (9,2)}; pair[] c={(11,0), (11,1), (11,2), (11,3), (11,4), (12,1), (12,2), (12,3), (12,4), (13,2), (13,3), (13,4), (14,3), (14,4), (15,4)}; pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)}; pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)}; int i; for(int i=0; i<15; i=i+1) { draw(shift(a[i])*p); draw(shift(b[i])*p); draw(shift(c[i])*p); draw(shift(d[i])*p); draw(shift(e[i])*p); } [/asy]\n\\[\\textbf{(A)}\\qquad\\qquad\\qquad\\textbf{(B)}\\quad\\qquad\\qquad\\textbf{(C)}\\:\\qquad\\qquad\\qquad\\textbf{(D)}\\quad\\qquad\\qquad\\textbf{(E)}\\]" + } + }, + { + "question": "Return your final response within \\boxed{}. Applied to a bill for $\\textdollar{10,000}$ the difference between a discount of $40$% and two successive discounts of $36$% and $4$%, \nexpressed in dollars, is:\n$\\textbf{(A)}0\\qquad \\textbf{(B)}144\\qquad \\textbf{(C)}256\\qquad \\textbf{(D)}400\\qquad \\textbf{(E)}416$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2353", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Applied to a bill for $\\textdollar{10,000}$ the difference between a discount of $40$% and two successive discounts of $36$% and $4$%, \nexpressed in dollars, is:\n$\\textbf{(A)}0\\qquad \\textbf{(B)}144\\qquad \\textbf{(C)}256\\qquad \\textbf{(D)}400\\qquad \\textbf{(E)}416$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?\n$\\textbf{(A) }36.2 \\qquad \\textbf{(B) }36.4 \\qquad \\textbf{(C) }36.6\\qquad \\textbf{(D) }36.8 \\qquad \\textbf{(E) }37$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2354", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?\n$\\textbf{(A) }36.2 \\qquad \\textbf{(B) }36.4 \\qquad \\textbf{(C) }36.6\\qquad \\textbf{(D) }36.8 \\qquad \\textbf{(E) }37$" + } + }, + { + "question": "Return your final response within \\boxed{}. The area of the ring between two concentric circles is $12\\tfrac{1}{2}\\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is:\n$\\text{(A) } \\frac{5}{\\sqrt{2}}\\quad \\text{(B) } 5\\quad \\text{(C) } 5\\sqrt{2}\\quad \\text{(D) } 10\\quad \\text{(E) } 10\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2355", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The area of the ring between two concentric circles is $12\\tfrac{1}{2}\\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is:\n$\\text{(A) } \\frac{5}{\\sqrt{2}}\\quad \\text{(B) } 5\\quad \\text{(C) } 5\\sqrt{2}\\quad \\text{(D) } 10\\quad \\text{(E) } 10\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is \n$\\textbf{(A)}\\ 20 \\qquad \\textbf{(B)}\\ 18 \\qquad \\textbf{(C)}\\ 12 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2356", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is \n$\\textbf{(A)}\\ 20 \\qquad \\textbf{(B)}\\ 18 \\qquad \\textbf{(C)}\\ 12 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many digits are in the product $4^5 \\cdot 5^{10}$?\n$\\textbf{(A) } 8 \\qquad\\textbf{(B) } 9 \\qquad\\textbf{(C) } 10 \\qquad\\textbf{(D) } 11 \\qquad\\textbf{(E) } 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2357", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many digits are in the product $4^5 \\cdot 5^{10}$?\n$\\textbf{(A) } 8 \\qquad\\textbf{(B) } 9 \\qquad\\textbf{(C) } 10 \\qquad\\textbf{(D) } 11 \\qquad\\textbf{(E) } 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. For positive integers $m$ and $n$ such that $m+10\\sqrt{ab}>\\frac{a+b}{2}\\qquad \\text{(B) } \\sqrt{ab}>\\frac{2ab}{a+b}>\\frac{a+b}{2} \\\\ \\text{(C) } \\frac{2ab}{a+b}>\\frac{a+b}{2}>\\sqrt{ab}\\qquad \\text{(D) } \\frac{a+b}{2}>\\frac{2ab}{a+b}>\\sqrt{ab} \\\\ \\text{(E) } \\frac {a + b}{2} > \\sqrt {ab} > \\frac {2ab}{a + b}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2366", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a$ and $b$ are two unequal positive numbers, then: \n$\\text{(A) } \\frac{2ab}{a+b}>\\sqrt{ab}>\\frac{a+b}{2}\\qquad \\text{(B) } \\sqrt{ab}>\\frac{2ab}{a+b}>\\frac{a+b}{2} \\\\ \\text{(C) } \\frac{2ab}{a+b}>\\frac{a+b}{2}>\\sqrt{ab}\\qquad \\text{(D) } \\frac{a+b}{2}>\\frac{2ab}{a+b}>\\sqrt{ab} \\\\ \\text{(E) } \\frac {a + b}{2} > \\sqrt {ab} > \\frac {2ab}{a + b}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Prove that the average of the numbers $n\\sin n^{\\circ}\\; (n = 2,4,6,\\ldots,180)$ is $\\cot 1^\\circ$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "\\cot 1^\\circ", + "index": "Sky-T1_10k_2367", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Prove that the average of the numbers $n\\sin n^{\\circ}\\; (n = 2,4,6,\\ldots,180)$ is $\\cot 1^\\circ$." + } + }, + { + "question": "Return your final response within \\boxed{}. Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 20\\qquad\\textbf{(E)}\\ 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2368", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 20\\qquad\\textbf{(E)}\\ 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the largest number of acute angles that a convex hexagon can have?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4\\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2369", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the largest number of acute angles that a convex hexagon can have?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4\\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "40", + "index": "Sky-T1_10k_2370", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$" + } + }, + { + "question": "Return your final response within \\boxed{}. Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,\\ldots$ and a sequence of points on the circles $A_1,A_2,A_3,\\ldots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \\frac{11}{60}$, the distance from the center $C_0$ to $B$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "110", + "index": "Sky-T1_10k_2371", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,\\ldots$ and a sequence of points on the circles $A_1,A_2,A_3,\\ldots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \\frac{11}{60}$, the distance from the center $C_0$ to $B$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$." + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?\n$\\textbf{(A)} ~10{,}272\\qquad\\textbf{(B)} ~11{,}700\\qquad\\textbf{(C)} ~13{,}362\\qquad\\textbf{(D)} ~14{,}238\\qquad\\textbf{(E)} ~15{,}426$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2372", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?\n$\\textbf{(A)} ~10{,}272\\qquad\\textbf{(B)} ~11{,}700\\qquad\\textbf{(C)} ~13{,}362\\qquad\\textbf{(D)} ~14{,}238\\qquad\\textbf{(E)} ~15{,}426$" + } + }, + { + "question": "Return your final response within \\boxed{}. An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is\n$\\text{(A)}\\ 150 \\qquad \\text{(B)}\\ 180 \\qquad \\text{(C)}\\ 200 \\qquad \\text{(D)}\\ 400 \\qquad \\text{(E)}\\ 460$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2373", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is\n$\\text{(A)}\\ 150 \\qquad \\text{(B)}\\ 180 \\qquad \\text{(C)}\\ 200 \\qquad \\text{(D)}\\ 400 \\qquad \\text{(E)}\\ 460$" + } + }, + { + "question": "Return your final response within \\boxed{}. Pat intended to multiply a number by $6$ but instead divided by $6$. Pat then meant to add $14$ but instead subtracted $14$. After these mistakes, the result was $16$. If the correct operations had been used, the value produced would have been\n$\\textbf{(A)}\\ \\text{less than 400} \\qquad\\textbf{(B)}\\ \\text{between 400 and 600} \\qquad\\textbf{(C)}\\ \\text{between 600 and 800} \\\\ \\textbf{(D)}\\ \\text{between 800 and 1000} \\qquad\\textbf{(E)}\\ \\text{greater than 1000}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2374", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Pat intended to multiply a number by $6$ but instead divided by $6$. Pat then meant to add $14$ but instead subtracted $14$. After these mistakes, the result was $16$. If the correct operations had been used, the value produced would have been\n$\\textbf{(A)}\\ \\text{less than 400} \\qquad\\textbf{(B)}\\ \\text{between 400 and 600} \\qquad\\textbf{(C)}\\ \\text{between 600 and 800} \\\\ \\textbf{(D)}\\ \\text{between 800 and 1000} \\qquad\\textbf{(E)}\\ \\text{greater than 1000}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $a$ and $b$ be distinct real numbers for which\n\\[\\frac{a}{b} + \\frac{a+10b}{b+10a} = 2.\\]\nFind $\\frac{a}{b}$\n$\\text{(A) }0.4 \\qquad \\text{(B) }0.5 \\qquad \\text{(C) }0.6 \\qquad \\text{(D) }0.7 \\qquad \\text{(E) }0.8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2375", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $a$ and $b$ be distinct real numbers for which\n\\[\\frac{a}{b} + \\frac{a+10b}{b+10a} = 2.\\]\nFind $\\frac{a}{b}$\n$\\text{(A) }0.4 \\qquad \\text{(B) }0.5 \\qquad \\text{(C) }0.6 \\qquad \\text{(D) }0.7 \\qquad \\text{(E) }0.8$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 19\\qquad \\textbf{(D)}\\ -1\\qquad \\textbf{(E)}\\ -19$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2376", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 19\\qquad \\textbf{(D)}\\ -1\\qquad \\textbf{(E)}\\ -19$" + } + }, + { + "question": "Return your final response within \\boxed{}. The bar graph shows the results of a survey on color preferences. What percent preferred blue?\n\n$\\text{(A)}\\ 20\\% \\qquad \\text{(B)}\\ 24\\% \\qquad \\text{(C)}\\ 30\\% \\qquad \\text{(D)}\\ 36\\% \\qquad \\text{(E)}\\ 42\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2377", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The bar graph shows the results of a survey on color preferences. What percent preferred blue?\n\n$\\text{(A)}\\ 20\\% \\qquad \\text{(B)}\\ 24\\% \\qquad \\text{(C)}\\ 30\\% \\qquad \\text{(D)}\\ 36\\% \\qquad \\text{(E)}\\ 42\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. The table below gives the percent of students in each grade at Annville and Cleona elementary schools:\n\\[\\begin{tabular}{rccccccc}&\\textbf{\\underline{K}}&\\textbf{\\underline{1}}&\\textbf{\\underline{2}}&\\textbf{\\underline{3}}&\\textbf{\\underline{4}}&\\textbf{\\underline{5}}&\\textbf{\\underline{6}}\\\\ \\textbf{Annville:}& 16\\% & 15\\% & 15\\% & 14\\% & 13\\% & 16\\% & 11\\%\\\\ \\textbf{Cleona:}& 12\\% & 15\\% & 14\\% & 13\\% & 15\\% & 14\\% & 17\\%\\end{tabular}\\]\nAnnville has 100 students and Cleona has 200 students. In the two schools combined, what percent of the students are in grade 6?\n$\\text{(A)}\\ 12\\% \\qquad \\text{(B)}\\ 13\\% \\qquad \\text{(C)}\\ 14\\% \\qquad \\text{(D)}\\ 15\\% \\qquad \\text{(E)}\\ 28\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2378", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The table below gives the percent of students in each grade at Annville and Cleona elementary schools:\n\\[\\begin{tabular}{rccccccc}&\\textbf{\\underline{K}}&\\textbf{\\underline{1}}&\\textbf{\\underline{2}}&\\textbf{\\underline{3}}&\\textbf{\\underline{4}}&\\textbf{\\underline{5}}&\\textbf{\\underline{6}}\\\\ \\textbf{Annville:}& 16\\% & 15\\% & 15\\% & 14\\% & 13\\% & 16\\% & 11\\%\\\\ \\textbf{Cleona:}& 12\\% & 15\\% & 14\\% & 13\\% & 15\\% & 14\\% & 17\\%\\end{tabular}\\]\nAnnville has 100 students and Cleona has 200 students. In the two schools combined, what percent of the students are in grade 6?\n$\\text{(A)}\\ 12\\% \\qquad \\text{(B)}\\ 13\\% \\qquad \\text{(C)}\\ 14\\% \\qquad \\text{(D)}\\ 15\\% \\qquad \\text{(E)}\\ 28\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. Sale prices at the Ajax Outlet Store are $50\\%$ below original prices. On Saturdays an additional discount of $20\\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $\\textdollar 180$?\n$\\text{(A)}$ $\\textdollar 54$\n$\\text{(B)}$ $\\textdollar 72$\n$\\text{(C)}$ $\\textdollar 90$\n$\\text{(D)}$ $\\textdollar 108$\n$\\text{(D)}$ $\\textdollar 110$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2379", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Sale prices at the Ajax Outlet Store are $50\\%$ below original prices. On Saturdays an additional discount of $20\\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $\\textdollar 180$?\n$\\text{(A)}$ $\\textdollar 54$\n$\\text{(B)}$ $\\textdollar 72$\n$\\text{(C)}$ $\\textdollar 90$\n$\\text{(D)}$ $\\textdollar 108$\n$\\text{(D)}$ $\\textdollar 110$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the statement, \"If $n$ is not prime, then $n-2$ is prime.\" Which of the following values of $n$ is a counterexample to this statement?\n$\\textbf{(A) } 11 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 19 \\qquad \\textbf{(D) } 21 \\qquad \\textbf{(E) } 27$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2380", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the statement, \"If $n$ is not prime, then $n-2$ is prime.\" Which of the following values of $n$ is a counterexample to this statement?\n$\\textbf{(A) } 11 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 19 \\qquad \\textbf{(D) } 21 \\qquad \\textbf{(E) } 27$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $i=\\sqrt{-1}$. The product of the real parts of the roots of $z^2-z=5-5i$ is\n$\\text{(A) } -25\\quad \\text{(B) } -6\\quad \\text{(C) } -5\\quad \\text{(D) } \\frac{1}{4}\\quad \\text{(E) } 25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2381", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $i=\\sqrt{-1}$. The product of the real parts of the roots of $z^2-z=5-5i$ is\n$\\text{(A) } -25\\quad \\text{(B) } -6\\quad \\text{(C) } -5\\quad \\text{(D) } \\frac{1}{4}\\quad \\text{(E) } 25$" + } + }, + { + "question": "Return your final response within \\boxed{}. $1-2-3+4+5-6-7+8+9-10-11+\\cdots + 1992+1993-1994-1995+1996=$\n$\\text{(A)}\\ -998 \\qquad \\text{(B)}\\ -1 \\qquad \\text{(C)}\\ 0 \\qquad \\text{(D)}\\ 1 \\qquad \\text{(E)}\\ 998$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2382", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $1-2-3+4+5-6-7+8+9-10-11+\\cdots + 1992+1993-1994-1995+1996=$\n$\\text{(A)}\\ -998 \\qquad \\text{(B)}\\ -1 \\qquad \\text{(C)}\\ 0 \\qquad \\text{(D)}\\ 1 \\qquad \\text{(E)}\\ 998$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of these numbers is largest?\n$\\text{(A) } \\sqrt{\\sqrt[3]{5\\cdot 6}}\\quad \\text{(B) } \\sqrt{6\\sqrt[3]{5}}\\quad \\text{(C) } \\sqrt{5\\sqrt[3]{6}}\\quad \\text{(D) } \\sqrt[3]{5\\sqrt{6}}\\quad \\text{(E) } \\sqrt[3]{6\\sqrt{5}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2383", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of these numbers is largest?\n$\\text{(A) } \\sqrt{\\sqrt[3]{5\\cdot 6}}\\quad \\text{(B) } \\sqrt{6\\sqrt[3]{5}}\\quad \\text{(C) } \\sqrt{5\\sqrt[3]{6}}\\quad \\text{(D) } \\sqrt[3]{5\\sqrt{6}}\\quad \\text{(E) } \\sqrt[3]{6\\sqrt{5}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mr. J left his entire estate to his wife, his daughter, his son, and the cook. \nHis daughter and son got half the estate, sharing in the ratio of $4$ to $3$. \nHis wife got twice as much as the son. If the cook received a bequest of $\\textdollar{500}$, then the entire estate was: \n$\\textbf{(A)}\\ \\textdollar{3500}\\qquad \\textbf{(B)}\\ \\textdollar{5500}\\qquad \\textbf{(C)}\\ \\textdollar{6500}\\qquad \\textbf{(D)}\\ \\textdollar{7000}\\qquad \\textbf{(E)}\\ \\textdollar{7500}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2384", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mr. J left his entire estate to his wife, his daughter, his son, and the cook. \nHis daughter and son got half the estate, sharing in the ratio of $4$ to $3$. \nHis wife got twice as much as the son. If the cook received a bequest of $\\textdollar{500}$, then the entire estate was: \n$\\textbf{(A)}\\ \\textdollar{3500}\\qquad \\textbf{(B)}\\ \\textdollar{5500}\\qquad \\textbf{(C)}\\ \\textdollar{6500}\\qquad \\textbf{(D)}\\ \\textdollar{7000}\\qquad \\textbf{(E)}\\ \\textdollar{7500}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Betty used a calculator to find the [product](https://artofproblemsolving.com/wiki/index.php/Product) $0.075 \\times 2.56$. She forgot to enter the decimal points. The calculator showed $19200$. If Betty had entered the decimal points correctly, the answer would have been\n$\\text{(A)}\\ .0192 \\qquad \\text{(B)}\\ .192 \\qquad \\text{(C)}\\ 1.92 \\qquad \\text{(D)}\\ 19.2 \\qquad \\text{(E)}\\ 192$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2385", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Betty used a calculator to find the [product](https://artofproblemsolving.com/wiki/index.php/Product) $0.075 \\times 2.56$. She forgot to enter the decimal points. The calculator showed $19200$. If Betty had entered the decimal points correctly, the answer would have been\n$\\text{(A)}\\ .0192 \\qquad \\text{(B)}\\ .192 \\qquad \\text{(C)}\\ 1.92 \\qquad \\text{(D)}\\ 19.2 \\qquad \\text{(E)}\\ 192$" + } + }, + { + "question": "Return your final response within \\boxed{}. Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?\n$\\textbf{(A) } \\text{Arn}\\qquad\\textbf{(B) }\\text{Bob}\\qquad\\textbf{(C) }\\text{Cyd}\\qquad\\textbf{(D) }\\text{Dan}\\qquad \\textbf{(E) }\\text{Eve}\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2386", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?\n$\\textbf{(A) } \\text{Arn}\\qquad\\textbf{(B) }\\text{Bob}\\qquad\\textbf{(C) }\\text{Cyd}\\qquad\\textbf{(D) }\\text{Dan}\\qquad \\textbf{(E) }\\text{Eve}\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose $a$, $b$, $c$ are positive integers such that \\[a+b+c=23\\] and \\[\\gcd(a,b)+\\gcd(b,c)+\\gcd(c,a)=9.\\] What is the sum of all possible distinct values of $a^2+b^2+c^2$? \n$\\textbf{(A)}\\: 259\\qquad\\textbf{(B)} \\: 438\\qquad\\textbf{(C)} \\: 516\\qquad\\textbf{(D)} \\: 625\\qquad\\textbf{(E)} \\: 687$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2387", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose $a$, $b$, $c$ are positive integers such that \\[a+b+c=23\\] and \\[\\gcd(a,b)+\\gcd(b,c)+\\gcd(c,a)=9.\\] What is the sum of all possible distinct values of $a^2+b^2+c^2$? \n$\\textbf{(A)}\\: 259\\qquad\\textbf{(B)} \\: 438\\qquad\\textbf{(C)} \\: 516\\qquad\\textbf{(D)} \\: 625\\qquad\\textbf{(E)} \\: 687$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:\n$\\textbf{(A)}\\ (x-2)^4 \\qquad \\textbf{(B)}\\ (x-1)^4 \\qquad \\textbf{(C)}\\ x^4 \\qquad \\textbf{(D)}\\ (x+1)^4 \\qquad \\textbf{(E)}\\ x^4+1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2388", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:\n$\\textbf{(A)}\\ (x-2)^4 \\qquad \\textbf{(B)}\\ (x-1)^4 \\qquad \\textbf{(C)}\\ x^4 \\qquad \\textbf{(D)}\\ (x+1)^4 \\qquad \\textbf{(E)}\\ x^4+1$" + } + }, + { + "question": "Return your final response within \\boxed{}. For a certain complex number $c$, the polynomial\n\\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\\]has exactly 4 distinct roots. What is $|c|$?\n$\\textbf{(A) } 2 \\qquad \\textbf{(B) } \\sqrt{6} \\qquad \\textbf{(C) } 2\\sqrt{2} \\qquad \\textbf{(D) } 3 \\qquad \\textbf{(E) } \\sqrt{10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2389", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a certain complex number $c$, the polynomial\n\\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\\]has exactly 4 distinct roots. What is $|c|$?\n$\\textbf{(A) } 2 \\qquad \\textbf{(B) } \\sqrt{6} \\qquad \\textbf{(C) } 2\\sqrt{2} \\qquad \\textbf{(D) } 3 \\qquad \\textbf{(E) } \\sqrt{10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L$, without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.)\n\nHow many different routes can Paula take?\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2390", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L$, without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.)\n\nHow many different routes can Paula take?\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$ ?\n$\\textbf{(A) }23\\qquad\\textbf{(B) }24\\qquad\\textbf{(C) }25\\qquad\\textbf{(D) }26\\qquad\\textbf{(E) }27$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2391", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$ ?\n$\\textbf{(A) }23\\qquad\\textbf{(B) }24\\qquad\\textbf{(C) }25\\qquad\\textbf{(D) }26\\qquad\\textbf{(E) }27$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.\n[asy] fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray); fill((1,2)--(2,2)--(2,3)--(1,3)--cycle,gray); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,linewidth(1)); draw((2,0)--(2,3),linewidth(1)); draw((0,1)--(3,1),linewidth(1)); draw((1,0)--(1,3),linewidth(1)); draw((0,2)--(3,2),linewidth(1)); fill((6,0)--(8,0)--(8,1)--(6,1)--cycle,gray); draw((6,0)--(9,0)--(9,3)--(6,3)--cycle,linewidth(1)); draw((8,0)--(8,3),linewidth(1)); draw((6,1)--(9,1),linewidth(1)); draw((7,0)--(7,3),linewidth(1)); draw((6,2)--(9,2),linewidth(1)); fill((14,1)--(15,1)--(15,3)--(14,3)--cycle,gray); draw((12,0)--(15,0)--(15,3)--(12,3)--cycle,linewidth(1)); draw((14,0)--(14,3),linewidth(1)); draw((12,1)--(15,1),linewidth(1)); draw((13,0)--(13,3),linewidth(1)); draw((12,2)--(15,2),linewidth(1)); fill((18,1)--(19,1)--(19,3)--(18,3)--cycle,gray); draw((18,0)--(21,0)--(21,3)--(18,3)--cycle,linewidth(1)); draw((20,0)--(20,3),linewidth(1)); draw((18,1)--(21,1),linewidth(1)); draw((19,0)--(19,3),linewidth(1)); draw((18,2)--(21,2),linewidth(1)); [/asy]\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 6 \\qquad \\text{(C)}\\ 8 \\qquad \\text{(D)}\\ 12 \\qquad \\text{(E)}\\ 18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2392", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.\n[asy] fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray); fill((1,2)--(2,2)--(2,3)--(1,3)--cycle,gray); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,linewidth(1)); draw((2,0)--(2,3),linewidth(1)); draw((0,1)--(3,1),linewidth(1)); draw((1,0)--(1,3),linewidth(1)); draw((0,2)--(3,2),linewidth(1)); fill((6,0)--(8,0)--(8,1)--(6,1)--cycle,gray); draw((6,0)--(9,0)--(9,3)--(6,3)--cycle,linewidth(1)); draw((8,0)--(8,3),linewidth(1)); draw((6,1)--(9,1),linewidth(1)); draw((7,0)--(7,3),linewidth(1)); draw((6,2)--(9,2),linewidth(1)); fill((14,1)--(15,1)--(15,3)--(14,3)--cycle,gray); draw((12,0)--(15,0)--(15,3)--(12,3)--cycle,linewidth(1)); draw((14,0)--(14,3),linewidth(1)); draw((12,1)--(15,1),linewidth(1)); draw((13,0)--(13,3),linewidth(1)); draw((12,2)--(15,2),linewidth(1)); fill((18,1)--(19,1)--(19,3)--(18,3)--cycle,gray); draw((18,0)--(21,0)--(21,3)--(18,3)--cycle,linewidth(1)); draw((20,0)--(20,3),linewidth(1)); draw((18,1)--(21,1),linewidth(1)); draw((19,0)--(19,3),linewidth(1)); draw((18,2)--(21,2),linewidth(1)); [/asy]\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 6 \\qquad \\text{(C)}\\ 8 \\qquad \\text{(D)}\\ 12 \\qquad \\text{(E)}\\ 18$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?\n$\\textbf{(A) } 1\\qquad\\textbf{(B) } 2\\qquad\\textbf{(C) } 3\\qquad\\textbf{(D) } 4\\qquad\\textbf{(E) } 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2393", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?\n$\\textbf{(A) } 1\\qquad\\textbf{(B) } 2\\qquad\\textbf{(C) } 3\\qquad\\textbf{(D) } 4\\qquad\\textbf{(E) } 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. With $ $1000$ a rancher is to buy steers at $ $25$ each and cows at $ $26$ each. If the number of steers $s$ and the number of cows $c$ are both positive integers, then:\n$\\textbf{(A)}\\ \\text{this problem has no solution}\\qquad\\\\ \\textbf{(B)}\\ \\text{there are two solutions with }{s}\\text{ exceeding }{c}\\qquad \\\\ \\textbf{(C)}\\ \\text{there are two solutions with }{c}\\text{ exceeding }{s}\\qquad \\\\ \\textbf{(D)}\\ \\text{there is one solution with }{s}\\text{ exceeding }{c}\\qquad \\\\ \\textbf{(E)}\\ \\text{there is one solution with }{c}\\text{ exceeding }{s}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2394", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. With $ $1000$ a rancher is to buy steers at $ $25$ each and cows at $ $26$ each. If the number of steers $s$ and the number of cows $c$ are both positive integers, then:\n$\\textbf{(A)}\\ \\text{this problem has no solution}\\qquad\\\\ \\textbf{(B)}\\ \\text{there are two solutions with }{s}\\text{ exceeding }{c}\\qquad \\\\ \\textbf{(C)}\\ \\text{there are two solutions with }{c}\\text{ exceeding }{s}\\qquad \\\\ \\textbf{(D)}\\ \\text{there is one solution with }{s}\\text{ exceeding }{c}\\qquad \\\\ \\textbf{(E)}\\ \\text{there is one solution with }{c}\\text{ exceeding }{s}$" + } + }, + { + "question": "Return your final response within \\boxed{}. An arbitrary [circle](https://artofproblemsolving.com/wiki/index.php/Circle) can intersect the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $y=\\sin x$ in\n$\\mathrm{(A) \\ } \\text{at most }2\\text{ points} \\qquad \\mathrm{(B) \\ }\\text{at most }4\\text{ points} \\qquad \\mathrm{(C) \\ } \\text{at most }6\\text{ points} \\qquad \\mathrm{(D) \\ } \\text{at most }8\\text{ points}\\qquad \\mathrm{(E) \\ }\\text{more than }16\\text{ points}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2395", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An arbitrary [circle](https://artofproblemsolving.com/wiki/index.php/Circle) can intersect the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $y=\\sin x$ in\n$\\mathrm{(A) \\ } \\text{at most }2\\text{ points} \\qquad \\mathrm{(B) \\ }\\text{at most }4\\text{ points} \\qquad \\mathrm{(C) \\ } \\text{at most }6\\text{ points} \\qquad \\mathrm{(D) \\ } \\text{at most }8\\text{ points}\\qquad \\mathrm{(E) \\ }\\text{more than }16\\text{ points}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The top of one tree is $16$ feet higher than the top of another tree. The heights of the two trees are in the ratio $3:4$. In feet, how tall is the taller tree? \n$\\textbf{(A)}\\ 48 \\qquad\\textbf{(B)}\\ 64 \\qquad\\textbf{(C)}\\ 80 \\qquad\\textbf{(D)}\\ 96\\qquad\\textbf{(E)}\\ 112$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2396", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The top of one tree is $16$ feet higher than the top of another tree. The heights of the two trees are in the ratio $3:4$. In feet, how tall is the taller tree? \n$\\textbf{(A)}\\ 48 \\qquad\\textbf{(B)}\\ 64 \\qquad\\textbf{(C)}\\ 80 \\qquad\\textbf{(D)}\\ 96\\qquad\\textbf{(E)}\\ 112$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1)+E(2)+E(3)+\\cdots+E(100)$\n$\\text{(A)}\\ 200\\qquad\\text{(B)}\\ 360\\qquad\\text{(C)}\\ 400\\qquad\\text{(D)}\\ 900\\qquad\\text{(E)}\\ 2250$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2397", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1)+E(2)+E(3)+\\cdots+E(100)$\n$\\text{(A)}\\ 200\\qquad\\text{(B)}\\ 360\\qquad\\text{(C)}\\ 400\\qquad\\text{(D)}\\ 900\\qquad\\text{(E)}\\ 2250$" + } + }, + { + "question": "Return your final response within \\boxed{}. The pairs of values of $x$ and $y$ that are the common solutions of the equations $y=(x+1)^2$ and $xy+y=1$ are: \n$\\textbf{(A)}\\ \\text{3 real pairs}\\qquad\\textbf{(B)}\\ \\text{4 real pairs}\\qquad\\textbf{(C)}\\ \\text{4 imaginary pairs}\\\\ \\textbf{(D)}\\ \\text{2 real and 2 imaginary pairs}\\qquad\\textbf{(E)}\\ \\text{1 real and 2 imaginary pairs}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2398", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The pairs of values of $x$ and $y$ that are the common solutions of the equations $y=(x+1)^2$ and $xy+y=1$ are: \n$\\textbf{(A)}\\ \\text{3 real pairs}\\qquad\\textbf{(B)}\\ \\text{4 real pairs}\\qquad\\textbf{(C)}\\ \\text{4 imaginary pairs}\\\\ \\textbf{(D)}\\ \\text{2 real and 2 imaginary pairs}\\qquad\\textbf{(E)}\\ \\text{1 real and 2 imaginary pairs}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $\\textdollar 2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\\textdollar 0.75$ for her to make. In dollars, what is her profit for the day?\n$\\textbf{(A)}\\ 24\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 44\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 52$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2399", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $\\textdollar 2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\\textdollar 0.75$ for her to make. In dollars, what is her profit for the day?\n$\\textbf{(A)}\\ 24\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 44\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 52$" + } + }, + { + "question": "Return your final response within \\boxed{}. A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$?\n\n$\\textbf{(A)}\\hspace{.05in}\\frac{1}5\\qquad\\textbf{(B)}\\hspace{.05in}\\frac{2}5\\qquad\\textbf{(C)}\\hspace{.05in}\\frac{1}2\\qquad\\textbf{(D)}\\hspace{.05in}1\\qquad\\textbf{(E)}\\hspace{.05in}4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2400", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$?\n\n$\\textbf{(A)}\\hspace{.05in}\\frac{1}5\\qquad\\textbf{(B)}\\hspace{.05in}\\frac{2}5\\qquad\\textbf{(C)}\\hspace{.05in}\\frac{1}2\\qquad\\textbf{(D)}\\hspace{.05in}1\\qquad\\textbf{(E)}\\hspace{.05in}4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following numbers is the largest?\n$\\text{(A)}\\ 0.97 \\qquad \\text{(B)}\\ 0.979 \\qquad \\text{(C)}\\ 0.9709 \\qquad \\text{(D)}\\ 0.907 \\qquad \\text{(E)}\\ 0.9089$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2401", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following numbers is the largest?\n$\\text{(A)}\\ 0.97 \\qquad \\text{(B)}\\ 0.979 \\qquad \\text{(C)}\\ 0.9709 \\qquad \\text{(D)}\\ 0.907 \\qquad \\text{(E)}\\ 0.9089$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the markings on the [number line](https://artofproblemsolving.com/wiki/index.php/Number_line) are equally spaced, what is the number $\\text{y}$?\n\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 10 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 15 \\qquad \\text{(E)}\\ 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2402", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the markings on the [number line](https://artofproblemsolving.com/wiki/index.php/Number_line) are equally spaced, what is the number $\\text{y}$?\n\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 10 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 15 \\qquad \\text{(E)}\\ 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Equilateral $\\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?\n\n$\\textbf{(A)}\\ \\dfrac{12+3\\sqrt3}4\\qquad\\textbf{(B)}\\ \\dfrac92\\qquad\\textbf{(C)}\\ 3+\\sqrt3\\qquad\\textbf{(D)}\\ \\dfrac{6+3\\sqrt3}2\\qquad\\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2403", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Equilateral $\\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?\n\n$\\textbf{(A)}\\ \\dfrac{12+3\\sqrt3}4\\qquad\\textbf{(B)}\\ \\dfrac92\\qquad\\textbf{(C)}\\ 3+\\sqrt3\\qquad\\textbf{(D)}\\ \\dfrac{6+3\\sqrt3}2\\qquad\\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many integers $n$ between $1$ and $50$, inclusive, is \\[\\frac{(n^2-1)!}{(n!)^n}\\] an integer? (Recall that $0! = 1$.)\n$\\textbf{(A) } 31 \\qquad \\textbf{(B) } 32 \\qquad \\textbf{(C) } 33 \\qquad \\textbf{(D) } 34 \\qquad \\textbf{(E) } 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2404", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many integers $n$ between $1$ and $50$, inclusive, is \\[\\frac{(n^2-1)!}{(n!)^n}\\] an integer? (Recall that $0! = 1$.)\n$\\textbf{(A) } 31 \\qquad \\textbf{(B) } 32 \\qquad \\textbf{(C) } 33 \\qquad \\textbf{(D) } 34 \\qquad \\textbf{(E) } 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?\n\n$\\begin{tabular}[t]{|c|c|c|c|} \\multicolumn{4}{c}{Average Scores}\\\\\\hline Category&Adams&Baker&Adams\\&Baker\\\\\\hline Boys&71&81&79\\\\ Girls&76&90&?\\\\ Boys\\&Girls&74&84& \\\\\\hline \\end{tabular}$\n\n$\\text{(A) } 81\\quad \\text{(B) } 82\\quad \\text{(C) } 83\\quad \\text{(D) } 84\\quad \\text{(E) } 85$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2405", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?\n\n$\\begin{tabular}[t]{|c|c|c|c|} \\multicolumn{4}{c}{Average Scores}\\\\\\hline Category&Adams&Baker&Adams\\&Baker\\\\\\hline Boys&71&81&79\\\\ Girls&76&90&?\\\\ Boys\\&Girls&74&84& \\\\\\hline \\end{tabular}$\n\n$\\text{(A) } 81\\quad \\text{(B) } 82\\quad \\text{(C) } 83\\quad \\text{(D) } 84\\quad \\text{(E) } 85$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\{a_1,a_2,a_3,\\ldots,a_n\\}$ is a [set](https://artofproblemsolving.com/wiki/index.php/Set) of [real numbers](https://artofproblemsolving.com/wiki/index.php/Real_numbers), indexed so that $a_1 < a_2 < a_3 < \\cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \\cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty [subsets](https://artofproblemsolving.com/wiki/index.php/Subset) of $\\{1,2,\\ldots,n\\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "368", + "index": "Sky-T1_10k_2406", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\{a_1,a_2,a_3,\\ldots,a_n\\}$ is a [set](https://artofproblemsolving.com/wiki/index.php/Set) of [real numbers](https://artofproblemsolving.com/wiki/index.php/Real_numbers), indexed so that $a_1 < a_2 < a_3 < \\cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \\cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty [subsets](https://artofproblemsolving.com/wiki/index.php/Subset) of $\\{1,2,\\ldots,n\\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$" + } + }, + { + "question": "Return your final response within \\boxed{}. A positive number $x$ satisfies the inequality $\\sqrt{x} < 2x$ if and only if\n$\\text{(A)} \\ x > \\frac{1}{4} \\qquad \\text{(B)} \\ x > 2 \\qquad \\text{(C)} \\ x > 4 \\qquad \\text{(D)} \\ x < \\frac{1}{4}\\qquad \\text{(E)} \\ x < 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2407", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A positive number $x$ satisfies the inequality $\\sqrt{x} < 2x$ if and only if\n$\\text{(A)} \\ x > \\frac{1}{4} \\qquad \\text{(B)} \\ x > 2 \\qquad \\text{(C)} \\ x > 4 \\qquad \\text{(D)} \\ x < \\frac{1}{4}\\qquad \\text{(E)} \\ x < 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $n$ be the smallest positive integer such that $n$ is divisible by $20$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2408", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $n$ be the smallest positive integer such that $n$ is divisible by $20$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. The digits $1$, $2$, $3$, $4$, and $5$ are each used once to write a five-digit number $PQRST$. The three-digit number $PQR$ is divisible by $4$, the three-digit number $QRS$ is divisible by $5$, and the three-digit number $RST$ is divisible by $3$. What is $P$?\n$\\textbf{(A) }1\\qquad\\textbf{(B) }2\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }4\\qquad \\textbf{(E) }5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2409", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The digits $1$, $2$, $3$, $4$, and $5$ are each used once to write a five-digit number $PQRST$. The three-digit number $PQR$ is divisible by $4$, the three-digit number $QRS$ is divisible by $5$, and the three-digit number $RST$ is divisible by $3$. What is $P$?\n$\\textbf{(A) }1\\qquad\\textbf{(B) }2\\qquad\\textbf{(C) }3\\qquad\\textbf{(D) }4\\qquad \\textbf{(E) }5$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many integers $x$ is the number $x^4-51x^2+50$ negative?\n$\\textbf {(A) } 8 \\qquad \\textbf {(B) } 10 \\qquad \\textbf {(C) } 12 \\qquad \\textbf {(D) } 14 \\qquad \\textbf {(E) } 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2410", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many integers $x$ is the number $x^4-51x^2+50$ negative?\n$\\textbf {(A) } 8 \\qquad \\textbf {(B) } 10 \\qquad \\textbf {(C) } 12 \\qquad \\textbf {(D) } 14 \\qquad \\textbf {(E) } 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Exactly three of the interior angles of a convex [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) are obtuse. What is the maximum number of sides of such a polygon?\n$\\mathrm{(A)\\ } 4 \\qquad \\mathrm{(B) \\ }5 \\qquad \\mathrm{(C) \\ } 6 \\qquad \\mathrm{(D) \\ } 7 \\qquad \\mathrm{(E) \\ }8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2411", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Exactly three of the interior angles of a convex [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) are obtuse. What is the maximum number of sides of such a polygon?\n$\\mathrm{(A)\\ } 4 \\qquad \\mathrm{(B) \\ }5 \\qquad \\mathrm{(C) \\ } 6 \\qquad \\mathrm{(D) \\ } 7 \\qquad \\mathrm{(E) \\ }8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?\n$\\textbf{(A) } \\frac14 \\qquad\\textbf{(B) } \\frac13 \\qquad\\textbf{(C) } \\frac12 \\qquad\\textbf{(D) } \\frac23 \\qquad\\textbf{(E) } \\frac34$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2412", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?\n$\\textbf{(A) } \\frac14 \\qquad\\textbf{(B) } \\frac13 \\qquad\\textbf{(C) } \\frac12 \\qquad\\textbf{(D) } \\frac23 \\qquad\\textbf{(E) } \\frac34$" + } + }, + { + "question": "Return your final response within \\boxed{}. Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?\n$\\textbf{(A) } 90 \\qquad \\textbf{(B) } 100 \\qquad \\textbf{(C) } 180 \\qquad \\textbf{(D) } 200 \\qquad \\textbf{(E) } 360$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2413", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?\n$\\textbf{(A) } 90 \\qquad \\textbf{(B) } 100 \\qquad \\textbf{(C) } 180 \\qquad \\textbf{(D) } 200 \\qquad \\textbf{(E) } 360$" + } + }, + { + "question": "Return your final response within \\boxed{}. Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?\n$\\textbf{(A)}\\ \\frac{1}{8} \\qquad \\textbf{(B)}\\ \\frac{1}{7} \\qquad \\textbf{(C)}\\ \\frac{1}{6} \\qquad \\textbf{(D)}\\ \\frac{1}{4} \\qquad \\textbf{(E)}\\ \\frac{1}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2414", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?\n$\\textbf{(A)}\\ \\frac{1}{8} \\qquad \\textbf{(B)}\\ \\frac{1}{7} \\qquad \\textbf{(C)}\\ \\frac{1}{6} \\qquad \\textbf{(D)}\\ \\frac{1}{4} \\qquad \\textbf{(E)}\\ \\frac{1}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \\tfrac{1}{2}$?\n$\\textbf{(A) } \\frac{1}{3} \\qquad \\textbf{(B) } \\frac{7}{16} \\qquad \\textbf{(C) } \\frac{1}{2} \\qquad \\textbf{(D) } \\frac{9}{16} \\qquad \\textbf{(E) } \\frac{2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2415", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \\tfrac{1}{2}$?\n$\\textbf{(A) } \\frac{1}{3} \\qquad \\textbf{(B) } \\frac{7}{16} \\qquad \\textbf{(C) } \\frac{1}{2} \\qquad \\textbf{(D) } \\frac{9}{16} \\qquad \\textbf{(E) } \\frac{2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The unit's digit (one's digit) of the product of any six consecutive positive whole numbers is\n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2416", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The unit's digit (one's digit) of the product of any six consecutive positive whole numbers is\n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 4 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. If \\[N=\\frac{\\sqrt{\\sqrt{5}+2}+\\sqrt{\\sqrt{5}-2}}{\\sqrt{\\sqrt{5}+1}}-\\sqrt{3-2\\sqrt{2}},\\] then $N$ equals\n$\\textbf{(A) }1\\qquad \\textbf{(B) }2\\sqrt{2}-1\\qquad \\textbf{(C) }\\frac{\\sqrt{5}}{2}\\qquad \\textbf{(D) }\\sqrt{\\frac{5}{2}}\\qquad \\textbf{(E) }\\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2417", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If \\[N=\\frac{\\sqrt{\\sqrt{5}+2}+\\sqrt{\\sqrt{5}-2}}{\\sqrt{\\sqrt{5}+1}}-\\sqrt{3-2\\sqrt{2}},\\] then $N$ equals\n$\\textbf{(A) }1\\qquad \\textbf{(B) }2\\sqrt{2}-1\\qquad \\textbf{(C) }\\frac{\\sqrt{5}}{2}\\qquad \\textbf{(D) }\\sqrt{\\frac{5}{2}}\\qquad \\textbf{(E) }\\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?\n$\\textbf{(A)}\\dfrac{47}{256}\\qquad\\textbf{(B)}\\dfrac{3}{16}\\qquad\\textbf{(C) }\\dfrac{49}{256}\\qquad\\textbf{(D) }\\dfrac{25}{128}\\qquad\\textbf{(E) }\\dfrac{51}{256}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2418", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?\n$\\textbf{(A)}\\dfrac{47}{256}\\qquad\\textbf{(B)}\\dfrac{3}{16}\\qquad\\textbf{(C) }\\dfrac{49}{256}\\qquad\\textbf{(D) }\\dfrac{25}{128}\\qquad\\textbf{(E) }\\dfrac{51}{256}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$\n$\\textbf{(A)}\\ 4:3 \\qquad\\textbf{(B)}\\ 3:2 \\qquad\\textbf{(C)}\\ 8:3 \\qquad\\textbf{(D)}\\ 4:1 \\qquad\\textbf{(E)}\\ 16:3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2419", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$\n$\\textbf{(A)}\\ 4:3 \\qquad\\textbf{(B)}\\ 3:2 \\qquad\\textbf{(C)}\\ 8:3 \\qquad\\textbf{(D)}\\ 4:1 \\qquad\\textbf{(E)}\\ 16:3$" + } + }, + { + "question": "Return your final response within \\boxed{}. You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $$1.02$, with at least one coin of each type. How many dimes must you have?\n$\\text{(A)}\\ 1\\qquad\\text{(B)}\\ 2\\qquad\\text{(C)}\\ 3\\qquad\\text{(D)}\\ 4\\qquad\\text{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2420", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $$1.02$, with at least one coin of each type. How many dimes must you have?\n$\\text{(A)}\\ 1\\qquad\\text{(B)}\\ 2\\qquad\\text{(C)}\\ 3\\qquad\\text{(D)}\\ 4\\qquad\\text{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each cell of an $m\\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:\n(i) The difference between any two adjacent numbers is either $0$ or $1$. \n(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$ .\nDetermine the number of distinct gardens in terms of $m$ and $n$ .", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "2^{mn} -1", + "index": "Sky-T1_10k_2421", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each cell of an $m\\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:\n(i) The difference between any two adjacent numbers is either $0$ or $1$. \n(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$ .\nDetermine the number of distinct gardens in terms of $m$ and $n$ ." + } + }, + { + "question": "Return your final response within \\boxed{}. Two real numbers are selected independently at random from the interval $[-20, 10]$. What is the probability that the product of those numbers is greater than zero?\n$\\textbf{(A)}\\ \\frac{1}{9} \\qquad\\textbf{(B)}\\ \\frac{1}{3} \\qquad\\textbf{(C)}\\ \\frac{4}{9} \\qquad\\textbf{(D)}\\ \\frac{5}{9} \\qquad\\textbf{(E)}\\ \\frac{2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2422", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two real numbers are selected independently at random from the interval $[-20, 10]$. What is the probability that the product of those numbers is greater than zero?\n$\\textbf{(A)}\\ \\frac{1}{9} \\qquad\\textbf{(B)}\\ \\frac{1}{3} \\qquad\\textbf{(C)}\\ \\frac{4}{9} \\qquad\\textbf{(D)}\\ \\frac{5}{9} \\qquad\\textbf{(E)}\\ \\frac{2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alice is making a batch of cookies and needs $2\\frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?\n$\\textbf{(A)}\\ 8 \\qquad\\textbf{(B)}\\ 10 \\qquad\\textbf{(C)}\\ 12 \\qquad\\textbf{(D)}\\ 16 \\qquad\\textbf{(E)}\\ 20$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2423", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alice is making a batch of cookies and needs $2\\frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?\n$\\textbf{(A)}\\ 8 \\qquad\\textbf{(B)}\\ 10 \\qquad\\textbf{(C)}\\ 12 \\qquad\\textbf{(D)}\\ 16 \\qquad\\textbf{(E)}\\ 20$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than the common value. Which of the following is a correct description of $S$?\n$\\textbf{(A)}\\ \\text{a single point} \\qquad\\textbf{(B)}\\ \\text{two intersecting lines} \\\\ \\qquad\\textbf{(C)}\\ \\text{three lines whose pairwise intersections are three distinct points} \\\\ \\qquad\\textbf{(D)}\\ \\text{a triangle}\\qquad\\textbf{(E)}\\ \\text{three rays with a common point}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2424", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than the common value. Which of the following is a correct description of $S$?\n$\\textbf{(A)}\\ \\text{a single point} \\qquad\\textbf{(B)}\\ \\text{two intersecting lines} \\\\ \\qquad\\textbf{(C)}\\ \\text{three lines whose pairwise intersections are three distinct points} \\\\ \\qquad\\textbf{(D)}\\ \\text{a triangle}\\qquad\\textbf{(E)}\\ \\text{three rays with a common point}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The set of $x$-values satisfying the equation $x^{\\log_{10} x} = \\frac{x^3}{100}$ consists of: \n$\\textbf{(A)}\\ \\frac{1}{10}\\qquad\\textbf{(B)}\\ \\text{10, only}\\qquad\\textbf{(C)}\\ \\text{100, only}\\qquad\\textbf{(D)}\\ \\text{10 or 100, only}\\qquad\\textbf{(E)}\\ \\text{more than two real numbers.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2425", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The set of $x$-values satisfying the equation $x^{\\log_{10} x} = \\frac{x^3}{100}$ consists of: \n$\\textbf{(A)}\\ \\frac{1}{10}\\qquad\\textbf{(B)}\\ \\text{10, only}\\qquad\\textbf{(C)}\\ \\text{100, only}\\qquad\\textbf{(D)}\\ \\text{10 or 100, only}\\qquad\\textbf{(E)}\\ \\text{more than two real numbers.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$ is inscribed in a circle, and $\\angle B = \\angle C = 4\\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$\n\n$\\textbf{(A)}\\ 5 \\qquad\\textbf{(B)}\\ 7 \\qquad\\textbf{(C)}\\ 9 \\qquad\\textbf{(D)}\\ 15 \\qquad\\textbf{(E)}\\ 18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2426", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ is inscribed in a circle, and $\\angle B = \\angle C = 4\\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$\n\n$\\textbf{(A)}\\ 5 \\qquad\\textbf{(B)}\\ 7 \\qquad\\textbf{(C)}\\ 9 \\qquad\\textbf{(D)}\\ 15 \\qquad\\textbf{(E)}\\ 18$" + } + }, + { + "question": "Return your final response within \\boxed{}. Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $$$1$ more than a pink pill, and Al's pills cost a total of $\\textdollar 546$ for the two weeks. How much does one green pill cost?\n$\\textbf{(A)}\\ \\textdollar 7 \\qquad\\textbf{(B) }\\textdollar 14 \\qquad\\textbf{(C) }\\textdollar 19\\qquad\\textbf{(D) }\\textdollar 20\\qquad\\textbf{(E) }\\textdollar 39$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2427", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $$$1$ more than a pink pill, and Al's pills cost a total of $\\textdollar 546$ for the two weeks. How much does one green pill cost?\n$\\textbf{(A)}\\ \\textdollar 7 \\qquad\\textbf{(B) }\\textdollar 14 \\qquad\\textbf{(C) }\\textdollar 19\\qquad\\textbf{(D) }\\textdollar 20\\qquad\\textbf{(E) }\\textdollar 39$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define the operation \"$\\circ$\" by $x\\circ y=4x-3y+xy$, for all real numbers $x$ and $y$. For how many real numbers $y$ does $3\\circ y=12$?\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 3\\quad \\text{(D) } 4\\quad \\text{(E) more than 4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2428", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define the operation \"$\\circ$\" by $x\\circ y=4x-3y+xy$, for all real numbers $x$ and $y$. For how many real numbers $y$ does $3\\circ y=12$?\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 3\\quad \\text{(D) } 4\\quad \\text{(E) more than 4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Show that for any positive real $x$, $[nx]\\ge \\sum_{1}^{n}\\left(\\frac{[kx]}{k}\\right)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "[nx] \\ge \\sum_{k=1}^{n} \\frac{[kx]}{k}", + "index": "Sky-T1_10k_2429", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Show that for any positive real $x$, $[nx]\\ge \\sum_{1}^{n}\\left(\\frac{[kx]}{k}\\right)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $Z$ be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of $Z$?\n$\\textbf{(A) }11\\qquad\\textbf{(B) }19\\qquad\\textbf{(C) }101\\qquad\\textbf{(D) }111\\qquad\\textbf{(E) }1111$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2430", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $Z$ be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of $Z$?\n$\\textbf{(A) }11\\qquad\\textbf{(B) }19\\qquad\\textbf{(C) }101\\qquad\\textbf{(D) }111\\qquad\\textbf{(E) }1111$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?\n(A) all 4 are boys\n(B) all 4 are girls\n(C) 2 are girls and 2 are boys\n(D) 3 are of one gender and 1 is of the other gender\n(E) all of these outcomes are equally likely", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2431", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?\n(A) all 4 are boys\n(B) all 4 are girls\n(C) 2 are girls and 2 are boys\n(D) 3 are of one gender and 1 is of the other gender\n(E) all of these outcomes are equally likely" + } + }, + { + "question": "Return your final response within \\boxed{}. Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?\n$\\textbf{(A)}\\ \\frac{1}{10}\\qquad\\textbf{(B)}\\ \\frac{1}{6}\\qquad\\textbf{(C)}\\ \\frac{11}{60}\\qquad\\textbf{(D)}\\ \\frac{1}{5}\\qquad\\textbf{(E)}\\ \\frac{7}{30}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2432", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?\n$\\textbf{(A)}\\ \\frac{1}{10}\\qquad\\textbf{(B)}\\ \\frac{1}{6}\\qquad\\textbf{(C)}\\ \\frac{11}{60}\\qquad\\textbf{(D)}\\ \\frac{1}{5}\\qquad\\textbf{(E)}\\ \\frac{7}{30}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\\{1, 2, 3,\\dots, 2010\\}$. What is the probability that $abc + ab + a$ is divisible by $3$?\n$\\textbf{(A)}\\ \\dfrac{1}{3} \\qquad \\textbf{(B)}\\ \\dfrac{29}{81} \\qquad \\textbf{(C)}\\ \\dfrac{31}{81} \\qquad \\textbf{(D)}\\ \\dfrac{11}{27} \\qquad \\textbf{(E)}\\ \\dfrac{13}{27}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2433", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\\{1, 2, 3,\\dots, 2010\\}$. What is the probability that $abc + ab + a$ is divisible by $3$?\n$\\textbf{(A)}\\ \\dfrac{1}{3} \\qquad \\textbf{(B)}\\ \\dfrac{29}{81} \\qquad \\textbf{(C)}\\ \\dfrac{31}{81} \\qquad \\textbf{(D)}\\ \\dfrac{11}{27} \\qquad \\textbf{(E)}\\ \\dfrac{13}{27}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by: \n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 4 \\qquad\\textbf{(D)}\\ 6 \\qquad\\textbf{(E)}\\ \\text{more than }6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2434", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by: \n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)}\\ 3 \\qquad\\textbf{(C)}\\ 4 \\qquad\\textbf{(D)}\\ 6 \\qquad\\textbf{(E)}\\ \\text{more than }6$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the figure, $\\triangle ABC$ has $\\angle A =45^{\\circ}$ and $\\angle B =30^{\\circ}$. A line $DE$, with $D$ on $AB$ \nand $\\angle ADE =60^{\\circ}$, divides $\\triangle ABC$ into two pieces of equal area. \n(Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$\nThe ratio $\\frac{AD}{AB}$ is\n\n$\\textbf{(A)}\\ \\frac{1}{\\sqrt{2}} \\qquad \\textbf{(B)}\\ \\frac{2}{2+\\sqrt{2}} \\qquad \\textbf{(C)}\\ \\frac{1}{\\sqrt{3}} \\qquad \\textbf{(D)}\\ \\frac{1}{\\sqrt[3]{6}}\\qquad \\textbf{(E)}\\ \\frac{1}{\\sqrt[4]{12}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2435", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the figure, $\\triangle ABC$ has $\\angle A =45^{\\circ}$ and $\\angle B =30^{\\circ}$. A line $DE$, with $D$ on $AB$ \nand $\\angle ADE =60^{\\circ}$, divides $\\triangle ABC$ into two pieces of equal area. \n(Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$\nThe ratio $\\frac{AD}{AB}$ is\n\n$\\textbf{(A)}\\ \\frac{1}{\\sqrt{2}} \\qquad \\textbf{(B)}\\ \\frac{2}{2+\\sqrt{2}} \\qquad \\textbf{(C)}\\ \\frac{1}{\\sqrt{3}} \\qquad \\textbf{(D)}\\ \\frac{1}{\\sqrt[3]{6}}\\qquad \\textbf{(E)}\\ \\frac{1}{\\sqrt[4]{12}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Reduced to lowest terms, $\\frac{a^{2}-b^{2}}{ab} - \\frac{ab-b^{2}}{ab-a^{2}}$ is equal to:\n$\\textbf{(A)}\\ \\frac{a}{b}\\qquad\\textbf{(B)}\\ \\frac{a^{2}-2b^{2}}{ab}\\qquad\\textbf{(C)}\\ a^{2}\\qquad\\textbf{(D)}\\ a-2b\\qquad\\textbf{(E)}\\ \\text{None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2436", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Reduced to lowest terms, $\\frac{a^{2}-b^{2}}{ab} - \\frac{ab-b^{2}}{ab-a^{2}}$ is equal to:\n$\\textbf{(A)}\\ \\frac{a}{b}\\qquad\\textbf{(B)}\\ \\frac{a^{2}-2b^{2}}{ab}\\qquad\\textbf{(C)}\\ a^{2}\\qquad\\textbf{(D)}\\ a-2b\\qquad\\textbf{(E)}\\ \\text{None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A customer who intends to purchase an appliance has three coupons, only one of which may be used:\nCoupon 1: $10\\%$ off the listed price if the listed price is at least $\\textdollar50$\nCoupon 2: $\\textdollar 20$ off the listed price if the listed price is at least $\\textdollar100$\nCoupon 3: $18\\%$ off the amount by which the listed price exceeds $\\textdollar100$\nFor which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?\n$\\textbf{(A) }\\textdollar179.95\\qquad \\textbf{(B) }\\textdollar199.95\\qquad \\textbf{(C) }\\textdollar219.95\\qquad \\textbf{(D) }\\textdollar239.95\\qquad \\textbf{(E) }\\textdollar259.95\\qquad$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2437", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A customer who intends to purchase an appliance has three coupons, only one of which may be used:\nCoupon 1: $10\\%$ off the listed price if the listed price is at least $\\textdollar50$\nCoupon 2: $\\textdollar 20$ off the listed price if the listed price is at least $\\textdollar100$\nCoupon 3: $18\\%$ off the amount by which the listed price exceeds $\\textdollar100$\nFor which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?\n$\\textbf{(A) }\\textdollar179.95\\qquad \\textbf{(B) }\\textdollar199.95\\qquad \\textbf{(C) }\\textdollar219.95\\qquad \\textbf{(D) }\\textdollar239.95\\qquad \\textbf{(E) }\\textdollar259.95\\qquad$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x-y>x$ and $x+y0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2438", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x-y>x$ and $x+y0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others. \nA tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals \n$\\text{(A)} \\ 4\\sqrt 2 \\qquad \\text{(B)} \\ 4\\sqrt 3 \\qquad \\text{(C)} \\ 2\\sqrt 6 \\qquad \\text{(D)}\\ 1+2\\sqrt 6\\qquad \\text{(E)}\\ 2+2\\sqrt 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2439", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others. \nA tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals \n$\\text{(A)} \\ 4\\sqrt 2 \\qquad \\text{(B)} \\ 4\\sqrt 3 \\qquad \\text{(C)} \\ 2\\sqrt 6 \\qquad \\text{(D)}\\ 1+2\\sqrt 6\\qquad \\text{(E)}\\ 2+2\\sqrt 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?\n[asy] size(0,50); draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle); dot((-1,1)); dot((-2,2)); dot((-3,1)); dot((-2,0)); draw((1,0){up}..{left}(0,1)); dot((1,0)); dot((0,1)); draw((0,1){right}..{up}(1,2)); dot((1,2)); draw((1,2){down}..{right}(2,1)); dot((2,1)); draw((2,1){left}..{down}(1,0));[/asy]\n\n$\\textbf{(A)}\\hspace{.05in}\\frac{4-\\pi}{\\pi}\\qquad\\textbf{(B)}\\hspace{.05in}\\frac{1}\\pi\\qquad\\textbf{(C)}\\hspace{.05in}\\frac{\\sqrt2}{\\pi}\\qquad\\textbf{(D)}\\hspace{.05in}\\frac{\\pi-1}{\\pi}\\qquad\\textbf{(E)}\\hspace{.05in}\\frac{3}\\pi$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2440", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?\n[asy] size(0,50); draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle); dot((-1,1)); dot((-2,2)); dot((-3,1)); dot((-2,0)); draw((1,0){up}..{left}(0,1)); dot((1,0)); dot((0,1)); draw((0,1){right}..{up}(1,2)); dot((1,2)); draw((1,2){down}..{right}(2,1)); dot((2,1)); draw((2,1){left}..{down}(1,0));[/asy]\n\n$\\textbf{(A)}\\hspace{.05in}\\frac{4-\\pi}{\\pi}\\qquad\\textbf{(B)}\\hspace{.05in}\\frac{1}\\pi\\qquad\\textbf{(C)}\\hspace{.05in}\\frac{\\sqrt2}{\\pi}\\qquad\\textbf{(D)}\\hspace{.05in}\\frac{\\pi-1}{\\pi}\\qquad\\textbf{(E)}\\hspace{.05in}\\frac{3}\\pi$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $78$ is divided into three parts which are proportional to $1, \\frac13, \\frac16,$ the middle part is:\n$\\textbf{(A)}\\ 9\\frac13 \\qquad\\textbf{(B)}\\ 13\\qquad\\textbf{(C)}\\ 17\\frac13 \\qquad\\textbf{(D)}\\ 18\\frac13\\qquad\\textbf{(E)}\\ 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2441", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $78$ is divided into three parts which are proportional to $1, \\frac13, \\frac16,$ the middle part is:\n$\\textbf{(A)}\\ 9\\frac13 \\qquad\\textbf{(B)}\\ 13\\qquad\\textbf{(C)}\\ 17\\frac13 \\qquad\\textbf{(D)}\\ 18\\frac13\\qquad\\textbf{(E)}\\ 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many positive two-digit integers are factors of $2^{24}-1$?\n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 10 \\qquad \\textbf{(D)}\\ 12 \\qquad \\textbf{(E)}\\ 14$\n~ pi_is_3.14", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2442", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many positive two-digit integers are factors of $2^{24}-1$?\n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 10 \\qquad \\textbf{(D)}\\ 12 \\qquad \\textbf{(E)}\\ 14$\n~ pi_is_3.14" + } + }, + { + "question": "Return your final response within \\boxed{}. Shauna takes five tests, each worth a maximum of $100$ points. Her scores on the first three tests are $76$ , $94$ , and $87$ . In order to average $81$ for all five tests, what is the lowest score she could earn on one of the other two tests?\n$\\textbf{(A) }48\\qquad\\textbf{(B) }52\\qquad\\textbf{(C) }66\\qquad\\textbf{(D) }70\\qquad\\textbf{(E) }74$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2443", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Shauna takes five tests, each worth a maximum of $100$ points. Her scores on the first three tests are $76$ , $94$ , and $87$ . In order to average $81$ for all five tests, what is the lowest score she could earn on one of the other two tests?\n$\\textbf{(A) }48\\qquad\\textbf{(B) }52\\qquad\\textbf{(C) }66\\qquad\\textbf{(D) }70\\qquad\\textbf{(E) }74$" + } + }, + { + "question": "Return your final response within \\boxed{}. An ATM password at Fred's Bank is composed of four digits from $0$ to $9$, with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible?\n$\\textbf{(A)}\\mbox{ }30\\qquad\\textbf{(B)}\\mbox{ }7290\\qquad\\textbf{(C)}\\mbox{ }9000\\qquad\\textbf{(D)}\\mbox{ }9990\\qquad\\textbf{(E)}\\mbox{ }9999$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2444", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An ATM password at Fred's Bank is composed of four digits from $0$ to $9$, with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible?\n$\\textbf{(A)}\\mbox{ }30\\qquad\\textbf{(B)}\\mbox{ }7290\\qquad\\textbf{(C)}\\mbox{ }9000\\qquad\\textbf{(D)}\\mbox{ }9990\\qquad\\textbf{(E)}\\mbox{ }9999$" + } + }, + { + "question": "Return your final response within \\boxed{}. A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1\\leq a \\leq b \\leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?\n$\\textbf{(A)}\\; 4 \\qquad\\textbf{(B)}\\; 10 \\qquad\\textbf{(C)}\\; 12 \\qquad\\textbf{(D)}\\; 21 \\qquad\\textbf{(E)}\\; 26$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2445", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1\\leq a \\leq b \\leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?\n$\\textbf{(A)}\\; 4 \\qquad\\textbf{(B)}\\; 10 \\qquad\\textbf{(C)}\\; 12 \\qquad\\textbf{(D)}\\; 21 \\qquad\\textbf{(E)}\\; 26$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the product of the [real](https://artofproblemsolving.com/wiki/index.php/Real) [roots](https://artofproblemsolving.com/wiki/index.php/Root) of the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}$?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "20", + "index": "Sky-T1_10k_2446", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the product of the [real](https://artofproblemsolving.com/wiki/index.php/Real) [roots](https://artofproblemsolving.com/wiki/index.php/Root) of the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}$?" + } + }, + { + "question": "Return your final response within \\boxed{}. If $(3x-1)^7 = a_7x^7 + a_6x^6 + \\cdots + a_0$, then $a_7 + a_6 + \\cdots + a_0$ equals \n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 1 \\qquad \\text{(C)}\\ 64 \\qquad \\text{(D)}\\ -64 \\qquad \\text{(E)}\\ 128$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2447", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $(3x-1)^7 = a_7x^7 + a_6x^6 + \\cdots + a_0$, then $a_7 + a_6 + \\cdots + a_0$ equals \n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 1 \\qquad \\text{(C)}\\ 64 \\qquad \\text{(D)}\\ -64 \\qquad \\text{(E)}\\ 128$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\\textdollar 1.43$. Sharona bought some of the same pencils and paid $\\textdollar 1.87$. How many more pencils did Sharona buy than Jamar?\n$\\textbf{(A)}\\hspace{.05in}2\\qquad\\textbf{(B)}\\hspace{.05in}3\\qquad\\textbf{(C)}\\hspace{.05in}4\\qquad\\textbf{(D)}\\hspace{.05in}5\\qquad\\textbf{(E)}\\hspace{.05in}6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2448", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\\textdollar 1.43$. Sharona bought some of the same pencils and paid $\\textdollar 1.87$. How many more pencils did Sharona buy than Jamar?\n$\\textbf{(A)}\\hspace{.05in}2\\qquad\\textbf{(B)}\\hspace{.05in}3\\qquad\\textbf{(C)}\\hspace{.05in}4\\qquad\\textbf{(D)}\\hspace{.05in}5\\qquad\\textbf{(E)}\\hspace{.05in}6$" + } + }, + { + "question": "Return your final response within \\boxed{}. The diagram shows an octagon consisting of $10$ unit squares. The portion below $\\overline{PQ}$ is a unit square and a triangle with base $5$. If $\\overline{PQ}$ bisects the area of the octagon, what is the ratio $\\dfrac{XQ}{QY}$?\n\n$\\textbf{(A)}\\ \\frac{2}{5}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ \\frac{3}{5}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{3}{4}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2449", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The diagram shows an octagon consisting of $10$ unit squares. The portion below $\\overline{PQ}$ is a unit square and a triangle with base $5$. If $\\overline{PQ}$ bisects the area of the octagon, what is the ratio $\\dfrac{XQ}{QY}$?\n\n$\\textbf{(A)}\\ \\frac{2}{5}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ \\frac{3}{5}\\qquad\\textbf{(D)}\\ \\frac{2}{3}\\qquad\\textbf{(E)}\\ \\frac{3}{4}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\frac {35x - 29}{x^2 - 3x + 2} = \\frac {N_1}{x - 1} + \\frac {N_2}{x - 2}$ be an [identity](https://artofproblemsolving.com/wiki/index.php/Identity) in $x$. The numerical value of $N_1N_2$ is:\n$\\text{(A)} \\ - 246 \\qquad \\text{(B)} \\ - 210 \\qquad \\text{(C)} \\ - 29 \\qquad \\text{(D)} \\ 210 \\qquad \\text{(E)} \\ 246$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2450", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\frac {35x - 29}{x^2 - 3x + 2} = \\frac {N_1}{x - 1} + \\frac {N_2}{x - 2}$ be an [identity](https://artofproblemsolving.com/wiki/index.php/Identity) in $x$. The numerical value of $N_1N_2$ is:\n$\\text{(A)} \\ - 246 \\qquad \\text{(B)} \\ - 210 \\qquad \\text{(C)} \\ - 29 \\qquad \\text{(D)} \\ 210 \\qquad \\text{(E)} \\ 246$" + } + }, + { + "question": "Return your final response within \\boxed{}. If rose bushes are spaced about $1$ foot apart, approximately how many bushes are needed to surround a [circular](https://artofproblemsolving.com/wiki/index.php/Circle) patio whose radius is $12$ feet?\n$\\text{(A)}\\ 12 \\qquad \\text{(B)}\\ 38 \\qquad \\text{(C)}\\ 48 \\qquad \\text{(D)}\\ 75 \\qquad \\text{(E)}\\ 450$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2451", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If rose bushes are spaced about $1$ foot apart, approximately how many bushes are needed to surround a [circular](https://artofproblemsolving.com/wiki/index.php/Circle) patio whose radius is $12$ feet?\n$\\text{(A)}\\ 12 \\qquad \\text{(B)}\\ 38 \\qquad \\text{(C)}\\ 48 \\qquad \\text{(D)}\\ 75 \\qquad \\text{(E)}\\ 450$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even? \n$\\textbf{(A) }\\frac{1}{9}\\qquad\\textbf{(B) }\\frac{2}{9}\\qquad\\textbf{(C) }\\frac{4}{9}\\qquad\\textbf{(D) }\\frac{1}{2}\\qquad \\textbf{(E) }\\frac{5}{9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2452", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even? \n$\\textbf{(A) }\\frac{1}{9}\\qquad\\textbf{(B) }\\frac{2}{9}\\qquad\\textbf{(C) }\\frac{4}{9}\\qquad\\textbf{(D) }\\frac{1}{2}\\qquad \\textbf{(E) }\\frac{5}{9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jack had a bag of $128$ apples. He sold $25\\%$ of them to Jill. Next he sold $25\\%$ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?\n$\\text{(A)}\\ 7 \\qquad \\text{(B)}\\ 63 \\qquad \\text{(C)}\\ 65 \\qquad \\text{(D)}\\ 71 \\qquad \\text{(E)}\\ 111$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2453", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jack had a bag of $128$ apples. He sold $25\\%$ of them to Jill. Next he sold $25\\%$ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?\n$\\text{(A)}\\ 7 \\qquad \\text{(B)}\\ 63 \\qquad \\text{(C)}\\ 65 \\qquad \\text{(D)}\\ 71 \\qquad \\text{(E)}\\ 111$" + } + }, + { + "question": "Return your final response within \\boxed{}. Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?\n$\\textbf{(A)}\\ 18 \\qquad \\textbf{(B)}\\ 21 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 30$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2454", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?\n$\\textbf{(A)}\\ 18 \\qquad \\textbf{(B)}\\ 21 \\qquad \\textbf{(C)}\\ 24 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 30$" + } + }, + { + "question": "Return your final response within \\boxed{}. An equivalent of the expression\n$\\left(\\frac{x^2+1}{x}\\right)\\left(\\frac{y^2+1}{y}\\right)+\\left(\\frac{x^2-1}{y}\\right)\\left(\\frac{y^2-1}{x}\\right)$, $xy \\not= 0$,\nis:\n$\\text{(A)}\\ 1\\qquad\\text{(B)}\\ 2xy\\qquad\\text{(C)}\\ 2x^2y^2+2\\qquad\\text{(D)}\\ 2xy+\\frac{2}{xy}\\qquad\\text{(E)}\\ \\frac{2x}{y}+\\frac{2y}{x}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2455", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An equivalent of the expression\n$\\left(\\frac{x^2+1}{x}\\right)\\left(\\frac{y^2+1}{y}\\right)+\\left(\\frac{x^2-1}{y}\\right)\\left(\\frac{y^2-1}{x}\\right)$, $xy \\not= 0$,\nis:\n$\\text{(A)}\\ 1\\qquad\\text{(B)}\\ 2xy\\qquad\\text{(C)}\\ 2x^2y^2+2\\qquad\\text{(D)}\\ 2xy+\\frac{2}{xy}\\qquad\\text{(E)}\\ \\frac{2x}{y}+\\frac{2y}{x}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The product $(1.8)(40.3+.07)$ is closest to \n$\\text{(A)}\\ 7 \\qquad \\text{(B)}\\ 42 \\qquad \\text{(C)}\\ 74 \\qquad \\text{(D)}\\ 84 \\qquad \\text{(E)}\\ 737$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2456", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The product $(1.8)(40.3+.07)$ is closest to \n$\\text{(A)}\\ 7 \\qquad \\text{(B)}\\ 42 \\qquad \\text{(C)}\\ 74 \\qquad \\text{(D)}\\ 84 \\qquad \\text{(E)}\\ 737$" + } + }, + { + "question": "Return your final response within \\boxed{}. The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\\angle ACB$ intersect $AB$ at $D$ and $E$. What is the area of $\\bigtriangleup CDE$?\n$\\textbf{(A) }\\dfrac{5\\sqrt{2}}{3}\\qquad\\textbf{(B) }\\dfrac{50\\sqrt{3}-75}{4}\\qquad\\textbf{(C) }\\dfrac{15\\sqrt{3}}{8}\\qquad\\textbf{(D) }\\dfrac{50-25\\sqrt{3}}{2}\\qquad\\textbf{(E) }\\dfrac{25}{6}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2457", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\\angle ACB$ intersect $AB$ at $D$ and $E$. What is the area of $\\bigtriangleup CDE$?\n$\\textbf{(A) }\\dfrac{5\\sqrt{2}}{3}\\qquad\\textbf{(B) }\\dfrac{50\\sqrt{3}-75}{4}\\qquad\\textbf{(C) }\\dfrac{15\\sqrt{3}}{8}\\qquad\\textbf{(D) }\\dfrac{50-25\\sqrt{3}}{2}\\qquad\\textbf{(E) }\\dfrac{25}{6}$" + } + }, + { + "question": "Return your final response within \\boxed{}. From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are \n$\\textbf{(A) } 3^{1/3},\\ 2^{1/2}\\quad \\textbf{(B) } 3^{1/3},\\ 8^{1/8}\\quad \\textbf{(C) } 3^{1/3},\\ 9^{1/9}\\quad \\textbf{(D) } 8^{1/8},\\ 9^{1/9}\\quad \\\\ \\text{(E) None of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2458", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are \n$\\textbf{(A) } 3^{1/3},\\ 2^{1/2}\\quad \\textbf{(B) } 3^{1/3},\\ 8^{1/8}\\quad \\textbf{(C) } 3^{1/3},\\ 9^{1/9}\\quad \\textbf{(D) } 8^{1/8},\\ 9^{1/9}\\quad \\\\ \\text{(E) None of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "At some moment, three mathematicians were sleeping simultaneously.", + "index": "Sky-T1_10k_2459", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously." + } + }, + { + "question": "Return your final response within \\boxed{}. In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?\n[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy]\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 5 \\qquad \\textbf{(C) } 6 \\qquad \\textbf{(D) } 7 \\qquad \\textbf{(E) } 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2460", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?\n[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy]\n$\\textbf{(A) } 4 \\qquad \\textbf{(B) } 5 \\qquad \\textbf{(C) } 6 \\qquad \\textbf{(D) } 7 \\qquad \\textbf{(E) } 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\sqrt{3+2\\sqrt{2}}-\\sqrt{3-2\\sqrt{2}}$ is equal to\n$\\text{(A) } 2\\quad \\text{(B) } 2\\sqrt{3}\\quad \\text{(C) } 4\\sqrt{2}\\quad \\text{(D) } \\sqrt{6}\\quad \\text{(E) } 2\\sqrt{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2461", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\sqrt{3+2\\sqrt{2}}-\\sqrt{3-2\\sqrt{2}}$ is equal to\n$\\text{(A) } 2\\quad \\text{(B) } 2\\sqrt{3}\\quad \\text{(C) } 4\\sqrt{2}\\quad \\text{(D) } \\sqrt{6}\\quad \\text{(E) } 2\\sqrt{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?\n[asy] defaultpen(linewidth(0.6)); pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle); draw(O--D, dashed); draw(A--C, dashed);[/asy]\n$\\textbf{(A)}\\ 21 \\qquad \\textbf{(B)}\\ 23 \\qquad \\textbf{(C)}\\ 25 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 29$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2462", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?\n[asy] defaultpen(linewidth(0.6)); pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle); draw(O--D, dashed); draw(A--C, dashed);[/asy]\n$\\textbf{(A)}\\ 21 \\qquad \\textbf{(B)}\\ 23 \\qquad \\textbf{(C)}\\ 25 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 29$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\\triangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$?\n$\\textbf{(A)}\\ (-8, 9)\\qquad\\textbf{(B)}\\ (-4, 8)\\qquad\\textbf{(C)}\\ (-4, 9)\\qquad\\textbf{(D)}\\ (-2, 3)\\qquad\\textbf{(E)}\\ (-1, 0)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2463", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\\triangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$?\n$\\textbf{(A)}\\ (-8, 9)\\qquad\\textbf{(B)}\\ (-4, 8)\\qquad\\textbf{(C)}\\ (-4, 9)\\qquad\\textbf{(D)}\\ (-2, 3)\\qquad\\textbf{(E)}\\ (-1, 0)$" + } + }, + { + "question": "Return your final response within \\boxed{}. Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?\n$\\textbf{(A)}\\ 256\\qquad\\textbf{(B)}\\ 336\\qquad\\textbf{(C)}\\ 384\\qquad\\textbf{(D)}\\ 448\\qquad\\textbf{(E)}\\ 512$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2464", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?\n$\\textbf{(A)}\\ 256\\qquad\\textbf{(B)}\\ 336\\qquad\\textbf{(C)}\\ 384\\qquad\\textbf{(D)}\\ 448\\qquad\\textbf{(E)}\\ 512$" + } + }, + { + "question": "Return your final response within \\boxed{}. For what value of $x$ does $10^{x}\\cdot 100^{2x}=1000^{5}$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2465", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For what value of $x$ does $10^{x}\\cdot 100^{2x}=1000^{5}$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose $f(x)$ is defined for all real numbers $x; f(x) > 0$ for all $x;$ and $f(a)f(b) = f(a + b)$ for all $a$ and $b$. Which of the following statements are true?\n$I.\\ f(0) = 1 \\qquad \\qquad \\ \\ \\qquad \\qquad \\qquad II.\\ f(-a) = \\frac{1}{f(a)}\\ \\text{for all}\\ a \\\\ III.\\ f(a) = \\sqrt[3]{f(3a)}\\ \\text{for all}\\ a \\qquad IV.\\ f(b) > f(a)\\ \\text{if}\\ b > a$\n$\\textbf{(A)}\\ \\text{III and IV only} \\qquad \\textbf{(B)}\\ \\text{I, III, and IV only} \\\\ \\textbf{(C)}\\ \\text{I, II, and IV only} \\qquad \\textbf{(D)}\\ \\text{I, II, and III only} \\qquad \\textbf{(E)}\\ \\text{All are true.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2466", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose $f(x)$ is defined for all real numbers $x; f(x) > 0$ for all $x;$ and $f(a)f(b) = f(a + b)$ for all $a$ and $b$. Which of the following statements are true?\n$I.\\ f(0) = 1 \\qquad \\qquad \\ \\ \\qquad \\qquad \\qquad II.\\ f(-a) = \\frac{1}{f(a)}\\ \\text{for all}\\ a \\\\ III.\\ f(a) = \\sqrt[3]{f(3a)}\\ \\text{for all}\\ a \\qquad IV.\\ f(b) > f(a)\\ \\text{if}\\ b > a$\n$\\textbf{(A)}\\ \\text{III and IV only} \\qquad \\textbf{(B)}\\ \\text{I, III, and IV only} \\\\ \\textbf{(C)}\\ \\text{I, II, and IV only} \\qquad \\textbf{(D)}\\ \\text{I, II, and III only} \\qquad \\textbf{(E)}\\ \\text{All are true.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 60$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2467", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 36\\qquad\\textbf{(C)}\\ 42\\qquad\\textbf{(D)}\\ 48\\qquad\\textbf{(E)}\\ 60$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n+1):(4n+27)$ for all $n$, the ratio of the eleventh term of the first series to the eleventh term of the second series is:\n$\\text{(A) } 4:3\\quad \\text{(B) } 3:2\\quad \\text{(C) } 7:4\\quad \\text{(D) } 78:71\\quad \\text{(E) undetermined}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2468", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n+1):(4n+27)$ for all $n$, the ratio of the eleventh term of the first series to the eleventh term of the second series is:\n$\\text{(A) } 4:3\\quad \\text{(B) } 3:2\\quad \\text{(C) } 7:4\\quad \\text{(D) } 78:71\\quad \\text{(E) undetermined}$" + } + }, + { + "question": "Return your final response within \\boxed{}. To satisfy the equation $\\frac{a+b}{a}=\\frac{b}{a+b}$, $a$ and $b$ must be:\n$\\textbf{(A)}\\ \\text{both rational}\\qquad\\textbf{(B)}\\ \\text{both real but not rational}\\qquad\\textbf{(C)}\\ \\text{both not real}\\qquad$\n$\\textbf{(D)}\\ \\text{one real, one not real}\\qquad\\textbf{(E)}\\ \\text{one real, one not real or both not real}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2469", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. To satisfy the equation $\\frac{a+b}{a}=\\frac{b}{a+b}$, $a$ and $b$ must be:\n$\\textbf{(A)}\\ \\text{both rational}\\qquad\\textbf{(B)}\\ \\text{both real but not rational}\\qquad\\textbf{(C)}\\ \\text{both not real}\\qquad$\n$\\textbf{(D)}\\ \\text{one real, one not real}\\qquad\\textbf{(E)}\\ \\text{one real, one not real or both not real}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. \nFor all real values of $t$ the expression $\\sqrt{t^4+t^2}$ is equal to?\n$\\textbf{(A)}\\ t^3\\qquad \\textbf{(B)}\\ t^2+t\\qquad \\textbf{(C)}\\ |t^2+t|\\qquad \\textbf{(D)}\\ t\\sqrt{t^2+1}\\qquad \\textbf{(E)}\\ |t|\\sqrt{1+t^2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2470", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. \nFor all real values of $t$ the expression $\\sqrt{t^4+t^2}$ is equal to?\n$\\textbf{(A)}\\ t^3\\qquad \\textbf{(B)}\\ t^2+t\\qquad \\textbf{(C)}\\ |t^2+t|\\qquad \\textbf{(D)}\\ t\\sqrt{t^2+1}\\qquad \\textbf{(E)}\\ |t|\\sqrt{1+t^2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many ordered triples $(x,y,z)$ of positive integers satisfy $\\text{lcm}(x,y) = 72, \\text{lcm}(x,z) = 600 \\text{ and lcm}(y,z)=900$?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 27\\qquad\\textbf{(E)}\\ 64$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2471", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many ordered triples $(x,y,z)$ of positive integers satisfy $\\text{lcm}(x,y) = 72, \\text{lcm}(x,z) = 600 \\text{ and lcm}(y,z)=900$?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 27\\qquad\\textbf{(E)}\\ 64$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\\angle CED$?\n$\\textbf{(A) }90\\qquad\\textbf{(B) }105\\qquad\\textbf{(C) }120\\qquad\\textbf{(D) }135\\qquad \\textbf{(E) }150$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2472", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\\angle CED$?\n$\\textbf{(A) }90\\qquad\\textbf{(B) }105\\qquad\\textbf{(C) }120\\qquad\\textbf{(D) }135\\qquad \\textbf{(E) }150$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $t = \\frac{1}{1 - \\sqrt[4]{2}}$, then $t$ equals \n$\\text{(A)}\\ (1-\\sqrt[4]{2})(2-\\sqrt{2})\\qquad \\text{(B)}\\ (1-\\sqrt[4]{2})(1+\\sqrt{2})\\qquad \\text{(C)}\\ (1+\\sqrt[4]{2})(1-\\sqrt{2}) \\qquad \\\\ \\text{(D)}\\ (1+\\sqrt[4]{2})(1+\\sqrt{2})\\qquad \\text{(E)}-(1+\\sqrt[4]{2})(1+\\sqrt{2})$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2473", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $t = \\frac{1}{1 - \\sqrt[4]{2}}$, then $t$ equals \n$\\text{(A)}\\ (1-\\sqrt[4]{2})(2-\\sqrt{2})\\qquad \\text{(B)}\\ (1-\\sqrt[4]{2})(1+\\sqrt{2})\\qquad \\text{(C)}\\ (1+\\sqrt[4]{2})(1-\\sqrt{2}) \\qquad \\\\ \\text{(D)}\\ (1+\\sqrt[4]{2})(1+\\sqrt{2})\\qquad \\text{(E)}-(1+\\sqrt[4]{2})(1+\\sqrt{2})$" + } + }, + { + "question": "Return your final response within \\boxed{}. Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\\triangle ABC$.\n\n[AIME 1989 Problem 15.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1989_Problem_15.png)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "108", + "index": "Sky-T1_10k_2474", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\\triangle ABC$.\n\n[AIME 1989 Problem 15.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1989_Problem_15.png)" + } + }, + { + "question": "Return your final response within \\boxed{}. If the [arithmetic mean](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean) of two [numbers](https://artofproblemsolving.com/wiki/index.php/Number) is $6$ and their [geometric mean](https://artofproblemsolving.com/wiki/index.php/Geometric_mean) is $10$, then an [equation](https://artofproblemsolving.com/wiki/index.php/Equation) with the given two numbers as [roots](https://artofproblemsolving.com/wiki/index.php/Root) is:\n$\\text{(A)} \\ x^2 + 12x + 100 = 0 ~~ \\text{(B)} \\ x^2 + 6x + 100 = 0 ~~ \\text{(C)} \\ x^2 - 12x - 10 = 0$\n$\\text{(D)} \\ x^2 - 12x + 100 = 0 \\qquad \\text{(E)} \\ x^2 - 6x + 100 = 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2475", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the [arithmetic mean](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean) of two [numbers](https://artofproblemsolving.com/wiki/index.php/Number) is $6$ and their [geometric mean](https://artofproblemsolving.com/wiki/index.php/Geometric_mean) is $10$, then an [equation](https://artofproblemsolving.com/wiki/index.php/Equation) with the given two numbers as [roots](https://artofproblemsolving.com/wiki/index.php/Root) is:\n$\\text{(A)} \\ x^2 + 12x + 100 = 0 ~~ \\text{(B)} \\ x^2 + 6x + 100 = 0 ~~ \\text{(C)} \\ x^2 - 12x - 10 = 0$\n$\\text{(D)} \\ x^2 - 12x + 100 = 0 \\qquad \\text{(E)} \\ x^2 - 6x + 100 = 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?\n\\[\\begin{tabular}{cccccc}&A&B&B&C&B\\\\ +&B&C&A&D&A\\\\ \\hline &D&B&D&D&D\\end{tabular}\\]\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 7\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 9$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2476", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?\n\\[\\begin{tabular}{cccccc}&A&B&B&C&B\\\\ +&B&C&A&D&A\\\\ \\hline &D&B&D&D&D\\end{tabular}\\]\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 7\\qquad\\textbf{(D)}\\ 8\\qquad\\textbf{(E)}\\ 9$" + } + }, + { + "question": "Return your final response within \\boxed{}. The cost $C$ of sending a parcel post package weighing $P$ pounds, $P$ an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. The formula for the cost is:\n$\\textbf{(A) \\ }C=10+3P \\qquad \\textbf{(B) \\ }C=10P+3 \\qquad \\textbf{(C) \\ }C=10+3(P-1) \\qquad$\n$\\textbf{(D) \\ }C=9+3P \\qquad \\textbf{(E) \\ }C=10P-7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2477", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The cost $C$ of sending a parcel post package weighing $P$ pounds, $P$ an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. The formula for the cost is:\n$\\textbf{(A) \\ }C=10+3P \\qquad \\textbf{(B) \\ }C=10P+3 \\qquad \\textbf{(C) \\ }C=10+3(P-1) \\qquad$\n$\\textbf{(D) \\ }C=9+3P \\qquad \\textbf{(E) \\ }C=10P-7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Six different digits from the set\n\\[\\{ 1,2,3,4,5,6,7,8,9\\}\\]\nare placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12.\nThe sum of the six digits used is\n\n$\\text{(A)}\\ 27 \\qquad \\text{(B)}\\ 29 \\qquad \\text{(C)}\\ 31 \\qquad \\text{(D)}\\ 33 \\qquad \\text{(E)}\\ 35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2478", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Six different digits from the set\n\\[\\{ 1,2,3,4,5,6,7,8,9\\}\\]\nare placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12.\nThe sum of the six digits used is\n\n$\\text{(A)}\\ 27 \\qquad \\text{(B)}\\ 29 \\qquad \\text{(C)}\\ 31 \\qquad \\text{(D)}\\ 33 \\qquad \\text{(E)}\\ 35$" + } + }, + { + "question": "Return your final response within \\boxed{}. A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid. \n\n\n$\\mathrm{(A) \\ } \\frac 35 \\qquad \\mathrm{(B) \\ } \\frac 23 \\qquad \\mathrm{(C) \\ } \\frac 34 \\qquad \\mathrm{(D) \\ } \\frac 56 \\qquad \\mathrm{(E) \\ } \\frac 78$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2479", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid. \n\n\n$\\mathrm{(A) \\ } \\frac 35 \\qquad \\mathrm{(B) \\ } \\frac 23 \\qquad \\mathrm{(C) \\ } \\frac 34 \\qquad \\mathrm{(D) \\ } \\frac 56 \\qquad \\mathrm{(E) \\ } \\frac 78$" + } + }, + { + "question": "Return your final response within \\boxed{}. A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is:\n$\\textbf{(A)}\\ 800 \\qquad \\textbf{(B)}\\ 640\\qquad \\textbf{(C)}\\ 400 \\qquad \\textbf{(D)}\\ 320 \\qquad \\textbf{(E)}\\ 160$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2480", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is:\n$\\textbf{(A)}\\ 800 \\qquad \\textbf{(B)}\\ 640\\qquad \\textbf{(C)}\\ 400 \\qquad \\textbf{(D)}\\ 320 \\qquad \\textbf{(E)}\\ 160$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider the sequence \n$1,-2,3,-4,5,-6,\\ldots,$\nwhose $n$th term is $(-1)^{n+1}\\cdot n$. What is the average of the first $200$ terms of the sequence?\n$\\textbf{(A)}-\\!1\\qquad\\textbf{(B)}-\\!0.5\\qquad\\textbf{(C)}\\ 0\\qquad\\textbf{(D)}\\ 0.5\\qquad\\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2481", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider the sequence \n$1,-2,3,-4,5,-6,\\ldots,$\nwhose $n$th term is $(-1)^{n+1}\\cdot n$. What is the average of the first $200$ terms of the sequence?\n$\\textbf{(A)}-\\!1\\qquad\\textbf{(B)}-\\!0.5\\qquad\\textbf{(C)}\\ 0\\qquad\\textbf{(D)}\\ 0.5\\qquad\\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?\n\n[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p); [/asy]\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ \\sqrt {10} \\qquad \\textbf{(C)}\\ 2 + \\sqrt2 \\qquad \\textbf{(D)}\\ 2\\sqrt3 \\qquad \\textbf{(E)}\\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2482", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?\n\n[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p); [/asy]\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ \\sqrt {10} \\qquad \\textbf{(C)}\\ 2 + \\sqrt2 \\qquad \\textbf{(D)}\\ 2\\sqrt3 \\qquad \\textbf{(E)}\\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. The percent that $M$ is greater than $N$ is:\n$(\\mathrm{A})\\ \\frac{100(M-N)}{M} \\qquad (\\mathrm{B})\\ \\frac{100(M-N)}{N} \\qquad (\\mathrm{C})\\ \\frac{M-N}{N} \\qquad (\\mathrm{D})\\ \\frac{M-N}{N} \\qquad (\\mathrm{E})\\ \\frac{100(M+N)}{N}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2483", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The percent that $M$ is greater than $N$ is:\n$(\\mathrm{A})\\ \\frac{100(M-N)}{M} \\qquad (\\mathrm{B})\\ \\frac{100(M-N)}{N} \\qquad (\\mathrm{C})\\ \\frac{M-N}{N} \\qquad (\\mathrm{D})\\ \\frac{M-N}{N} \\qquad (\\mathrm{E})\\ \\frac{100(M+N)}{N}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $23$,$14$, $11$, and $20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $18$, what is the least number of points she could have scored in the tenth game?\n$\\textbf{(A)}\\ 26\\qquad\\textbf{(B)}\\ 27\\qquad\\textbf{(C)}\\ 28\\qquad\\textbf{(D)}\\ 29\\qquad\\textbf{(E)}\\ 30$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2484", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $23$,$14$, $11$, and $20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $18$, what is the least number of points she could have scored in the tenth game?\n$\\textbf{(A)}\\ 26\\qquad\\textbf{(B)}\\ 27\\qquad\\textbf{(C)}\\ 28\\qquad\\textbf{(D)}\\ 29\\qquad\\textbf{(E)}\\ 30$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\begin{tabular}{r|l}a&b \\\\ \\hline c&d\\end{tabular} = \\text{a}\\cdot \\text{d} - \\text{b}\\cdot \\text{c}$, what is the value of $\\begin{tabular}{r|l}3&4 \\\\ \\hline 1&2\\end{tabular}$?\n$\\text{(A)}\\ -2 \\qquad \\text{(B)}\\ -1 \\qquad \\text{(C)}\\ 0 \\qquad \\text{(D)}\\ 1 \\qquad \\text{(E)}\\ 2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2485", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\begin{tabular}{r|l}a&b \\\\ \\hline c&d\\end{tabular} = \\text{a}\\cdot \\text{d} - \\text{b}\\cdot \\text{c}$, what is the value of $\\begin{tabular}{r|l}3&4 \\\\ \\hline 1&2\\end{tabular}$?\n$\\text{(A)}\\ -2 \\qquad \\text{(B)}\\ -1 \\qquad \\text{(C)}\\ 0 \\qquad \\text{(D)}\\ 1 \\qquad \\text{(E)}\\ 2$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during $1988$. For example, about $.5$ million had been spent by the beginning of February and approximately $2$ million by the end of April. Approximately how many millions of dollars were spent during the summer months of June, July, and August?\n\n\n$\\text{(A)}\\ 1.5 \\qquad \\text{(B)}\\ 2.5 \\qquad \\text{(C)}\\ 3.5 \\qquad \\text{(D)}\\ 4.5 \\qquad \\text{(E)}\\ 5.5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2486", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during $1988$. For example, about $.5$ million had been spent by the beginning of February and approximately $2$ million by the end of April. Approximately how many millions of dollars were spent during the summer months of June, July, and August?\n\n\n$\\text{(A)}\\ 1.5 \\qquad \\text{(B)}\\ 2.5 \\qquad \\text{(C)}\\ 3.5 \\qquad \\text{(D)}\\ 4.5 \\qquad \\text{(E)}\\ 5.5$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ is real and positive and grows beyond all bounds, then $\\log_3{(6x-5)}-\\log_3{(2x+1)}$ approaches:\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 3\\qquad \\textbf{(D)}\\ 4\\qquad \\textbf{(E)}\\ \\text{no finite number}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2487", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ is real and positive and grows beyond all bounds, then $\\log_3{(6x-5)}-\\log_3{(2x+1)}$ approaches:\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 3\\qquad \\textbf{(D)}\\ 4\\qquad \\textbf{(E)}\\ \\text{no finite number}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2488", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 5\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the product $\\dfrac{3}{2}\\cdot \\dfrac{4}{3}\\cdot \\dfrac{5}{4}\\cdot \\dfrac{6}{5}\\cdot \\ldots\\cdot \\dfrac{a}{b} = 9$, what is the sum of $a$ and $b$?\n$\\text{(A)}\\ 11 \\qquad \\text{(B)}\\ 13 \\qquad \\text{(C)}\\ 17 \\qquad \\text{(D)}\\ 35 \\qquad \\text{(E)}\\ 37$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2489", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the product $\\dfrac{3}{2}\\cdot \\dfrac{4}{3}\\cdot \\dfrac{5}{4}\\cdot \\dfrac{6}{5}\\cdot \\ldots\\cdot \\dfrac{a}{b} = 9$, what is the sum of $a$ and $b$?\n$\\text{(A)}\\ 11 \\qquad \\text{(B)}\\ 13 \\qquad \\text{(C)}\\ 17 \\qquad \\text{(D)}\\ 35 \\qquad \\text{(E)}\\ 37$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following numbers is not a perfect square?\n$\\textbf{(A) }1^{2016}\\qquad\\textbf{(B) }2^{2017}\\qquad\\textbf{(C) }3^{2018}\\qquad\\textbf{(D) }4^{2019}\\qquad \\textbf{(E) }5^{2020}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2490", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following numbers is not a perfect square?\n$\\textbf{(A) }1^{2016}\\qquad\\textbf{(B) }2^{2017}\\qquad\\textbf{(C) }3^{2018}\\qquad\\textbf{(D) }4^{2019}\\qquad \\textbf{(E) }5^{2020}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2491", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x=\\frac{a}{b}$, $a\\neq b$ and $b\\neq 0$, then $\\frac{a+b}{a-b}=$\n(A) $\\frac{x}{x+1}$ (B) $\\frac{x+1}{x-1}$ (C) $1$ (D) $x-\\frac{1}{x}$ (E) $x+\\frac{1}{x}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2492", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x=\\frac{a}{b}$, $a\\neq b$ and $b\\neq 0$, then $\\frac{a+b}{a-b}=$\n(A) $\\frac{x}{x+1}$ (B) $\\frac{x+1}{x-1}$ (C) $1$ (D) $x-\\frac{1}{x}$ (E) $x+\\frac{1}{x}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Simplify $\\left(\\sqrt[6]{27} - \\sqrt{6 \\frac{3}{4} }\\right)^2$\n$\\textbf{(A)}\\ \\frac{3}{4} \\qquad \\textbf{(B)}\\ \\frac{\\sqrt 3}{2} \\qquad \\textbf{(C)}\\ \\frac{3\\sqrt 3}{4}\\qquad \\textbf{(D)}\\ \\frac{3}{2}\\qquad \\textbf{(E)}\\ \\frac{3\\sqrt 3}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2493", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Simplify $\\left(\\sqrt[6]{27} - \\sqrt{6 \\frac{3}{4} }\\right)^2$\n$\\textbf{(A)}\\ \\frac{3}{4} \\qquad \\textbf{(B)}\\ \\frac{\\sqrt 3}{2} \\qquad \\textbf{(C)}\\ \\frac{3\\sqrt 3}{4}\\qquad \\textbf{(D)}\\ \\frac{3}{2}\\qquad \\textbf{(E)}\\ \\frac{3\\sqrt 3}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A number which when divided by $10$ leaves a remainder of $9$, when divided by $9$ leaves a remainder of $8$, by $8$ leaves a remainder of $7$, etc., down to where, when divided by $2$, it leaves a remainder of $1$, is: \n$\\textbf{(A)}\\ 59\\qquad\\textbf{(B)}\\ 419\\qquad\\textbf{(C)}\\ 1259\\qquad\\textbf{(D)}\\ 2519\\qquad\\textbf{(E)}\\ \\text{none of these answers}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2494", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A number which when divided by $10$ leaves a remainder of $9$, when divided by $9$ leaves a remainder of $8$, by $8$ leaves a remainder of $7$, etc., down to where, when divided by $2$, it leaves a remainder of $1$, is: \n$\\textbf{(A)}\\ 59\\qquad\\textbf{(B)}\\ 419\\qquad\\textbf{(C)}\\ 1259\\qquad\\textbf{(D)}\\ 2519\\qquad\\textbf{(E)}\\ \\text{none of these answers}$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the smallest positive odd integer $n$ such that the product $2^{1/7}2^{3/7}\\cdots2^{(2n+1)/7}$ is greater than $1000$? \n(In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.)\n$\\textbf{(A) }7\\qquad \\textbf{(B) }9\\qquad \\textbf{(C) }11\\qquad \\textbf{(D) }17\\qquad \\textbf{(E) }19$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2495", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the smallest positive odd integer $n$ such that the product $2^{1/7}2^{3/7}\\cdots2^{(2n+1)/7}$ is greater than $1000$? \n(In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.)\n$\\textbf{(A) }7\\qquad \\textbf{(B) }9\\qquad \\textbf{(C) }11\\qquad \\textbf{(D) }17\\qquad \\textbf{(E) }19$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ten points are selected on the positive $x$-axis,$X^+$, and five points are selected on the positive $y$-axis,$Y^+$. The fifty segments connecting the ten points on $X^+$ to the five points on $Y^+$ are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?\n$\\text{(A) } 250\\quad \\text{(B) } 450\\quad \\text{(C) } 500\\quad \\text{(D) } 1250\\quad \\text{(E) } 2500$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2496", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ten points are selected on the positive $x$-axis,$X^+$, and five points are selected on the positive $y$-axis,$Y^+$. The fifty segments connecting the ten points on $X^+$ to the five points on $Y^+$ are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?\n$\\text{(A) } 250\\quad \\text{(B) } 450\\quad \\text{(C) } 500\\quad \\text{(D) } 1250\\quad \\text{(E) } 2500$" + } + }, + { + "question": "Return your final response within \\boxed{}. Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?\n$\\textbf{(A)}\\ \\frac{\\pi}{3} \\qquad \\textbf{(B)}\\ \\frac{2\\pi}{3} \\qquad \\textbf{(C)}\\ \\pi \\qquad \\textbf{(D)}\\ \\frac{4\\pi}{3} \\qquad \\textbf{(E)}\\ \\frac{5\\pi}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2497", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?\n$\\textbf{(A)}\\ \\frac{\\pi}{3} \\qquad \\textbf{(B)}\\ \\frac{2\\pi}{3} \\qquad \\textbf{(C)}\\ \\pi \\qquad \\textbf{(D)}\\ \\frac{4\\pi}{3} \\qquad \\textbf{(E)}\\ \\frac{5\\pi}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Given the areas of the three squares in the figure, what is the area of the interior triangle?\n\n$\\mathrm{(A)}\\ 13 \\qquad\\mathrm{(B)}\\ 30 \\qquad\\mathrm{(C)}\\ 60 \\qquad\\mathrm{(D)}\\ 300 \\qquad\\mathrm{(E)}\\ 1800$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2498", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Given the areas of the three squares in the figure, what is the area of the interior triangle?\n\n$\\mathrm{(A)}\\ 13 \\qquad\\mathrm{(B)}\\ 30 \\qquad\\mathrm{(C)}\\ 60 \\qquad\\mathrm{(D)}\\ 300 \\qquad\\mathrm{(E)}\\ 1800$" + } + }, + { + "question": "Return your final response within \\boxed{}. Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals\n$\\text{(A) } \\sqrt{4q+1}\\quad \\text{(B) } q-1\\quad \\text{(C) } -\\sqrt{4q+1}\\quad \\text{(D) } q+1\\quad \\text{(E) } \\sqrt{4q-1}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2499", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals\n$\\text{(A) } \\sqrt{4q+1}\\quad \\text{(B) } q-1\\quad \\text{(C) } -\\sqrt{4q+1}\\quad \\text{(D) } q+1\\quad \\text{(E) } \\sqrt{4q-1}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?\n$\\textbf{(A)}\\ 62 \\qquad \\textbf{(B)}\\ 82 \\qquad \\textbf{(C)}\\ 83 \\qquad \\textbf{(D)}\\ 102 \\qquad \\textbf{(E)}\\ 103$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2500", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?\n$\\textbf{(A)}\\ 62 \\qquad \\textbf{(B)}\\ 82 \\qquad \\textbf{(C)}\\ 83 \\qquad \\textbf{(D)}\\ 102 \\qquad \\textbf{(E)}\\ 103$" + } + }, + { + "question": "Return your final response within \\boxed{}. A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$?\n$\\textbf{(A)}\\ 120\\qquad\\textbf{(B)}\\ 160\\qquad\\textbf{(C)}\\ 200\\qquad\\textbf{(D)}\\ 240\\qquad\\textbf{(E)}\\ 280$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2501", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$?\n$\\textbf{(A)}\\ 120\\qquad\\textbf{(B)}\\ 160\\qquad\\textbf{(C)}\\ 200\\qquad\\textbf{(D)}\\ 240\\qquad\\textbf{(E)}\\ 280$" + } + }, + { + "question": "Return your final response within \\boxed{}. A [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer) is called ascending if, in its [decimal representation](https://artofproblemsolving.com/wiki/index.php?title=Decimal_representation&action=edit&redlink=1), there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "502", + "index": "Sky-T1_10k_2502", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer) is called ascending if, in its [decimal representation](https://artofproblemsolving.com/wiki/index.php?title=Decimal_representation&action=edit&redlink=1), there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?" + } + }, + { + "question": "Return your final response within \\boxed{}. When simplified $\\sqrt{1+ \\left (\\frac{x^4-1}{2x^2} \\right )^2}$ equals: \n$\\textbf{(A)}\\ \\frac{x^4+2x^2-1}{2x^2} \\qquad \\textbf{(B)}\\ \\frac{x^4-1}{2x^2} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{x^2+1}}{2}\\\\ \\textbf{(D)}\\ \\frac{x^2}{\\sqrt{2}}\\qquad\\textbf{(E)}\\ \\frac{x^2}{2}+\\frac{1}{2x^2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2503", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When simplified $\\sqrt{1+ \\left (\\frac{x^4-1}{2x^2} \\right )^2}$ equals: \n$\\textbf{(A)}\\ \\frac{x^4+2x^2-1}{2x^2} \\qquad \\textbf{(B)}\\ \\frac{x^4-1}{2x^2} \\qquad \\textbf{(C)}\\ \\frac{\\sqrt{x^2+1}}{2}\\\\ \\textbf{(D)}\\ \\frac{x^2}{\\sqrt{2}}\\qquad\\textbf{(E)}\\ \\frac{x^2}{2}+\\frac{1}{2x^2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A man walked a certain distance at a constant rate. If he had gone $\\textstyle\\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\\textstyle\\frac{1}{2}$ mile per hour slower, he would have been $2\\textstyle\\frac{1}{2}$ hours longer on the road. The distance in miles he walked was\n$\\textbf{(A) }13\\textstyle\\frac{1}{2}\\qquad \\textbf{(B) }15\\qquad \\textbf{(C) }17\\frac{1}{2}\\qquad \\textbf{(D) }20\\qquad \\textbf{(E) }25$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2504", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A man walked a certain distance at a constant rate. If he had gone $\\textstyle\\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\\textstyle\\frac{1}{2}$ mile per hour slower, he would have been $2\\textstyle\\frac{1}{2}$ hours longer on the road. The distance in miles he walked was\n$\\textbf{(A) }13\\textstyle\\frac{1}{2}\\qquad \\textbf{(B) }15\\qquad \\textbf{(C) }17\\frac{1}{2}\\qquad \\textbf{(D) }20\\qquad \\textbf{(E) }25$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\\frac yx$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$?\n$\\mathrm{(A)}\\ 3 \\qquad\\mathrm{(B)}\\ \\sqrt{10} \\qquad\\mathrm{(C)}\\ \\frac 72 \\qquad\\mathrm{(D)}\\ \\frac 92 \\qquad\\mathrm{(E)}\\ 2\\sqrt{14}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2505", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\\frac yx$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$?\n$\\mathrm{(A)}\\ 3 \\qquad\\mathrm{(B)}\\ \\sqrt{10} \\qquad\\mathrm{(C)}\\ \\frac 72 \\qquad\\mathrm{(D)}\\ \\frac 92 \\qquad\\mathrm{(E)}\\ 2\\sqrt{14}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x$ is a real number and $|x-4|+|x-3|0$, then:\n$\\textbf{(A)}\\ 01$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2506", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x$ is a real number and $|x-4|+|x-3|0$, then:\n$\\textbf{(A)}\\ 01$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum\n\\[f \\left(\\frac{1}{2019} \\right)-f \\left(\\frac{2}{2019} \\right)+f \\left(\\frac{3}{2019} \\right)-f \\left(\\frac{4}{2019} \\right)+\\cdots + f \\left(\\frac{2017}{2019} \\right) - f \\left(\\frac{2018}{2019} \\right)?\\]\n$\\textbf{(A) }0\\qquad\\textbf{(B) }\\frac{1}{2019^{4}}\\qquad\\textbf{(C) }\\frac{2018^{2}}{2019^{4}}\\qquad\\textbf{(D) }\\frac{2020^{2}}{2019^{4}}\\qquad\\textbf{(E) }1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2507", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum\n\\[f \\left(\\frac{1}{2019} \\right)-f \\left(\\frac{2}{2019} \\right)+f \\left(\\frac{3}{2019} \\right)-f \\left(\\frac{4}{2019} \\right)+\\cdots + f \\left(\\frac{2017}{2019} \\right) - f \\left(\\frac{2018}{2019} \\right)?\\]\n$\\textbf{(A) }0\\qquad\\textbf{(B) }\\frac{1}{2019^{4}}\\qquad\\textbf{(C) }\\frac{2018^{2}}{2019^{4}}\\qquad\\textbf{(D) }\\frac{2020^{2}}{2019^{4}}\\qquad\\textbf{(E) }1$" + } + }, + { + "question": "Return your final response within \\boxed{}. A line $x=k$ intersects the graph of $y=\\log_5 x$ and the graph of $y=\\log_5 (x + 4)$. The distance between the points of intersection is $0.5$. Given that $k = a + \\sqrt{b}$, where $a$ and $b$ are integers, what is $a+b$?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2508", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A line $x=k$ intersects the graph of $y=\\log_5 x$ and the graph of $y=\\log_5 (x + 4)$. The distance between the points of intersection is $0.5$. Given that $k = a + \\sqrt{b}$, where $a$ and $b$ are integers, what is $a+b$?\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 7\\qquad\\textbf{(C)}\\ 8\\qquad\\textbf{(D)}\\ 9\\qquad\\textbf{(E)}\\ 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Four whole numbers, when added three at a time, give the sums $180,197,208$ and $222$. What is the largest of the four numbers?\n$\\text{(A) } 77\\quad \\text{(B) } 83\\quad \\text{(C) } 89\\quad \\text{(D) } 95\\quad \\text{(E) cannot be determined from the given information}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2509", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Four whole numbers, when added three at a time, give the sums $180,197,208$ and $222$. What is the largest of the four numbers?\n$\\text{(A) } 77\\quad \\text{(B) } 83\\quad \\text{(C) } 89\\quad \\text{(D) } 95\\quad \\text{(E) cannot be determined from the given information}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$?\n$\\textbf{(A) } 2 \\qquad\\textbf{(B) } 3 \\qquad\\textbf{(C) } 4 \\qquad\\textbf{(D) } 6 \\qquad\\textbf{(E) } 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2510", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$?\n$\\textbf{(A) } 2 \\qquad\\textbf{(B) } 3 \\qquad\\textbf{(C) } 4 \\qquad\\textbf{(D) } 6 \\qquad\\textbf{(E) } 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "Such infinitely many pairs exist, e.g., (2k+1, 2k+3) for k \\geq 1.", + "index": "Sky-T1_10k_2511", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$." + } + }, + { + "question": "Return your final response within \\boxed{}. Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either $12$ pieces of red candy, $14$ pieces of green candy, $15$ pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs $20$ cents. What is the smallest possible value of $n$?\n$\\textbf{(A) } 18 \\qquad \\textbf{(B) } 21 \\qquad \\textbf{(C) } 24\\qquad \\textbf{(D) } 25 \\qquad \\textbf{(E) } 28$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2512", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either $12$ pieces of red candy, $14$ pieces of green candy, $15$ pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs $20$ cents. What is the smallest possible value of $n$?\n$\\textbf{(A) } 18 \\qquad \\textbf{(B) } 21 \\qquad \\textbf{(C) } 24\\qquad \\textbf{(D) } 25 \\qquad \\textbf{(E) } 28$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$, sides $a,b$ and $c$ are opposite $\\angle{A},\\angle{B}$ and $\\angle{C}$ respectively. $AD$ bisects $\\angle{A}$ and meets $BC$ at $D$. \nThen if $x = \\overline{CD}$ and $y = \\overline{BD}$ the correct proportion is: \n$\\textbf{(A)}\\ \\frac {x}{a} = \\frac {a}{b + c} \\qquad \\textbf{(B)}\\ \\frac {x}{b} = \\frac {a}{a + c} \\qquad \\textbf{(C)}\\ \\frac{y}{c}=\\frac{c}{b+c}\\\\ \\textbf{(D)}\\ \\frac{y}{c}=\\frac{a}{b+c}\\qquad \\textbf{(E)}\\ \\frac{x}{y}=\\frac{c}{b}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2513", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$, sides $a,b$ and $c$ are opposite $\\angle{A},\\angle{B}$ and $\\angle{C}$ respectively. $AD$ bisects $\\angle{A}$ and meets $BC$ at $D$. \nThen if $x = \\overline{CD}$ and $y = \\overline{BD}$ the correct proportion is: \n$\\textbf{(A)}\\ \\frac {x}{a} = \\frac {a}{b + c} \\qquad \\textbf{(B)}\\ \\frac {x}{b} = \\frac {a}{a + c} \\qquad \\textbf{(C)}\\ \\frac{y}{c}=\\frac{c}{b+c}\\\\ \\textbf{(D)}\\ \\frac{y}{c}=\\frac{a}{b+c}\\qquad \\textbf{(E)}\\ \\frac{x}{y}=\\frac{c}{b}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For each integer $n\\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "245", + "index": "Sky-T1_10k_2514", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For each integer $n\\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$." + } + }, + { + "question": "Return your final response within \\boxed{}. If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, \nand for all $a$ and $b$ $f(a+b)+f(a-b)=2f(a)+2f(b)$, then for all $x$ and $y$\n$\\textbf{(A) }f(0)=1\\qquad \\textbf{(B) }f(-x)=-f(x)\\qquad \\textbf{(C) }f(-x)=f(x)\\qquad \\\\ \\textbf{(D) }f(x+y)=f(x)+f(y) \\qquad \\\\ \\textbf{(E) }\\text{there is a positive real number }T\\text{ such that }f(x+T)=f(x)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2515", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, \nand for all $a$ and $b$ $f(a+b)+f(a-b)=2f(a)+2f(b)$, then for all $x$ and $y$\n$\\textbf{(A) }f(0)=1\\qquad \\textbf{(B) }f(-x)=-f(x)\\qquad \\textbf{(C) }f(-x)=f(x)\\qquad \\\\ \\textbf{(D) }f(x+y)=f(x)+f(y) \\qquad \\\\ \\textbf{(E) }\\text{there is a positive real number }T\\text{ such that }f(x+T)=f(x)$" + } + }, + { + "question": "Return your final response within \\boxed{}. $(\\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\\le p \\le 11$?\n$(\\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\\le p \\le 13$?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "No", + "index": "Sky-T1_10k_2516", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $(\\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\\le p \\le 11$?\n$(\\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\\le p \\le 13$?" + } + }, + { + "question": "Return your final response within \\boxed{}. The coordinates of $A,B$ and $C$ are $(5,5),(2,1)$ and $(0,k)$ respectively. \nThe value of $k$ that makes $\\overline{AC}+\\overline{BC}$ as small as possible is: \n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4\\frac{1}{2} \\qquad \\textbf{(C)}\\ 3\\frac{6}{7} \\qquad \\textbf{(D)}\\ 4\\frac{5}{6}\\qquad \\textbf{(E)}\\ 2\\frac{1}{7}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2517", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The coordinates of $A,B$ and $C$ are $(5,5),(2,1)$ and $(0,k)$ respectively. \nThe value of $k$ that makes $\\overline{AC}+\\overline{BC}$ as small as possible is: \n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4\\frac{1}{2} \\qquad \\textbf{(C)}\\ 3\\frac{6}{7} \\qquad \\textbf{(D)}\\ 4\\frac{5}{6}\\qquad \\textbf{(E)}\\ 2\\frac{1}{7}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "195", + "index": "Sky-T1_10k_2518", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$." + } + }, + { + "question": "Return your final response within \\boxed{}. If $\\log_{10}2=a$ and $\\log_{10}3=b$, then $\\log_{5}12=$?\n$\\textbf{(A)}\\ \\frac{a+b}{a+1}\\qquad \\textbf{(B)}\\ \\frac{2a+b}{a+1}\\qquad \\textbf{(C)}\\ \\frac{a+2b}{1+a}\\qquad \\textbf{(D)}\\ \\frac{2a+b}{1-a}\\qquad \\textbf{(E)}\\ \\frac{a+2b}{1-a}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2519", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $\\log_{10}2=a$ and $\\log_{10}3=b$, then $\\log_{5}12=$?\n$\\textbf{(A)}\\ \\frac{a+b}{a+1}\\qquad \\textbf{(B)}\\ \\frac{2a+b}{a+1}\\qquad \\textbf{(C)}\\ \\frac{a+2b}{1+a}\\qquad \\textbf{(D)}\\ \\frac{2a+b}{1-a}\\qquad \\textbf{(E)}\\ \\frac{a+2b}{1-a}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, \n$31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is\n$\\textbf{(A) }5\\qquad \\textbf{(B) }7\\qquad \\textbf{(C) }9\\qquad \\textbf{(D) }11\\qquad \\textbf{(E) }13$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2520", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, \n$31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is\n$\\textbf{(A) }5\\qquad \\textbf{(B) }7\\qquad \\textbf{(C) }9\\qquad \\textbf{(D) }11\\qquad \\textbf{(E) }13$" + } + }, + { + "question": "Return your final response within \\boxed{}. In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?\n\n$\\textbf{(A)}\\ 11 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 13 \\qquad \\textbf{(D)}\\ 14 \\qquad \\textbf{(E)}\\ 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2521", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?\n\n$\\textbf{(A)}\\ 11 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 13 \\qquad \\textbf{(D)}\\ 14 \\qquad \\textbf{(E)}\\ 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\\%$, and the cost of the helmet increased by $10\\%$. The percent increase in the combined cost of the bicycle and the helmet is:\n$\\text{(A) } 6\\%\\quad \\text{(B) } 7\\%\\quad \\text{(C) } 7.5\\%\\quad \\text{(D) } 8\\%\\quad \\text{(E) } 15\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2522", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\\%$, and the cost of the helmet increased by $10\\%$. The percent increase in the combined cost of the bicycle and the helmet is:\n$\\text{(A) } 6\\%\\quad \\text{(B) } 7\\%\\quad \\text{(C) } 7.5\\%\\quad \\text{(D) } 8\\%\\quad \\text{(E) } 15\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. The number of real values of $x$ that satisfy the equation \\[(2^{6x+3})(4^{3x+6})=8^{4x+5}\\] is:\n$\\text{(A) zero} \\qquad \\text{(B) one} \\qquad \\text{(C) two} \\qquad \\text{(D) three} \\qquad \\text{(E) greater than 3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2523", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The number of real values of $x$ that satisfy the equation \\[(2^{6x+3})(4^{3x+6})=8^{4x+5}\\] is:\n$\\text{(A) zero} \\qquad \\text{(B) one} \\qquad \\text{(C) two} \\qquad \\text{(D) three} \\qquad \\text{(E) greater than 3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The state income tax where Kristin lives is levied at the rate of $p\\%$ of the first\n$\\textdollar 28000$ of annual income plus $(p + 2)\\%$ of any amount above $\\textdollar 28000$. Kristin\nnoticed that the state income tax she paid amounted to $(p + 0.25)\\%$ of her\nannual income. What was her annual income? \n$\\textbf{(A)}\\,\\textdollar 28000 \\qquad \\textbf{(B)}\\,\\textdollar 32000 \\qquad \\textbf{(C)}\\,\\textdollar 35000 \\qquad \\textbf{(D)}\\,\\textdollar 42000 \\qquad \\textbf{(E)}\\,\\textdollar 56000$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2524", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The state income tax where Kristin lives is levied at the rate of $p\\%$ of the first\n$\\textdollar 28000$ of annual income plus $(p + 2)\\%$ of any amount above $\\textdollar 28000$. Kristin\nnoticed that the state income tax she paid amounted to $(p + 0.25)\\%$ of her\nannual income. What was her annual income? \n$\\textbf{(A)}\\,\\textdollar 28000 \\qquad \\textbf{(B)}\\,\\textdollar 32000 \\qquad \\textbf{(C)}\\,\\textdollar 35000 \\qquad \\textbf{(D)}\\,\\textdollar 42000 \\qquad \\textbf{(E)}\\,\\textdollar 56000$" + } + }, + { + "question": "Return your final response within \\boxed{}. There are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2525", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the point $(x,-4)$ lies on the straight line joining the points $(0,8)$ and $(-4,0)$ in the $xy$-plane, then $x$ is equal to\n$\\textbf{(A) }-2\\qquad \\textbf{(B) }2\\qquad \\textbf{(C) }-8\\qquad \\textbf{(D) }6\\qquad \\textbf{(E) }-6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2526", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the point $(x,-4)$ lies on the straight line joining the points $(0,8)$ and $(-4,0)$ in the $xy$-plane, then $x$ is equal to\n$\\textbf{(A) }-2\\qquad \\textbf{(B) }2\\qquad \\textbf{(C) }-8\\qquad \\textbf{(D) }6\\qquad \\textbf{(E) }-6$" + } + }, + { + "question": "Return your final response within \\boxed{}. In this figure $AB$ is a diameter of a circle, centered at $O$, with radius $a$. A chord $AD$ is drawn and extended to meet the tangent to the circle at $B$ in point $C$. Point $E$ is taken on $AC$ so the $AE=DC$. Denoting the distances of $E$ from the tangent through $A$ and from the diameter $AB$ by $x$ and $y$, respectively, we can deduce the relation:\n$\\text{(A) } y^2=\\frac{x^3}{2a-x} \\quad \\text{(B) } y^2=\\frac{x^3}{2a+x} \\quad \\text{(C) } y^4=\\frac{x^2}{2a-x} \\\\ \\text{(D) } x^2=\\frac{y^2}{2a-x} \\quad \\text{(E) } x^2=\\frac{y^2}{2a+x}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2527", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In this figure $AB$ is a diameter of a circle, centered at $O$, with radius $a$. A chord $AD$ is drawn and extended to meet the tangent to the circle at $B$ in point $C$. Point $E$ is taken on $AC$ so the $AE=DC$. Denoting the distances of $E$ from the tangent through $A$ and from the diameter $AB$ by $x$ and $y$, respectively, we can deduce the relation:\n$\\text{(A) } y^2=\\frac{x^3}{2a-x} \\quad \\text{(B) } y^2=\\frac{x^3}{2a+x} \\quad \\text{(C) } y^4=\\frac{x^2}{2a-x} \\\\ \\text{(D) } x^2=\\frac{y^2}{2a-x} \\quad \\text{(E) } x^2=\\frac{y^2}{2a+x}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If the line $L$ in the $xy$-plane has half the slope and twice the $y$-intercept of the line $y = \\frac{2}{3} x + 4$, then an equation for $L$ is:\n$\\textbf{(A)}\\ y = \\frac{1}{3} x + 8 \\qquad \\textbf{(B)}\\ y = \\frac{4}{3} x + 2 \\qquad \\textbf{(C)}\\ y =\\frac{1}{3}x+4\\qquad\\\\ \\textbf{(D)}\\ y =\\frac{4}{3}x+4\\qquad \\textbf{(E)}\\ y =\\frac{1}{3}x+2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2528", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If the line $L$ in the $xy$-plane has half the slope and twice the $y$-intercept of the line $y = \\frac{2}{3} x + 4$, then an equation for $L$ is:\n$\\textbf{(A)}\\ y = \\frac{1}{3} x + 8 \\qquad \\textbf{(B)}\\ y = \\frac{4}{3} x + 2 \\qquad \\textbf{(C)}\\ y =\\frac{1}{3}x+4\\qquad\\\\ \\textbf{(D)}\\ y =\\frac{4}{3}x+4\\qquad \\textbf{(E)}\\ y =\\frac{1}{3}x+2$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is $10\\cdot\\left(\\tfrac{1}{2}+\\tfrac{1}{5}+\\tfrac{1}{10}\\right)^{-1}?$\n$\\textbf{(A)}\\ 3 \\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ \\frac{25}{2} \\qquad\\textbf{(D)}\\ \\frac{170}{3}\\qquad\\textbf{(E)}\\ 170$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2529", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is $10\\cdot\\left(\\tfrac{1}{2}+\\tfrac{1}{5}+\\tfrac{1}{10}\\right)^{-1}?$\n$\\textbf{(A)}\\ 3 \\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ \\frac{25}{2} \\qquad\\textbf{(D)}\\ \\frac{170}{3}\\qquad\\textbf{(E)}\\ 170$" + } + }, + { + "question": "Return your final response within \\boxed{}. Define $[a,b,c]$ to mean $\\frac {a+b}c$, where $c \\neq 0$. What is the value of \n\n$\\left[[60,30,90],[2,1,3],[10,5,15]\\right]?$\n$\\mathrm{(A) \\ }0 \\qquad \\mathrm{(B) \\ }0.5 \\qquad \\mathrm{(C) \\ }1 \\qquad \\mathrm{(D) \\ }1.5 \\qquad \\mathrm{(E) \\ }2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2530", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Define $[a,b,c]$ to mean $\\frac {a+b}c$, where $c \\neq 0$. What is the value of \n\n$\\left[[60,30,90],[2,1,3],[10,5,15]\\right]?$\n$\\mathrm{(A) \\ }0 \\qquad \\mathrm{(B) \\ }0.5 \\qquad \\mathrm{(C) \\ }1 \\qquad \\mathrm{(D) \\ }1.5 \\qquad \\mathrm{(E) \\ }2$" + } + }, + { + "question": "Return your final response within \\boxed{}. A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?\n$\\textbf{(A) } 75 \\qquad\\textbf{(B) } 76 \\qquad\\textbf{(C) } 79 \\qquad\\textbf{(D) } 84 \\qquad\\textbf{(E) } 91$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2531", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?\n$\\textbf{(A) } 75 \\qquad\\textbf{(B) } 76 \\qquad\\textbf{(C) } 79 \\qquad\\textbf{(D) } 84 \\qquad\\textbf{(E) } 91$" + } + }, + { + "question": "Return your final response within \\boxed{}. For distinct real numbers $x$ and $y$, let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y$. If $a - 2 \\qquad \\textbf{(C) }\\ x < 3 \\\\ \\textbf{(D) }\\ x > 3 \\text{ and }x < - 2 \\qquad \\textbf{(E) }\\ x > 3 \\text{ and }x < - 2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2539", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The statement $x^2 - x - 6 < 0$ is equivalent to the statement: \n$\\textbf{(A)}\\ - 2 < x < 3 \\qquad \\textbf{(B) }\\ x > - 2 \\qquad \\textbf{(C) }\\ x < 3 \\\\ \\textbf{(D) }\\ x > 3 \\text{ and }x < - 2 \\qquad \\textbf{(E) }\\ x > 3 \\text{ and }x < - 2$" + } + }, + { + "question": "Return your final response within \\boxed{}. The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?\n$\\textbf{(A) }20\\qquad\\textbf{(B) }30\\qquad\\textbf{(C) }60\\qquad\\textbf{(D) }120\\qquad \\textbf{(E) }180$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2540", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?\n$\\textbf{(A) }20\\qquad\\textbf{(B) }30\\qquad\\textbf{(C) }60\\qquad\\textbf{(D) }120\\qquad \\textbf{(E) }180$" + } + }, + { + "question": "Return your final response within \\boxed{}. The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely:\n$\\textbf{(A)}\\ 0, 12, -12\\qquad \\textbf{(B)}\\ 0, 12\\qquad \\textbf{(C)}\\ 12, -12\\qquad \\textbf{(D)}\\ 12\\qquad \\textbf{(E)}\\ 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2541", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely:\n$\\textbf{(A)}\\ 0, 12, -12\\qquad \\textbf{(B)}\\ 0, 12\\qquad \\textbf{(C)}\\ 12, -12\\qquad \\textbf{(D)}\\ 12\\qquad \\textbf{(E)}\\ 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other. \nThe man at $R$ walks uniformly at the rate of $4\\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant \nrate of $3\\tfrac{1}{4}$ miles per hour for the first hour, at $3\\tfrac{3}{4}$ miles per hour for the second hour,\nand so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is:\n$\\textbf{(A)}\\ 10 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2542", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other. \nThe man at $R$ walks uniformly at the rate of $4\\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant \nrate of $3\\tfrac{1}{4}$ miles per hour for the first hour, at $3\\tfrac{3}{4}$ miles per hour for the second hour,\nand so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is:\n$\\textbf{(A)}\\ 10 \\qquad \\textbf{(B)}\\ 8 \\qquad \\textbf{(C)}\\ 6 \\qquad \\textbf{(D)}\\ 4 \\qquad \\textbf{(E)}\\ 2$" + } + }, + { + "question": "Return your final response within \\boxed{}. A right circular cone has for its base a circle having the same radius as a given sphere. \nThe volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:\n$\\textbf{(A)}\\ \\frac{1}{1}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ \\frac{2}{3}\\qquad\\textbf{(D)}\\ \\frac{2}{1}\\qquad\\textbf{(E)}\\ \\sqrt{\\frac{5}{4}}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2543", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A right circular cone has for its base a circle having the same radius as a given sphere. \nThe volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:\n$\\textbf{(A)}\\ \\frac{1}{1}\\qquad\\textbf{(B)}\\ \\frac{1}{2}\\qquad\\textbf{(C)}\\ \\frac{2}{3}\\qquad\\textbf{(D)}\\ \\frac{2}{1}\\qquad\\textbf{(E)}\\ \\sqrt{\\frac{5}{4}}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four centers to label locker number $10$. If it costs $137.94 to label all the lockers, how many lockers are there at the school?\n$\\textbf{(A)}\\ 2001 \\qquad \\textbf{(B)}\\ 2010 \\qquad \\textbf{(C)}\\ 2100 \\qquad \\textbf{(D)}\\ 2726 \\qquad \\textbf{(E)}\\ 6897$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2544", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four centers to label locker number $10$. If it costs $137.94 to label all the lockers, how many lockers are there at the school?\n$\\textbf{(A)}\\ 2001 \\qquad \\textbf{(B)}\\ 2010 \\qquad \\textbf{(C)}\\ 2100 \\qquad \\textbf{(D)}\\ 2726 \\qquad \\textbf{(E)}\\ 6897$" + } + }, + { + "question": "Return your final response within \\boxed{}. For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?\n$\\textbf{(A) } 14 \\qquad \\textbf{(B) } 16 \\qquad \\textbf{(C) } 18 \\qquad \\textbf{(D) } 19 \\qquad \\textbf{(E) } 21$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2545", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?\n$\\textbf{(A) } 14 \\qquad \\textbf{(B) } 16 \\qquad \\textbf{(C) } 18 \\qquad \\textbf{(D) } 19 \\qquad \\textbf{(E) } 21$" + } + }, + { + "question": "Return your final response within \\boxed{}. A small [ square](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) is constructed inside a square of [area](https://artofproblemsolving.com/wiki/index.php/Area) 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the [ vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) to the division points closest to the opposite vertices. Find the value of $n$ if the the [area](https://artofproblemsolving.com/wiki/index.php/Area) of the small square is exactly $\\frac1{1985}$. \n[AIME 1985 Problem 4.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1985_Problem_4.png)", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "32", + "index": "Sky-T1_10k_2546", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A small [ square](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) is constructed inside a square of [area](https://artofproblemsolving.com/wiki/index.php/Area) 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the [ vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) to the division points closest to the opposite vertices. Find the value of $n$ if the the [area](https://artofproblemsolving.com/wiki/index.php/Area) of the small square is exactly $\\frac1{1985}$. \n[AIME 1985 Problem 4.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1985_Problem_4.png)" + } + }, + { + "question": "Return your final response within \\boxed{}. Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?\n$\\mathrm{(A)}\\ \\frac{5\\sqrt{2}-7}{3}\\qquad \\mathrm{(B)}\\ \\frac{10-7\\sqrt{2}}{3}\\qquad \\mathrm{(C)}\\ \\frac{3-2\\sqrt{2}}{3}\\qquad \\mathrm{(D)}\\ \\frac{8\\sqrt{2}-11}{3}\\qquad \\mathrm{(E)}\\ \\frac{6-4\\sqrt{2}}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2547", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?\n$\\mathrm{(A)}\\ \\frac{5\\sqrt{2}-7}{3}\\qquad \\mathrm{(B)}\\ \\frac{10-7\\sqrt{2}}{3}\\qquad \\mathrm{(C)}\\ \\frac{3-2\\sqrt{2}}{3}\\qquad \\mathrm{(D)}\\ \\frac{8\\sqrt{2}-11}{3}\\qquad \\mathrm{(E)}\\ \\frac{6-4\\sqrt{2}}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The negation of the statement \"No slow learners attend this school\" is: \n$\\textbf{(A)}\\ \\text{All slow learners attend this school}\\\\ \\textbf{(B)}\\ \\text{All slow learners do not attend this school}\\\\ \\textbf{(C)}\\ \\text{Some slow learners attend this school}\\\\ \\textbf{(D)}\\ \\text{Some slow learners do not attend this school}\\\\ \\textbf{(E)}\\ \\text{No slow learners do not attend this school}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2548", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The negation of the statement \"No slow learners attend this school\" is: \n$\\textbf{(A)}\\ \\text{All slow learners attend this school}\\\\ \\textbf{(B)}\\ \\text{All slow learners do not attend this school}\\\\ \\textbf{(C)}\\ \\text{Some slow learners attend this school}\\\\ \\textbf{(D)}\\ \\text{Some slow learners do not attend this school}\\\\ \\textbf{(E)}\\ \\text{No slow learners do not attend this school}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following numbers is a perfect square?\n$\\textbf{(A)}\\ \\dfrac{14!15!}2\\qquad\\textbf{(B)}\\ \\dfrac{15!16!}2\\qquad\\textbf{(C)}\\ \\dfrac{16!17!}2\\qquad\\textbf{(D)}\\ \\dfrac{17!18!}2\\qquad\\textbf{(E)}\\ \\dfrac{18!19!}2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2549", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following numbers is a perfect square?\n$\\textbf{(A)}\\ \\dfrac{14!15!}2\\qquad\\textbf{(B)}\\ \\dfrac{15!16!}2\\qquad\\textbf{(C)}\\ \\dfrac{16!17!}2\\qquad\\textbf{(D)}\\ \\dfrac{17!18!}2\\qquad\\textbf{(E)}\\ \\dfrac{18!19!}2$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $2^{1998}-2^{1997}-2^{1996}+2^{1995} = k \\cdot 2^{1995},$ what is the value of $k$?\n$\\mathrm{(A) \\ } 1 \\qquad \\mathrm{(B) \\ } 2 \\qquad \\mathrm{(C) \\ } 3 \\qquad \\mathrm{(D) \\ } 4 \\qquad \\mathrm{(E) \\ } 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2550", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $2^{1998}-2^{1997}-2^{1996}+2^{1995} = k \\cdot 2^{1995},$ what is the value of $k$?\n$\\mathrm{(A) \\ } 1 \\qquad \\mathrm{(B) \\ } 2 \\qquad \\mathrm{(C) \\ } 3 \\qquad \\mathrm{(D) \\ } 4 \\qquad \\mathrm{(E) \\ } 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. The glass gauge on a [cylindrical](https://artofproblemsolving.com/wiki/index.php/Cylinder) coffee maker shows that there are $45$ cups left when the coffee maker is $36\\%$ full. How many cups of coffee does it hold when it is full?\n$\\text{(A)}\\ 80 \\qquad \\text{(B)}\\ 100 \\qquad \\text{(C)}\\ 125 \\qquad \\text{(D)}\\ 130 \\qquad \\text{(E)}\\ 262$\n[asy] draw((5,0)..(0,-1.3)..(-5,0)); draw((5,0)--(5,10)); draw((-5,0)--(-5,10)); draw(ellipse((0,10),5,1.3)); draw(circle((.3,1.3),.4)); draw((-.1,1.7)--(-.1,7.9)--(.7,7.9)--(.7,1.7)--cycle); fill((-.1,1.7)--(-.1,4)--(.7,4)--(.7,1.7)--cycle,black); draw((-2,11.3)--(2,11.3)..(2.6,11.9)..(2,12.2)--(-2,12.2)..(-2.6,11.9)..cycle); [/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2551", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The glass gauge on a [cylindrical](https://artofproblemsolving.com/wiki/index.php/Cylinder) coffee maker shows that there are $45$ cups left when the coffee maker is $36\\%$ full. How many cups of coffee does it hold when it is full?\n$\\text{(A)}\\ 80 \\qquad \\text{(B)}\\ 100 \\qquad \\text{(C)}\\ 125 \\qquad \\text{(D)}\\ 130 \\qquad \\text{(E)}\\ 262$\n[asy] draw((5,0)..(0,-1.3)..(-5,0)); draw((5,0)--(5,10)); draw((-5,0)--(-5,10)); draw(ellipse((0,10),5,1.3)); draw(circle((.3,1.3),.4)); draw((-.1,1.7)--(-.1,7.9)--(.7,7.9)--(.7,1.7)--cycle); fill((-.1,1.7)--(-.1,4)--(.7,4)--(.7,1.7)--cycle,black); draw((-2,11.3)--(2,11.3)..(2.6,11.9)..(2,12.2)--(-2,12.2)..(-2.6,11.9)..cycle); [/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. If $p$ is a positive integer, then $\\frac {3p + 25}{2p - 5}$ can be a positive integer, if and only if $p$ is: \n$\\textbf{(A)}\\ \\text{at least }3\\qquad \\textbf{(B)}\\ \\text{at least }3\\text{ and no more than }35\\qquad \\\\ \\textbf{(C)}\\ \\text{no more than }35 \\qquad \\textbf{(D)}\\ \\text{equal to }35 \\qquad \\textbf{(E)}\\ \\text{equal to }3\\text{ or }35$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2552", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $p$ is a positive integer, then $\\frac {3p + 25}{2p - 5}$ can be a positive integer, if and only if $p$ is: \n$\\textbf{(A)}\\ \\text{at least }3\\qquad \\textbf{(B)}\\ \\text{at least }3\\text{ and no more than }35\\qquad \\\\ \\textbf{(C)}\\ \\text{no more than }35 \\qquad \\textbf{(D)}\\ \\text{equal to }35 \\qquad \\textbf{(E)}\\ \\text{equal to }3\\text{ or }35$" + } + }, + { + "question": "Return your final response within \\boxed{}. Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2) + 12$ is prime that properly divides the positive number $a^2 + b^2 + c^2 + abc - 2017$?", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "No such triples exist", + "index": "Sky-T1_10k_2553", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2) + 12$ is prime that properly divides the positive number $a^2 + b^2 + c^2 + abc - 2017$?" + } + }, + { + "question": "Return your final response within \\boxed{}. Suppose the sequence of nonnegative integers $a_1,a_2,...,a_{1997}$ satisfies \n$a_i+a_j \\le a_{i+j} \\le a_i+a_j+1$\nfor all $i, j \\ge 1$ with $i+j \\le 1997$. Show that there exists a real number $x$ such that $a_n=\\lfloor{nx}\\rfloor$ (the greatest integer $\\le nx$) for all $1 \\le n \\le 1997$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "x", + "index": "Sky-T1_10k_2554", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Suppose the sequence of nonnegative integers $a_1,a_2,...,a_{1997}$ satisfies \n$a_i+a_j \\le a_{i+j} \\le a_i+a_j+1$\nfor all $i, j \\ge 1$ with $i+j \\le 1997$. Show that there exists a real number $x$ such that $a_n=\\lfloor{nx}\\rfloor$ (the greatest integer $\\le nx$) for all $1 \\le n \\le 1997$." + } + }, + { + "question": "Return your final response within \\boxed{}. What is the tens digit of $2015^{2016}-2017?$\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2555", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the tens digit of $2015^{2016}-2017?$\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 3 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the [origin](https://artofproblemsolving.com/wiki/index.php/Origin). What is the volume of [tetrahedron](https://artofproblemsolving.com/wiki/index.php/Tetrahedron) $OABC$?\n$\\mathrm{(A)}\\ \\sqrt{85}\\qquad\\mathrm{(B)}\\ \\sqrt{90}\\qquad\\mathrm{(C)}\\ \\sqrt{95}\\qquad\\mathrm{(D)}\\ 10\\qquad\\mathrm{(E)}\\ \\sqrt{105}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2556", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the [origin](https://artofproblemsolving.com/wiki/index.php/Origin). What is the volume of [tetrahedron](https://artofproblemsolving.com/wiki/index.php/Tetrahedron) $OABC$?\n$\\mathrm{(A)}\\ \\sqrt{85}\\qquad\\mathrm{(B)}\\ \\sqrt{90}\\qquad\\mathrm{(C)}\\ \\sqrt{95}\\qquad\\mathrm{(D)}\\ 10\\qquad\\mathrm{(E)}\\ \\sqrt{105}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The solutions to the system of equations\n\n$\\log_{225}x+\\log_{64}y=4$\n$\\log_{x}225-\\log_{y}64=1$\nare $(x_1,y_1)$ and $(x_2,y_2)$. Find $\\log_{30}\\left(x_1y_1x_2y_2\\right)$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "12", + "index": "Sky-T1_10k_2557", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The solutions to the system of equations\n\n$\\log_{225}x+\\log_{64}y=4$\n$\\log_{x}225-\\log_{y}64=1$\nare $(x_1,y_1)$ and $(x_2,y_2)$. Find $\\log_{30}\\left(x_1y_1x_2y_2\\right)$." + } + }, + { + "question": "Return your final response within \\boxed{}. If $i^2=-1$, then $(i-i^{-1})^{-1}=$\n$\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ -2i \\qquad\\textbf{(C)}\\ 2i \\qquad\\textbf{(D)}\\ -\\frac{i}{2} \\qquad\\textbf{(E)}\\ \\frac{i}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2558", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $i^2=-1$, then $(i-i^{-1})^{-1}=$\n$\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ -2i \\qquad\\textbf{(C)}\\ 2i \\qquad\\textbf{(D)}\\ -\\frac{i}{2} \\qquad\\textbf{(E)}\\ \\frac{i}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$?\n$\\textbf{(A)}\\ 50 \\qquad\\textbf{(B)}\\ 60 \\qquad\\textbf{(C)}\\ 75 \\qquad\\textbf{(D)}\\ 90 \\qquad\\textbf{(E)}\\ 100$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2559", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$?\n$\\textbf{(A)}\\ 50 \\qquad\\textbf{(B)}\\ 60 \\qquad\\textbf{(C)}\\ 75 \\qquad\\textbf{(D)}\\ 90 \\qquad\\textbf{(E)}\\ 100$" + } + }, + { + "question": "Return your final response within \\boxed{}. Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$?\n$\\textbf{(A)}\\ \\frac{23}{8} \\qquad\\textbf{(B)}\\ \\frac{29}{10} \\qquad\\textbf{(C)}\\ \\frac{35}{12} \\qquad\\textbf{(D)}\\ \\frac{73}{25} \\qquad\\textbf{(E)}\\ 3$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2560", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$?\n$\\textbf{(A)}\\ \\frac{23}{8} \\qquad\\textbf{(B)}\\ \\frac{29}{10} \\qquad\\textbf{(C)}\\ \\frac{35}{12} \\qquad\\textbf{(D)}\\ \\frac{73}{25} \\qquad\\textbf{(E)}\\ 3$" + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{2\\sqrt{6}}{\\sqrt{2}+\\sqrt{3}+\\sqrt{5}}$ equals\n$\\mathrm{(A) \\ }\\sqrt{2}+\\sqrt{3}-\\sqrt{5} \\qquad \\mathrm{(B) \\ }4-\\sqrt{2}-\\sqrt{3} \\qquad \\mathrm{(C) \\ } \\sqrt{2}+\\sqrt{3}+\\sqrt{6}-5 \\qquad$\n$\\mathrm{(D) \\ }\\frac{1}{2}(\\sqrt{2}+\\sqrt{5}-\\sqrt{3}) \\qquad \\mathrm{(E) \\ } \\frac{1}{3}(\\sqrt{3}+\\sqrt{5}-\\sqrt{2})$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2561", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{2\\sqrt{6}}{\\sqrt{2}+\\sqrt{3}+\\sqrt{5}}$ equals\n$\\mathrm{(A) \\ }\\sqrt{2}+\\sqrt{3}-\\sqrt{5} \\qquad \\mathrm{(B) \\ }4-\\sqrt{2}-\\sqrt{3} \\qquad \\mathrm{(C) \\ } \\sqrt{2}+\\sqrt{3}+\\sqrt{6}-5 \\qquad$\n$\\mathrm{(D) \\ }\\frac{1}{2}(\\sqrt{2}+\\sqrt{5}-\\sqrt{3}) \\qquad \\mathrm{(E) \\ } \\frac{1}{3}(\\sqrt{3}+\\sqrt{5}-\\sqrt{2})$" + } + }, + { + "question": "Return your final response within \\boxed{}. Distinct lines $\\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. What is the equation of line $m?$\n$(\\textbf{A})\\: 5x+2y=0\\qquad(\\textbf{B}) \\: 3x+2y=0\\qquad(\\textbf{C}) \\: x-3y=0\\qquad(\\textbf{D}) \\: 2x-3y=0\\qquad(\\textbf{E}) \\: 5x-3y=0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2562", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Distinct lines $\\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. What is the equation of line $m?$\n$(\\textbf{A})\\: 5x+2y=0\\qquad(\\textbf{B}) \\: 3x+2y=0\\qquad(\\textbf{C}) \\: x-3y=0\\qquad(\\textbf{D}) \\: 2x-3y=0\\qquad(\\textbf{E}) \\: 5x-3y=0$" + } + }, + { + "question": "Return your final response within \\boxed{}. Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?\n$\\textbf {(A) } 33 \\qquad \\textbf {(B) } 34 \\qquad \\textbf {(C) } 36 \\qquad \\textbf {(D) } 38 \\qquad \\textbf {(E) } 39$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2563", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?\n$\\textbf {(A) } 33 \\qquad \\textbf {(B) } 34 \\qquad \\textbf {(C) } 36 \\qquad \\textbf {(D) } 38 \\qquad \\textbf {(E) } 39$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\\omega$ in the plane. Suppose that the tangent lines to $\\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\\omega$?\n$\\textbf{(A) }\\frac{83\\pi}{8}\\qquad\\textbf{(B) }\\frac{21\\pi}{2}\\qquad\\textbf{(C) } \\frac{85\\pi}{8}\\qquad\\textbf{(D) }\\frac{43\\pi}{4}\\qquad\\textbf{(E) }\\frac{87\\pi}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2564", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\\omega$ in the plane. Suppose that the tangent lines to $\\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\\omega$?\n$\\textbf{(A) }\\frac{83\\pi}{8}\\qquad\\textbf{(B) }\\frac{21\\pi}{2}\\qquad\\textbf{(C) } \\frac{85\\pi}{8}\\qquad\\textbf{(D) }\\frac{43\\pi}{4}\\qquad\\textbf{(E) }\\frac{87\\pi}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 22\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 32\\qquad\\textbf{(E)}\\ 34$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2565", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 22\\qquad\\textbf{(C)}\\ 30\\qquad\\textbf{(D)}\\ 32\\qquad\\textbf{(E)}\\ 34$" + } + }, + { + "question": "Return your final response within \\boxed{}. When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?\n$\\textbf{(A)}\\ \\frac{3}{8} \\qquad\\textbf{(B)}\\ \\frac{4}{9} \\qquad\\textbf{(C)}\\ \\frac{5}{9} \\qquad\\textbf{(D)}\\ \\frac{9}{16} \\qquad\\textbf{(E)}\\ \\frac{5}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2566", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?\n$\\textbf{(A)}\\ \\frac{3}{8} \\qquad\\textbf{(B)}\\ \\frac{4}{9} \\qquad\\textbf{(C)}\\ \\frac{5}{9} \\qquad\\textbf{(D)}\\ \\frac{9}{16} \\qquad\\textbf{(E)}\\ \\frac{5}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is\n$\\textbf{(A) }\\text{the line segment from }A\\text{ to }B\\qquad \\textbf{(B) }\\text{the line passing through }A\\text{ and }B\\qquad\\\\ \\textbf{(C) }\\text{the perpendicular bisector of the line segment from }A\\text{ to }B\\qquad\\\\ \\textbf{(D) }\\text{an ellipse having positive area}\\qquad \\textbf{(E) }\\text{a parabola}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2567", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is\n$\\textbf{(A) }\\text{the line segment from }A\\text{ to }B\\qquad \\textbf{(B) }\\text{the line passing through }A\\text{ and }B\\qquad\\\\ \\textbf{(C) }\\text{the perpendicular bisector of the line segment from }A\\text{ to }B\\qquad\\\\ \\textbf{(D) }\\text{an ellipse having positive area}\\qquad \\textbf{(E) }\\text{a parabola}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $W,X,Y$ and $Z$ be four different digits selected from the set\n$\\{ 1,2,3,4,5,6,7,8,9\\}.$\nIf the sum $\\dfrac{W}{X} + \\dfrac{Y}{Z}$ is to be as small as possible, then $\\dfrac{W}{X} + \\dfrac{Y}{Z}$ must equal\n$\\text{(A)}\\ \\dfrac{2}{17} \\qquad \\text{(B)}\\ \\dfrac{3}{17} \\qquad \\text{(C)}\\ \\dfrac{17}{72} \\qquad \\text{(D)}\\ \\dfrac{25}{72} \\qquad \\text{(E)}\\ \\dfrac{13}{36}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2568", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $W,X,Y$ and $Z$ be four different digits selected from the set\n$\\{ 1,2,3,4,5,6,7,8,9\\}.$\nIf the sum $\\dfrac{W}{X} + \\dfrac{Y}{Z}$ is to be as small as possible, then $\\dfrac{W}{X} + \\dfrac{Y}{Z}$ must equal\n$\\text{(A)}\\ \\dfrac{2}{17} \\qquad \\text{(B)}\\ \\dfrac{3}{17} \\qquad \\text{(C)}\\ \\dfrac{17}{72} \\qquad \\text{(D)}\\ \\dfrac{25}{72} \\qquad \\text{(E)}\\ \\dfrac{13}{36}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The product of the 9 factors $\\Big(1 - \\frac12\\Big)\\Big(1 - \\frac13\\Big)\\Big(1 - \\frac14\\Big)\\cdots\\Big(1 - \\frac {1}{10}\\Big) =$\n$\\text{(A)}\\ \\frac {1}{10} \\qquad \\text{(B)}\\ \\frac {1}{9} \\qquad \\text{(C)}\\ \\frac {1}{2} \\qquad \\text{(D)}\\ \\frac {10}{11} \\qquad \\text{(E)}\\ \\frac {11}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2569", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The product of the 9 factors $\\Big(1 - \\frac12\\Big)\\Big(1 - \\frac13\\Big)\\Big(1 - \\frac14\\Big)\\cdots\\Big(1 - \\frac {1}{10}\\Big) =$\n$\\text{(A)}\\ \\frac {1}{10} \\qquad \\text{(B)}\\ \\frac {1}{9} \\qquad \\text{(C)}\\ \\frac {1}{2} \\qquad \\text{(D)}\\ \\frac {10}{11} \\qquad \\text{(E)}\\ \\frac {11}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \\[132_A+43_B=69_{A+B}.\\]\nWhat is $A+B$?\n$\\textbf{(A)}\\ 9\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 13\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 17$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2570", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \\[132_A+43_B=69_{A+B}.\\]\nWhat is $A+B$?\n$\\textbf{(A)}\\ 9\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 13\\qquad\\textbf{(D)}\\ 15\\qquad\\textbf{(E)}\\ 17$" + } + }, + { + "question": "Return your final response within \\boxed{}. Raashan, Sylvia, and Ted play the following game. Each starts with $$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $$1$? (For example, Raashan and Ted may each decide to give $$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $$0$, Sylvia will have $$2$, and Ted will have $$1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $$1$ to, and the holdings will be the same at the end of the second round.)\n$\\textbf{(A) } \\frac{1}{7} \\qquad\\textbf{(B) } \\frac{1}{4} \\qquad\\textbf{(C) } \\frac{1}{3} \\qquad\\textbf{(D) } \\frac{1}{2} \\qquad\\textbf{(E) } \\frac{2}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2571", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Raashan, Sylvia, and Ted play the following game. Each starts with $$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $$1$? (For example, Raashan and Ted may each decide to give $$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $$0$, Sylvia will have $$2$, and Ted will have $$1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $$1$ to, and the holdings will be the same at the end of the second round.)\n$\\textbf{(A) } \\frac{1}{7} \\qquad\\textbf{(B) } \\frac{1}{4} \\qquad\\textbf{(C) } \\frac{1}{3} \\qquad\\textbf{(D) } \\frac{1}{2} \\qquad\\textbf{(E) } \\frac{2}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of is a perfect square?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2572", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of is a perfect square?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$" + } + }, + { + "question": "Return your final response within \\boxed{}. A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?\n$\\textbf{(A)}\\ 30 \\qquad \\textbf{(B)}\\ 31 \\qquad \\textbf{(C)}\\ 32 \\qquad \\textbf{(D)}\\ 33 \\qquad \\textbf{(E)}\\ 34$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2573", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?\n$\\textbf{(A)}\\ 30 \\qquad \\textbf{(B)}\\ 31 \\qquad \\textbf{(C)}\\ 32 \\qquad \\textbf{(D)}\\ 33 \\qquad \\textbf{(E)}\\ 34$" + } + }, + { + "question": "Return your final response within \\boxed{}. The sides of a triangle are $30$, $70$, and $80$ units. If an altitude is dropped upon the side of length $80$, the larger segment cut off on this side is:\n$\\textbf{(A)}\\ 62\\qquad \\textbf{(B)}\\ 63\\qquad \\textbf{(C)}\\ 64\\qquad \\textbf{(D)}\\ 65\\qquad \\textbf{(E)}\\ 66$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2574", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The sides of a triangle are $30$, $70$, and $80$ units. If an altitude is dropped upon the side of length $80$, the larger segment cut off on this side is:\n$\\textbf{(A)}\\ 62\\qquad \\textbf{(B)}\\ 63\\qquad \\textbf{(C)}\\ 64\\qquad \\textbf{(D)}\\ 65\\qquad \\textbf{(E)}\\ 66$" + } + }, + { + "question": "Return your final response within \\boxed{}. There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "307", + "index": "Sky-T1_10k_2575", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$." + } + }, + { + "question": "Return your final response within \\boxed{}. If $N$, written in base $2$, is $11000$, the integer immediately preceding $N$, written in base $2$, is:\n$\\text{(A) } 10001\\quad \\text{(B) } 10010\\quad \\text{(C) } 10011\\quad \\text{(D) } 10110\\quad \\text{(E) } 10111$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2576", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $N$, written in base $2$, is $11000$, the integer immediately preceding $N$, written in base $2$, is:\n$\\text{(A) } 10001\\quad \\text{(B) } 10010\\quad \\text{(C) } 10011\\quad \\text{(D) } 10110\\quad \\text{(E) } 10111$" + } + }, + { + "question": "Return your final response within \\boxed{}. The graph of $y=\\log x$\n$\\textbf{(A)}\\ \\text{Cuts the }y\\text{-axis} \\qquad\\\\ \\textbf{(B)}\\ \\text{Cuts all lines perpendicular to the }x\\text{-axis} \\qquad\\\\ \\textbf{(C)}\\ \\text{Cuts the }x\\text{-axis} \\qquad\\\\ \\textbf{(D)}\\ \\text{Cuts neither axis} \\qquad\\\\ \\textbf{(E)}\\ \\text{Cuts all circles whose center is at the origin}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2577", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The graph of $y=\\log x$\n$\\textbf{(A)}\\ \\text{Cuts the }y\\text{-axis} \\qquad\\\\ \\textbf{(B)}\\ \\text{Cuts all lines perpendicular to the }x\\text{-axis} \\qquad\\\\ \\textbf{(C)}\\ \\text{Cuts the }x\\text{-axis} \\qquad\\\\ \\textbf{(D)}\\ \\text{Cuts neither axis} \\qquad\\\\ \\textbf{(E)}\\ \\text{Cuts all circles whose center is at the origin}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $f(x)=|x-2|+|x-4|-|2x-6|$ for $2 \\leq x\\leq 8$. The sum of the largest and smallest values of $f(x)$ is \n$\\textbf {(A)}\\ 1 \\qquad \\textbf {(B)}\\ 2 \\qquad \\textbf {(C)}\\ 4 \\qquad \\textbf {(D)}\\ 6 \\qquad \\textbf {(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2578", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $f(x)=|x-2|+|x-4|-|2x-6|$ for $2 \\leq x\\leq 8$. The sum of the largest and smallest values of $f(x)$ is \n$\\textbf {(A)}\\ 1 \\qquad \\textbf {(B)}\\ 2 \\qquad \\textbf {(C)}\\ 4 \\qquad \\textbf {(D)}\\ 6 \\qquad \\textbf {(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the [average](https://artofproblemsolving.com/wiki/index.php/Average) rainfall in inches per hour during that month?\n$\\text{(A)}\\ \\frac{366}{31\\times 24}$\n$\\text{(B)}\\ \\frac{366\\times 31}{24}$\n$\\text{(C)}\\ \\frac{366\\times 24}{31}$\n$\\text{(D)}\\ \\frac{31\\times 24}{366}$\n$\\text{(E)}\\ 366\\times 31\\times 24$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2579", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the [average](https://artofproblemsolving.com/wiki/index.php/Average) rainfall in inches per hour during that month?\n$\\text{(A)}\\ \\frac{366}{31\\times 24}$\n$\\text{(B)}\\ \\frac{366\\times 31}{24}$\n$\\text{(C)}\\ \\frac{366\\times 24}{31}$\n$\\text{(D)}\\ \\frac{31\\times 24}{366}$\n$\\text{(E)}\\ 366\\times 31\\times 24$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of the following is equivalent to\n\\[(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?\\]\n$\\textbf{(A)} ~3^{127} + 2^{127} \\qquad\\textbf{(B)} ~3^{127} + 2^{127} + 2 \\cdot 3^{63} + 3 \\cdot 2^{63} \\qquad\\textbf{(C)} ~3^{128}-2^{128} \\qquad\\textbf{(D)} ~3^{128} + 2^{128} \\qquad\\textbf{(E)} ~5^{127}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2580", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of the following is equivalent to\n\\[(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?\\]\n$\\textbf{(A)} ~3^{127} + 2^{127} \\qquad\\textbf{(B)} ~3^{127} + 2^{127} + 2 \\cdot 3^{63} + 3 \\cdot 2^{63} \\qquad\\textbf{(C)} ~3^{128}-2^{128} \\qquad\\textbf{(D)} ~3^{128} + 2^{128} \\qquad\\textbf{(E)} ~5^{127}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the non-decreasing sequence of odd integers $\\{a_1,a_2,a_3,\\ldots \\}=\\{1,3,3,3,5,5,5,5,5,\\ldots \\}$ each odd positive integer $k$\nappears $k$ times. It is a fact that there are integers $b, c$, and $d$ such that for all positive integers $n$, \n$a_n=b\\lfloor \\sqrt{n+c} \\rfloor +d$, \nwhere $\\lfloor x \\rfloor$ denotes the largest integer not exceeding $x$. The sum $b+c+d$ equals \n$\\text{(A)} \\ 0 \\qquad \\text{(B)} \\ 1 \\qquad \\text{(C)} \\ 2 \\qquad \\text{(D)} \\ 3 \\qquad \\text{(E)} \\ 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2581", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the non-decreasing sequence of odd integers $\\{a_1,a_2,a_3,\\ldots \\}=\\{1,3,3,3,5,5,5,5,5,\\ldots \\}$ each odd positive integer $k$\nappears $k$ times. It is a fact that there are integers $b, c$, and $d$ such that for all positive integers $n$, \n$a_n=b\\lfloor \\sqrt{n+c} \\rfloor +d$, \nwhere $\\lfloor x \\rfloor$ denotes the largest integer not exceeding $x$. The sum $b+c+d$ equals \n$\\text{(A)} \\ 0 \\qquad \\text{(B)} \\ 1 \\qquad \\text{(C)} \\ 2 \\qquad \\text{(D)} \\ 3 \\qquad \\text{(E)} \\ 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 13 \\qquad \\textbf{(C)}\\ 14 \\qquad \\textbf{(D)}\\ 15 \\qquad \\textbf{(E)} 16$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2582", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?\n$\\textbf{(A)}\\ 12 \\qquad \\textbf{(B)}\\ 13 \\qquad \\textbf{(C)}\\ 14 \\qquad \\textbf{(D)}\\ 15 \\qquad \\textbf{(E)} 16$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let a sequence $\\{u_n\\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:\n$\\text{(A) 3} \\quad \\text{(B) 4} \\quad \\text{(C) 5} \\quad \\text{(D) 6} \\quad \\text{(E) 11}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2583", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let a sequence $\\{u_n\\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:\n$\\text{(A) 3} \\quad \\text{(B) 4} \\quad \\text{(C) 5} \\quad \\text{(D) 6} \\quad \\text{(E) 11}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?\n$\\textbf{(A)}\\ 33\\qquad\\textbf{(B)}\\ 35\\qquad\\textbf{(C)}\\ 37\\qquad\\textbf{(D)}\\ 39\\qquad\\textbf{(E)}\\ 41$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2584", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?\n$\\textbf{(A)}\\ 33\\qquad\\textbf{(B)}\\ 35\\qquad\\textbf{(C)}\\ 37\\qquad\\textbf{(D)}\\ 39\\qquad\\textbf{(E)}\\ 41$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many three-digit numbers satisfy the property that the middle digit is the [average](https://artofproblemsolving.com/wiki/index.php/Mean) of the first and the last digits?\n$(\\mathrm {A}) \\ 41 \\qquad (\\mathrm {B}) \\ 42 \\qquad (\\mathrm {C})\\ 43 \\qquad (\\mathrm {D}) \\ 44 \\qquad (\\mathrm {E})\\ 45$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2585", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many three-digit numbers satisfy the property that the middle digit is the [average](https://artofproblemsolving.com/wiki/index.php/Mean) of the first and the last digits?\n$(\\mathrm {A}) \\ 41 \\qquad (\\mathrm {B}) \\ 42 \\qquad (\\mathrm {C})\\ 43 \\qquad (\\mathrm {D}) \\ 44 \\qquad (\\mathrm {E})\\ 45$" + } + }, + { + "question": "Return your final response within \\boxed{}. A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end?\n$\\text{(A)}\\ \\text{10:00 p.m.} \\qquad \\text{(B)}\\ \\text{midnight} \\qquad \\text{(C)}\\ \\text{2:30 a.m.} \\qquad \\text{(D)}\\ \\text{4:40 a.m.} \\qquad \\text{(E)}\\ \\text{6:40 a.m.}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2586", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end?\n$\\text{(A)}\\ \\text{10:00 p.m.} \\qquad \\text{(B)}\\ \\text{midnight} \\qquad \\text{(C)}\\ \\text{2:30 a.m.} \\qquad \\text{(D)}\\ \\text{4:40 a.m.} \\qquad \\text{(E)}\\ \\text{6:40 a.m.}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is \n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 16 \\qquad \\text{(C)}\\ 20 \\qquad \\text{(D)}\\ 22 \\qquad \\text{(E)}\\ 24$\n[asy] unitsize(36); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5.2,1.4)--(5.2,4.4)--(3,3)); draw((0,3)--(2.2,4.4)--(5.2,4.4)); fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black); fill((0,2)--(0,3)--(1,3)--(1,2)--cycle,black); fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black); fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black); draw((1,3)--(3.2,4.4)); draw((2,3)--(4.2,4.4)); draw((.733333333,3.4666666666)--(3.73333333333,3.466666666666)); draw((1.466666666,3.9333333333)--(4.466666666,3.9333333333)); fill((1.73333333,3.46666666666)--(2.7333333333,3.46666666666)--(3.46666666666,3.93333333333)--(2.46666666666,3.93333333333)--cycle,black); fill((3,1)--(3.733333333333,1.466666666666)--(3.73333333333,2.46666666666)--(3,2)--cycle,black); fill((3.73333333333,.466666666666)--(4.466666666666,.93333333333)--(4.46666666666,1.93333333333)--(3.733333333333,1.46666666666)--cycle,black); fill((3.73333333333,2.466666666666)--(4.466666666666,2.93333333333)--(4.46666666666,3.93333333333)--(3.733333333333,3.46666666666)--cycle,black); fill((4.466666666666,1.9333333333333)--(5.2,2.4)--(5.2,3.4)--(4.4666666666666,2.9333333333333)--cycle,black); [/asy]", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2587", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is \n$\\text{(A)}\\ 10 \\qquad \\text{(B)}\\ 16 \\qquad \\text{(C)}\\ 20 \\qquad \\text{(D)}\\ 22 \\qquad \\text{(E)}\\ 24$\n[asy] unitsize(36); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5.2,1.4)--(5.2,4.4)--(3,3)); draw((0,3)--(2.2,4.4)--(5.2,4.4)); fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black); fill((0,2)--(0,3)--(1,3)--(1,2)--cycle,black); fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black); fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black); draw((1,3)--(3.2,4.4)); draw((2,3)--(4.2,4.4)); draw((.733333333,3.4666666666)--(3.73333333333,3.466666666666)); draw((1.466666666,3.9333333333)--(4.466666666,3.9333333333)); fill((1.73333333,3.46666666666)--(2.7333333333,3.46666666666)--(3.46666666666,3.93333333333)--(2.46666666666,3.93333333333)--cycle,black); fill((3,1)--(3.733333333333,1.466666666666)--(3.73333333333,2.46666666666)--(3,2)--cycle,black); fill((3.73333333333,.466666666666)--(4.466666666666,.93333333333)--(4.46666666666,1.93333333333)--(3.733333333333,1.46666666666)--cycle,black); fill((3.73333333333,2.466666666666)--(4.466666666666,2.93333333333)--(4.46666666666,3.93333333333)--(3.733333333333,3.46666666666)--cycle,black); fill((4.466666666666,1.9333333333333)--(5.2,2.4)--(5.2,3.4)--(4.4666666666666,2.9333333333333)--cycle,black); [/asy]" + } + }, + { + "question": "Return your final response within \\boxed{}. If $a^{x}= c^{q}= b$ and $c^{y}= a^{z}= d$, then\n$\\textbf{(A)}\\ xy = qz\\qquad\\textbf{(B)}\\ \\frac{x}{y}=\\frac{q}{z}\\qquad\\textbf{(C)}\\ x+y = q+z\\qquad\\textbf{(D)}\\ x-y = q-z$\n$\\textbf{(E)}\\ x^{y}= q^{z}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2588", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a^{x}= c^{q}= b$ and $c^{y}= a^{z}= d$, then\n$\\textbf{(A)}\\ xy = qz\\qquad\\textbf{(B)}\\ \\frac{x}{y}=\\frac{q}{z}\\qquad\\textbf{(C)}\\ x+y = q+z\\qquad\\textbf{(D)}\\ x-y = q-z$\n$\\textbf{(E)}\\ x^{y}= q^{z}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $\\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\\clubsuit(8)=8$ and $\\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\\clubsuit(\\clubsuit(x))=3$?\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 6 \\qquad\\textbf{(D) } 9 \\qquad\\textbf{(E) } 10$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2589", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $\\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\\clubsuit(8)=8$ and $\\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\\clubsuit(\\clubsuit(x))=3$?\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 6 \\qquad\\textbf{(D) } 9 \\qquad\\textbf{(E) } 10$" + } + }, + { + "question": "Return your final response within \\boxed{}. Triangle $I$ is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle). \nTriangle $II$ is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then: \n$\\textbf{(A)}\\ P:p = R:r \\text{ } \\text{only sometimes} \\qquad \\\\ \\textbf{(B)}\\ P:p = R:r \\text{ } \\text{always}\\qquad \\\\ \\textbf{(C)}\\ P:p = K:k \\text{ } \\text{only sometimes} \\qquad \\\\ \\textbf{(D)}\\ R:r = K:k \\text{ } \\text{always}\\qquad \\\\ \\textbf{(E)}\\ R:r = K:k \\text{ } \\text{only sometimes}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2590", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $I$ is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle). \nTriangle $II$ is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then: \n$\\textbf{(A)}\\ P:p = R:r \\text{ } \\text{only sometimes} \\qquad \\\\ \\textbf{(B)}\\ P:p = R:r \\text{ } \\text{always}\\qquad \\\\ \\textbf{(C)}\\ P:p = K:k \\text{ } \\text{only sometimes} \\qquad \\\\ \\textbf{(D)}\\ R:r = K:k \\text{ } \\text{always}\\qquad \\\\ \\textbf{(E)}\\ R:r = K:k \\text{ } \\text{only sometimes}$" + } + }, + { + "question": "Return your final response within \\boxed{}. For integers $a,b,$ and $c$ define $\\fbox{a,b,c}$ to mean $a^b-b^c+c^a$. Then $\\fbox{1,-1,2}$ equals:\n$\\text{(A) } -4\\quad \\text{(B) } -2\\quad \\text{(C) } 0\\quad \\text{(D) } 2\\quad \\text{(E) } 4$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2591", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For integers $a,b,$ and $c$ define $\\fbox{a,b,c}$ to mean $a^b-b^c+c^a$. Then $\\fbox{1,-1,2}$ equals:\n$\\text{(A) } -4\\quad \\text{(B) } -2\\quad \\text{(C) } 0\\quad \\text{(D) } 2\\quad \\text{(E) } 4$" + } + }, + { + "question": "Return your final response within \\boxed{}. For $p=1, 2, \\cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\\cdots+S_{10}$ is\n$\\mathrm{(A)\\ } 80000 \\qquad \\mathrm{(B) \\ }80200 \\qquad \\mathrm{(C) \\ } 80400 \\qquad \\mathrm{(D) \\ } 80600 \\qquad \\mathrm{(E) \\ }80800$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2592", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For $p=1, 2, \\cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\\cdots+S_{10}$ is\n$\\mathrm{(A)\\ } 80000 \\qquad \\mathrm{(B) \\ }80200 \\qquad \\mathrm{(C) \\ } 80400 \\qquad \\mathrm{(D) \\ } 80600 \\qquad \\mathrm{(E) \\ }80800$" + } + }, + { + "question": "Return your final response within \\boxed{}. In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. The ratio $A:B$ is $1:18$. What is $N$?\n\n\n\n$\\textbf{(A) } 16 \\qquad \\textbf{(B) } 17 \\qquad \\textbf{(C) } 18 \\qquad \\textbf{(D) } 19 \\qquad \\textbf{(E) } 36$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2593", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. The ratio $A:B$ is $1:18$. What is $N$?\n\n\n\n$\\textbf{(A) } 16 \\qquad \\textbf{(B) } 17 \\qquad \\textbf{(C) } 18 \\qquad \\textbf{(D) } 19 \\qquad \\textbf{(E) } 36$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?\n$\\text{(A) } 6 \\qquad \\text{(B) } 12 \\qquad \\text{(C) } 24 \\qquad \\text{(D) } 36 \\qquad \\text{(E) } 48$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2594", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?\n$\\text{(A) } 6 \\qquad \\text{(B) } 12 \\qquad \\text{(C) } 24 \\qquad \\text{(D) } 36 \\qquad \\text{(E) } 48$" + } + }, + { + "question": "Return your final response within \\boxed{}. Segments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\sqrt{10}$, $CP=60\\sqrt{5}$, $DP=120\\sqrt{2}$, and $GP=36\\sqrt{7}$. Find $AP.$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "192", + "index": "Sky-T1_10k_2595", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Segments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\sqrt{10}$, $CP=60\\sqrt{5}$, $DP=120\\sqrt{2}$, and $GP=36\\sqrt{7}$. Find $AP.$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $x<0$, then $\\left|x-\\sqrt{(x-1)^2}\\right|$ equals\n$\\textbf{(A) }1\\qquad \\textbf{(B) }1-2x\\qquad \\textbf{(C) }-2x-1\\qquad \\textbf{(D) }1+2x\\qquad \\textbf{(E) }2x-1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2596", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $x<0$, then $\\left|x-\\sqrt{(x-1)^2}\\right|$ equals\n$\\textbf{(A) }1\\qquad \\textbf{(B) }1-2x\\qquad \\textbf{(C) }-2x-1\\qquad \\textbf{(D) }1+2x\\qquad \\textbf{(E) }2x-1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?\n$\\textbf{(A)}\\ \\frac{5}{72}\\qquad\\textbf{(B)}\\ \\frac{5}{36}\\qquad\\textbf{(C)}\\ \\frac{1}{6}\\qquad\\textbf{(D)}\\ \\frac{1}{3}\\qquad\\textbf{(E)}\\ 1$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2597", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?\n$\\textbf{(A)}\\ \\frac{5}{72}\\qquad\\textbf{(B)}\\ \\frac{5}{36}\\qquad\\textbf{(C)}\\ \\frac{1}{6}\\qquad\\textbf{(D)}\\ \\frac{1}{3}\\qquad\\textbf{(E)}\\ 1$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ana's monthly salary was $$2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was\n$\\text{(A)}\\ 1920\\text{ dollars} \\qquad \\text{(B)}\\ 1980\\text{ dollars} \\qquad \\text{(C)}\\ 2000\\text{ dollars} \\qquad \\text{(D)}\\ 2020\\text{ dollars} \\qquad \\text{(E)}\\ 2040\\text{ dollars}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2598", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ana's monthly salary was $$2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was\n$\\text{(A)}\\ 1920\\text{ dollars} \\qquad \\text{(B)}\\ 1980\\text{ dollars} \\qquad \\text{(C)}\\ 2000\\text{ dollars} \\qquad \\text{(D)}\\ 2020\\text{ dollars} \\qquad \\text{(E)}\\ 2040\\text{ dollars}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which of these five numbers is the largest?\n$\\text{(A)}\\ 13579+\\frac{1}{2468} \\qquad \\text{(B)}\\ 13579-\\frac{1}{2468} \\qquad \\text{(C)}\\ 13579\\times \\frac{1}{2468}$\n$\\text{(D)}\\ 13579\\div \\frac{1}{2468} \\qquad \\text{(E)}\\ 13579.2468$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2599", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which of these five numbers is the largest?\n$\\text{(A)}\\ 13579+\\frac{1}{2468} \\qquad \\text{(B)}\\ 13579-\\frac{1}{2468} \\qquad \\text{(C)}\\ 13579\\times \\frac{1}{2468}$\n$\\text{(D)}\\ 13579\\div \\frac{1}{2468} \\qquad \\text{(E)}\\ 13579.2468$" + } + }, + { + "question": "Return your final response within \\boxed{}. A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?\n$\\mathrm{(A)}\\ 3\\qquad\\mathrm{(B)}\\ 4\\qquad\\mathrm{(C)}\\ 5\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 7$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2601", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?\n$\\mathrm{(A)}\\ 3\\qquad\\mathrm{(B)}\\ 4\\qquad\\mathrm{(C)}\\ 5\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 7$" + } + }, + { + "question": "Return your final response within \\boxed{}. Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is: \n$\\textbf{(A)}\\ x > 4 \\qquad \\textbf{(B)}\\ x < \\frac {10}{3} \\qquad \\textbf{(C)}\\ x = \\frac {10}{3} \\qquad \\textbf{(D)}\\ x >\\frac{10}{3}\\qquad\\textbf{(E)}\\ x < 0$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2602", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is: \n$\\textbf{(A)}\\ x > 4 \\qquad \\textbf{(B)}\\ x < \\frac {10}{3} \\qquad \\textbf{(C)}\\ x = \\frac {10}{3} \\qquad \\textbf{(D)}\\ x >\\frac{10}{3}\\qquad\\textbf{(E)}\\ x < 0$" + } + }, + { + "question": "Return your final response within \\boxed{}. An athlete's target heart rate, in beats per minute, is $80\\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old?\n$\\textbf{(A)}\\ 134\\qquad\\textbf{(B)}\\ 155\\qquad\\textbf{(C)}\\ 176\\qquad\\textbf{(D)}\\ 194\\qquad\\textbf{(E)}\\ 243$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2603", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. An athlete's target heart rate, in beats per minute, is $80\\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old?\n$\\textbf{(A)}\\ 134\\qquad\\textbf{(B)}\\ 155\\qquad\\textbf{(C)}\\ 176\\qquad\\textbf{(D)}\\ 194\\qquad\\textbf{(E)}\\ 243$" + } + }, + { + "question": "Return your final response within \\boxed{}. Mindy made three purchases for $\\textdollar 1.98$ dollars, $\\textdollar 5.04$ dollars, and $\\textdollar 9.89$ dollars. What was her total, to the nearest dollar? \n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 15\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 17\\qquad\\textbf{(E)}\\ 18$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2604", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Mindy made three purchases for $\\textdollar 1.98$ dollars, $\\textdollar 5.04$ dollars, and $\\textdollar 9.89$ dollars. What was her total, to the nearest dollar? \n$\\textbf{(A)}\\ 10\\qquad\\textbf{(B)}\\ 15\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 17\\qquad\\textbf{(E)}\\ 18$" + } + }, + { + "question": "Return your final response within \\boxed{}. The letters $P$, $Q$, $R$, $S$, and $T$ represent numbers located on the number line as shown.\n\nWhich of the following expressions represents a negative number?\n$\\text{(A)}\\ P-Q \\qquad \\text{(B)}\\ P\\cdot Q \\qquad \\text{(C)}\\ \\dfrac{S}{Q}\\cdot P \\qquad \\text{(D)}\\ \\dfrac{R}{P\\cdot Q} \\qquad \\text{(E)}\\ \\dfrac{S+T}{R}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2605", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The letters $P$, $Q$, $R$, $S$, and $T$ represent numbers located on the number line as shown.\n\nWhich of the following expressions represents a negative number?\n$\\text{(A)}\\ P-Q \\qquad \\text{(B)}\\ P\\cdot Q \\qquad \\text{(C)}\\ \\dfrac{S}{Q}\\cdot P \\qquad \\text{(D)}\\ \\dfrac{R}{P\\cdot Q} \\qquad \\text{(E)}\\ \\dfrac{S+T}{R}$" + } + }, + { + "question": "Return your final response within \\boxed{}. A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to $(1,0)$. Then it makes a $90^\\circ$ counterclockwise and travels $\\frac 12$ a unit to $\\left(1, \\frac 12 \\right)$. If it continues in this fashion, each time making a $90^\\circ$ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest? \n$\\text{(A)} \\ \\left(\\frac 23, \\frac 23 \\right) \\qquad \\text{(B)} \\ \\left( \\frac 45, \\frac 25 \\right) \\qquad \\text{(C)} \\ \\left( \\frac 23, \\frac 45 \\right) \\qquad \\text{(D)} \\ \\left(\\frac 23, \\frac 13 \\right) \\qquad \\text{(E)} \\ \\left(\\frac 25, \\frac 45 \\right)$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2606", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to $(1,0)$. Then it makes a $90^\\circ$ counterclockwise and travels $\\frac 12$ a unit to $\\left(1, \\frac 12 \\right)$. If it continues in this fashion, each time making a $90^\\circ$ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest? \n$\\text{(A)} \\ \\left(\\frac 23, \\frac 23 \\right) \\qquad \\text{(B)} \\ \\left( \\frac 45, \\frac 25 \\right) \\qquad \\text{(C)} \\ \\left( \\frac 23, \\frac 45 \\right) \\qquad \\text{(D)} \\ \\left(\\frac 23, \\frac 13 \\right) \\qquad \\text{(E)} \\ \\left(\\frac 25, \\frac 45 \\right)$" + } + }, + { + "question": "Return your final response within \\boxed{}. A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?\n\n$\\textbf{(A) }148\\qquad\\textbf{(B) }324\\qquad\\textbf{(C) }361\\qquad\\textbf{(D) }1296\\qquad\\textbf{(E) }1369$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2607", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?\n\n$\\textbf{(A) }148\\qquad\\textbf{(B) }324\\qquad\\textbf{(C) }361\\qquad\\textbf{(D) }1296\\qquad\\textbf{(E) }1369$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a certain school, there are $3$ times as many boys as girls and $9$ times as many girls as teachers. Using the letters $b, g, t$ to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the [expression](https://artofproblemsolving.com/wiki/index.php/Expression)\n$\\mathrm{(A) \\ }31b \\qquad \\mathrm{(B) \\ }\\frac{37b}{27} \\qquad \\mathrm{(C) \\ } 13g \\qquad \\mathrm{(D) \\ }\\frac{37g}{27} \\qquad \\mathrm{(E) \\ } \\frac{37t}{27}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2608", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a certain school, there are $3$ times as many boys as girls and $9$ times as many girls as teachers. Using the letters $b, g, t$ to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the [expression](https://artofproblemsolving.com/wiki/index.php/Expression)\n$\\mathrm{(A) \\ }31b \\qquad \\mathrm{(B) \\ }\\frac{37b}{27} \\qquad \\mathrm{(C) \\ } 13g \\qquad \\mathrm{(D) \\ }\\frac{37g}{27} \\qquad \\mathrm{(E) \\ } \\frac{37t}{27}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.\n\n$2x_1+x_2+x_3+x_4+x_5=6$\n$x_1+2x_2+x_3+x_4+x_5=12$\n$x_1+x_2+2x_3+x_4+x_5=24$\n$x_1+x_2+x_3+2x_4+x_5=48$\n$x_1+x_2+x_3+x_4+2x_5=96$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "181", + "index": "Sky-T1_10k_2609", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.\n\n$2x_1+x_2+x_3+x_4+x_5=6$\n$x_1+2x_2+x_3+x_4+x_5=12$\n$x_1+x_2+2x_3+x_4+x_5=24$\n$x_1+x_2+x_3+2x_4+x_5=48$\n$x_1+x_2+x_3+x_4+2x_5=96$" + } + }, + { + "question": "Return your final response within \\boxed{}. The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?\n$\\mathrm{(A)\\ }25\\%\\qquad\\mathrm{(B)\\ }30\\%\\qquad\\mathrm{(C)\\ }35\\%\\qquad\\mathrm{(D)\\ }60\\%\\qquad\\mathrm{(E)\\ }65\\%$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2610", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?\n$\\mathrm{(A)\\ }25\\%\\qquad\\mathrm{(B)\\ }30\\%\\qquad\\mathrm{(C)\\ }35\\%\\qquad\\mathrm{(D)\\ }60\\%\\qquad\\mathrm{(E)\\ }65\\%$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $y=(\\log_23)(\\log_34)\\cdots(\\log_n[n+1])\\cdots(\\log_{31}32)$ then \n$\\textbf{(A) }49, b>6 \\qquad \\textbf{(B)}\\ a>9, b<6 \\qquad \\textbf{(C)}\\ a>9, b=6\\qquad \\textbf{(D)}\\ a>9, \\text{but we can put no bounds on} \\text{ } b\\qquad \\textbf{(E)}\\ 2a=3b$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2649", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Five times $A$'s money added to $B$'s money is more than $$51.00$. Three times $A$'s money minus $B$'s money is $$21.00$.\nIf $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then:\n$\\textbf{(A)}\\ a>9, b>6 \\qquad \\textbf{(B)}\\ a>9, b<6 \\qquad \\textbf{(C)}\\ a>9, b=6\\qquad \\textbf{(D)}\\ a>9, \\text{but we can put no bounds on} \\text{ } b\\qquad \\textbf{(E)}\\ 2a=3b$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?\n$\\mathrm{(A)}\\frac 18\\qquad \\mathrm{(B)}\\frac {3}{16}\\qquad \\mathrm{(C)}\\frac 14\\qquad \\mathrm{(D)}\\frac 38\\qquad \\mathrm{(E)}\\frac 12$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2650", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?\n$\\mathrm{(A)}\\frac 18\\qquad \\mathrm{(B)}\\frac {3}{16}\\qquad \\mathrm{(C)}\\frac 14\\qquad \\mathrm{(D)}\\frac 38\\qquad \\mathrm{(E)}\\frac 12$" + } + }, + { + "question": "Return your final response within \\boxed{}. For how many $n$ in $\\{1, 2, 3, ..., 100 \\}$ is the tens digit of $n^2$ odd?\n$\\textbf{(A)}\\ 10 \\qquad\\textbf{(B)}\\ 20 \\qquad\\textbf{(C)}\\ 30 \\qquad\\textbf{(D)}\\ 40 \\qquad\\textbf{(E)}\\ 50$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2651", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For how many $n$ in $\\{1, 2, 3, ..., 100 \\}$ is the tens digit of $n^2$ odd?\n$\\textbf{(A)}\\ 10 \\qquad\\textbf{(B)}\\ 20 \\qquad\\textbf{(C)}\\ 30 \\qquad\\textbf{(D)}\\ 40 \\qquad\\textbf{(E)}\\ 50$" + } + }, + { + "question": "Return your final response within \\boxed{}. The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?\n\n$\\textbf{(A)}\\ 401 \\qquad \\textbf{(B)}\\ 485 \\qquad \\textbf{(C)}\\ 585 \\qquad \\textbf{(D)}\\ 626 \\qquad \\textbf{(E)}\\ 761$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2652", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?\n\n$\\textbf{(A)}\\ 401 \\qquad \\textbf{(B)}\\ 485 \\qquad \\textbf{(C)}\\ 585 \\qquad \\textbf{(D)}\\ 626 \\qquad \\textbf{(E)}\\ 761$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then $c$ is\n$\\textbf{(A)}\\ -5\\qquad \\textbf{(B)}\\ -3\\qquad \\textbf{(C)}\\ -1\\qquad \\textbf{(D)}\\ 3\\qquad \\textbf{(E)}\\ 5$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2653", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then $c$ is\n$\\textbf{(A)}\\ -5\\qquad \\textbf{(B)}\\ -3\\qquad \\textbf{(C)}\\ -1\\qquad \\textbf{(D)}\\ 3\\qquad \\textbf{(E)}\\ 5$" + } + }, + { + "question": "Return your final response within \\boxed{}. It takes $A$ algebra books (all the same thickness) and $H$ geometry books (all the same thickness, \nwhich is greater than that of an algebra book) to completely fill a certain shelf. \nAlso, $S$ of the algebra books and $M$ of the geometry books would fill the same shelf. \nFinally, $E$ of the algebra books alone would fill this shelf. Given that $A, H, S, M, E$ are distinct positive integers, \nit follows that $E$ is\n$\\textbf{(A)}\\ \\frac{AM+SH}{M+H} \\qquad \\textbf{(B)}\\ \\frac{AM^2+SH^2}{M^2+H^2} \\qquad \\textbf{(C)}\\ \\frac{AH-SM}{M-H}\\qquad \\textbf{(D)}\\ \\frac{AM-SH}{M-H}\\qquad \\textbf{(E)}\\ \\frac{AM^2-SH^2}{M^2-H^2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2654", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. It takes $A$ algebra books (all the same thickness) and $H$ geometry books (all the same thickness, \nwhich is greater than that of an algebra book) to completely fill a certain shelf. \nAlso, $S$ of the algebra books and $M$ of the geometry books would fill the same shelf. \nFinally, $E$ of the algebra books alone would fill this shelf. Given that $A, H, S, M, E$ are distinct positive integers, \nit follows that $E$ is\n$\\textbf{(A)}\\ \\frac{AM+SH}{M+H} \\qquad \\textbf{(B)}\\ \\frac{AM^2+SH^2}{M^2+H^2} \\qquad \\textbf{(C)}\\ \\frac{AH-SM}{M-H}\\qquad \\textbf{(D)}\\ \\frac{AM-SH}{M-H}\\qquad \\textbf{(E)}\\ \\frac{AM^2-SH^2}{M^2-H^2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are:\n$\\textbf{(A)}\\ \\frac{3}{5} < x \\le 2\\qquad \\textbf{(B)}\\ \\frac{3}{5} \\le x \\le 2\\qquad \\textbf{(C)}\\ 0 < x \\le 2\\qquad \\\\ \\textbf{(D)}\\ 0 \\le x \\le 2\\qquad \\textbf{(E)}\\ -2 \\le x \\le 2$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2655", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are:\n$\\textbf{(A)}\\ \\frac{3}{5} < x \\le 2\\qquad \\textbf{(B)}\\ \\frac{3}{5} \\le x \\le 2\\qquad \\textbf{(C)}\\ 0 < x \\le 2\\qquad \\\\ \\textbf{(D)}\\ 0 \\le x \\le 2\\qquad \\textbf{(E)}\\ -2 \\le x \\le 2$" + } + }, + { + "question": "Return your final response within \\boxed{}. When simplified, $(x^{-1}+y^{-1})^{-1}$ is equal to:\n$\\textbf{(A) \\ }x+y \\qquad \\textbf{(B) \\ }\\frac{xy}{x+y} \\qquad \\textbf{(C) \\ }xy \\qquad \\textbf{(D) \\ }\\frac{1}{xy} \\qquad \\textbf{(E) \\ }\\frac{x+y}{xy}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2656", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. When simplified, $(x^{-1}+y^{-1})^{-1}$ is equal to:\n$\\textbf{(A) \\ }x+y \\qquad \\textbf{(B) \\ }\\frac{xy}{x+y} \\qquad \\textbf{(C) \\ }xy \\qquad \\textbf{(D) \\ }\\frac{1}{xy} \\qquad \\textbf{(E) \\ }\\frac{x+y}{xy}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which one of the following is not equivalent to $0.000000375$? \n$\\textbf{(A)}\\ 3.75\\times 10^{-7}\\qquad\\textbf{(B)}\\ 3\\frac{3}{4}\\times 10^{-7}\\qquad\\textbf{(C)}\\ 375\\times 10^{-9}\\qquad \\textbf{(D)}\\ \\frac{3}{8}\\times 10^{-7}\\qquad\\textbf{(E)}\\ \\frac{3}{80000000}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2657", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Which one of the following is not equivalent to $0.000000375$? \n$\\textbf{(A)}\\ 3.75\\times 10^{-7}\\qquad\\textbf{(B)}\\ 3\\frac{3}{4}\\times 10^{-7}\\qquad\\textbf{(C)}\\ 375\\times 10^{-9}\\qquad \\textbf{(D)}\\ \\frac{3}{8}\\times 10^{-7}\\qquad\\textbf{(E)}\\ \\frac{3}{80000000}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is\n[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2)); draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2)); draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8)); draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2)); draw((2,0)--(3,2)--(1,3)--(0,1)--cycle); [/asy]\n$\\text{(A)}\\ \\dfrac{\\sqrt{3}}{3} \\qquad \\text{(B)}\\ \\dfrac{5}{9} \\qquad \\text{(C)}\\ \\dfrac{2}{3} \\qquad \\text{(D)}\\ \\dfrac{\\sqrt{5}}{3} \\qquad \\text{(E)}\\ \\dfrac{7}{9}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2658", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is\n[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2)); draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2)); draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8)); draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2)); draw((2,0)--(3,2)--(1,3)--(0,1)--cycle); [/asy]\n$\\text{(A)}\\ \\dfrac{\\sqrt{3}}{3} \\qquad \\text{(B)}\\ \\dfrac{5}{9} \\qquad \\text{(C)}\\ \\dfrac{2}{3} \\qquad \\text{(D)}\\ \\dfrac{\\sqrt{5}}{3} \\qquad \\text{(E)}\\ \\dfrac{7}{9}$" + } + }, + { + "question": "Return your final response within \\boxed{}. How many polynomial functions $f$ of degree $\\ge 1$ satisfy\n$f(x^2)=[f(x)]^2=f(f(x))$  ?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ \\text{finitely many but more than 2}\\\\ \\qquad \\textbf{(E)}\\ \\infty$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2659", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. How many polynomial functions $f$ of degree $\\ge 1$ satisfy\n$f(x^2)=[f(x)]^2=f(f(x))$  ?\n$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ \\text{finitely many but more than 2}\\\\ \\qquad \\textbf{(E)}\\ \\infty$" + } + }, + { + "question": "Return your final response within \\boxed{}. Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\\omega$ in the plane. Suppose that the tangent lines to $\\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\\omega$?\n$\\textbf{(A) }\\frac{83\\pi}{8}\\qquad\\textbf{(B) }\\frac{21\\pi}{2}\\qquad\\textbf{(C) } \\frac{85\\pi}{8}\\qquad\\textbf{(D) }\\frac{43\\pi}{4}\\qquad\\textbf{(E) }\\frac{87\\pi}{8}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2660", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\\omega$ in the plane. Suppose that the tangent lines to $\\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\\omega$?\n$\\textbf{(A) }\\frac{83\\pi}{8}\\qquad\\textbf{(B) }\\frac{21\\pi}{2}\\qquad\\textbf{(C) } \\frac{85\\pi}{8}\\qquad\\textbf{(D) }\\frac{43\\pi}{4}\\qquad\\textbf{(E) }\\frac{87\\pi}{8}$" + } + }, + { + "question": "Return your final response within \\boxed{}. In a certain sequence of numbers, the first number is $1$, and, for all $n\\ge 2$, the product of the first $n$ numbers in the sequence is $n^2$. \nThe sum of the third and the fifth numbers in the sequence is\n$\\textbf{(A) }\\frac{25}{9}\\qquad \\textbf{(B) }\\frac{31}{15}\\qquad \\textbf{(C) }\\frac{61}{16}\\qquad \\textbf{(D) }\\frac{576}{225}\\qquad \\textbf{(E) }34$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2661", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In a certain sequence of numbers, the first number is $1$, and, for all $n\\ge 2$, the product of the first $n$ numbers in the sequence is $n^2$. \nThe sum of the third and the fifth numbers in the sequence is\n$\\textbf{(A) }\\frac{25}{9}\\qquad \\textbf{(B) }\\frac{31}{15}\\qquad \\textbf{(C) }\\frac{61}{16}\\qquad \\textbf{(D) }\\frac{576}{225}\\qquad \\textbf{(E) }34$" + } + }, + { + "question": "Return your final response within \\boxed{}. The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means \"less than\"; \nthe symbol $>$ means \"greater than.\"\nThe set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that:\n$\\textbf{(A)}\\ x^2<49 \\qquad\\textbf{(B)}\\ x^2>1 \\qquad\\textbf{(C)}\\ 1$ means \"greater than.\"\nThe set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that:\n$\\textbf{(A)}\\ x^2<49 \\qquad\\textbf{(B)}\\ x^2>1 \\qquad\\textbf{(C)}\\ 11$ is an integer, then $\\log_an$, $\\log_bn$, $\\log_cn$ form a sequence \n$\\textbf{(A)}\\ \\text{which is a G.P} \\qquad$\n$\\textbf{(B)}\\ \\text{which is an arithmetic progression (A.P)} \\qquad$\n$\\textbf{(C)}\\ \\text{in which the reciprocals of the terms form an A.P} \\qquad$\n$\\textbf{(D)}\\ \\text{in which the second and third terms are the }n\\text{th powers of the first and second respectively} \\qquad$\n$\\textbf{(E)}\\ \\text{none of these}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2663", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $a$, $b$, and $c$ are in geometric progression (G.P.) with $1 < a < b < c$ and $n>1$ is an integer, then $\\log_an$, $\\log_bn$, $\\log_cn$ form a sequence \n$\\textbf{(A)}\\ \\text{which is a G.P} \\qquad$\n$\\textbf{(B)}\\ \\text{which is an arithmetic progression (A.P)} \\qquad$\n$\\textbf{(C)}\\ \\text{in which the reciprocals of the terms form an A.P} \\qquad$\n$\\textbf{(D)}\\ \\text{in which the second and third terms are the }n\\text{th powers of the first and second respectively} \\qquad$\n$\\textbf{(E)}\\ \\text{none of these}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Parallelogram $ABCD$ has area $1,\\!000,\\!000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? (A lattice point is any point whose coordinates are both integers.)\n$\\textbf{(A)}\\ 49\\qquad \\textbf{(B)}\\ 720\\qquad \\textbf{(C)}\\ 784\\qquad \\textbf{(D)}\\ 2009\\qquad \\textbf{(E)}\\ 2048$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2664", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Parallelogram $ABCD$ has area $1,\\!000,\\!000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? (A lattice point is any point whose coordinates are both integers.)\n$\\textbf{(A)}\\ 49\\qquad \\textbf{(B)}\\ 720\\qquad \\textbf{(C)}\\ 784\\qquad \\textbf{(D)}\\ 2009\\qquad \\textbf{(E)}\\ 2048$" + } + }, + { + "question": "Return your final response within \\boxed{}. In $\\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the \nmidpoints of $BF$. The point $E$ divides side $BC$ in the ratio\n$\\textbf{(A) }1:4\\qquad \\textbf{(B) }1:3\\qquad \\textbf{(C) }2:5\\qquad \\textbf{(D) }4:11\\qquad \\textbf{(E) }3:8$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2665", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. In $\\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the \nmidpoints of $BF$. The point $E$ divides side $BC$ in the ratio\n$\\textbf{(A) }1:4\\qquad \\textbf{(B) }1:3\\qquad \\textbf{(C) }2:5\\qquad \\textbf{(D) }4:11\\qquad \\textbf{(E) }3:8$" + } + }, + { + "question": "Return your final response within \\boxed{}. The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?\n\n\n$\\textbf{(A) }$The mean increases by $1$ and the median does not change.\n$\\textbf{(B) }$The mean increases by $1$ and the median increases by $1$.\n$\\textbf{(C) }$The mean increases by $1$ and the median increases by $5$.\n$\\textbf{(D) }$The mean increases by $5$ and the median increases by $1$.\n$\\textbf{(E) }$The mean increases by $5$ and the median increases by $5$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2666", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?\n\n\n$\\textbf{(A) }$The mean increases by $1$ and the median does not change.\n$\\textbf{(B) }$The mean increases by $1$ and the median increases by $1$.\n$\\textbf{(C) }$The mean increases by $1$ and the median increases by $5$.\n$\\textbf{(D) }$The mean increases by $5$ and the median increases by $1$.\n$\\textbf{(E) }$The mean increases by $5$ and the median increases by $5$." + } + }, + { + "question": "Return your final response within \\boxed{}. Find the smallest prime that is the fifth term of an increasing [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence), all four preceding terms also being [prime](https://artofproblemsolving.com/wiki/index.php/Prime_number).", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "29", + "index": "Sky-T1_10k_2667", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Find the smallest prime that is the fifth term of an increasing [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence), all four preceding terms also being [prime](https://artofproblemsolving.com/wiki/index.php/Prime_number)." + } + }, + { + "question": "Return your final response within \\boxed{}. Let $@$ denote the \"averaged with\" operation: $a @ b = \\frac{a+b}{2}$. Which of the following distributive laws hold for all numbers $x, y,$ and $z$? \\[\\text{I. x @ (y + z) = (x @ y) + (x @ z)}\\] \\[\\text{II. x + (y @ z) = (x + y) @ (x + z)}\\] \\[\\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}\\]\n$\\textbf{(A)}\\ \\text{I only} \\qquad\\textbf{(B)}\\ \\text{II only} \\qquad\\textbf{(C)}\\ \\text{III only} \\qquad\\textbf{(D)}\\ \\text{I and III only} \\qquad\\textbf{(E)}\\ \\text{II and III only}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2668", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Let $@$ denote the \"averaged with\" operation: $a @ b = \\frac{a+b}{2}$. Which of the following distributive laws hold for all numbers $x, y,$ and $z$? \\[\\text{I. x @ (y + z) = (x @ y) + (x @ z)}\\] \\[\\text{II. x + (y @ z) = (x + y) @ (x + z)}\\] \\[\\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}\\]\n$\\textbf{(A)}\\ \\text{I only} \\qquad\\textbf{(B)}\\ \\text{II only} \\qquad\\textbf{(C)}\\ \\text{III only} \\qquad\\textbf{(D)}\\ \\text{I and III only} \\qquad\\textbf{(E)}\\ \\text{II and III only}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The value of $\\frac{3}{a+b}$ when $a=4$ and $b=-4$ is: \n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ \\frac{3}{8}\\qquad\\textbf{(C)}\\ 0\\qquad\\textbf{(D)}\\ \\text{any finite number}\\qquad\\textbf{(E)}\\ \\text{meaningless}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2669", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The value of $\\frac{3}{a+b}$ when $a=4$ and $b=-4$ is: \n$\\textbf{(A)}\\ 3\\qquad\\textbf{(B)}\\ \\frac{3}{8}\\qquad\\textbf{(C)}\\ 0\\qquad\\textbf{(D)}\\ \\text{any finite number}\\qquad\\textbf{(E)}\\ \\text{meaningless}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The function $4x^2-12x-1$: \n$\\textbf{(A)}\\ \\text{always increases as }x\\text{ increases}\\\\ \\textbf{(B)}\\ \\text{always decreases as }x\\text{ decreases to 1}\\\\ \\textbf{(C)}\\ \\text{cannot equal 0}\\\\ \\textbf{(D)}\\ \\text{has a maximum value when }x\\text{ is negative}\\\\ \\textbf{(E)}\\ \\text{has a minimum value of-10}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2670", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The function $4x^2-12x-1$: \n$\\textbf{(A)}\\ \\text{always increases as }x\\text{ increases}\\\\ \\textbf{(B)}\\ \\text{always decreases as }x\\text{ decreases to 1}\\\\ \\textbf{(C)}\\ \\text{cannot equal 0}\\\\ \\textbf{(D)}\\ \\text{has a maximum value when }x\\text{ is negative}\\\\ \\textbf{(E)}\\ \\text{has a minimum value of-10}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Maya lists all the positive divisors of $2010^2$. She then randomly selects two distinct divisors from this list. Let $p$ be the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that exactly one of the selected divisors is a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). The probability $p$ can be expressed in the form $\\frac {m}{n}$, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find $m + n$.", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "107", + "index": "Sky-T1_10k_2671", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Maya lists all the positive divisors of $2010^2$. She then randomly selects two distinct divisors from this list. Let $p$ be the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that exactly one of the selected divisors is a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). The probability $p$ can be expressed in the form $\\frac {m}{n}$, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find $m + n$." + } + }, + { + "question": "Return your final response within \\boxed{}. $\\frac{1}{1+\\frac{1}{2+\\frac{1}{3}}}=$\n$\\text{(A)}\\ \\dfrac{1}{6} \\qquad \\text{(B)}\\ \\dfrac{3}{10} \\qquad \\text{(C)}\\ \\dfrac{7}{10} \\qquad \\text{(D)}\\ \\dfrac{5}{6} \\qquad \\text{(E)}\\ \\dfrac{10}{3}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2672", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. $\\frac{1}{1+\\frac{1}{2+\\frac{1}{3}}}=$\n$\\text{(A)}\\ \\dfrac{1}{6} \\qquad \\text{(B)}\\ \\dfrac{3}{10} \\qquad \\text{(C)}\\ \\dfrac{7}{10} \\qquad \\text{(D)}\\ \\dfrac{5}{6} \\qquad \\text{(E)}\\ \\dfrac{10}{3}$" + } + }, + { + "question": "Return your final response within \\boxed{}. On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible?\n$\\text{(A)}\\ 90 \\qquad \\text{(B)}\\ 91 \\qquad \\text{(C)}\\ 92 \\qquad \\text{(D)}\\ 95 \\qquad \\text{(E)}\\ 97$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "E", + "index": "Sky-T1_10k_2673", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible?\n$\\text{(A)}\\ 90 \\qquad \\text{(B)}\\ 91 \\qquad \\text{(C)}\\ 92 \\qquad \\text{(D)}\\ 95 \\qquad \\text{(E)}\\ 97$" + } + }, + { + "question": "Return your final response within \\boxed{}. The mean of three numbers is $10$ more than the least of the numbers and $15$\nless than the greatest. The median of the three numbers is $5$. What is their\nsum?\n$\\textbf{(A)}\\ 5\\qquad \\textbf{(B)}\\ 20\\qquad \\textbf{(C)}\\ 25\\qquad \\textbf{(D)}\\ 30\\qquad \\textbf{(E)}\\ 36$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2674", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The mean of three numbers is $10$ more than the least of the numbers and $15$\nless than the greatest. The median of the three numbers is $5$. What is their\nsum?\n$\\textbf{(A)}\\ 5\\qquad \\textbf{(B)}\\ 20\\qquad \\textbf{(C)}\\ 25\\qquad \\textbf{(D)}\\ 30\\qquad \\textbf{(E)}\\ 36$" + } + }, + { + "question": "Return your final response within \\boxed{}. What is the radius of a circle inscribed in a rhombus with diagonals of length $10$ and $24$?\n$\\mathrm{(A) \\ }4 \\qquad \\mathrm{(B) \\ }\\frac {58}{13} \\qquad \\mathrm{(C) \\ }\\frac{60}{13} \\qquad \\mathrm{(D) \\ }5 \\qquad \\mathrm{(E) \\ }6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2675", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. What is the radius of a circle inscribed in a rhombus with diagonals of length $10$ and $24$?\n$\\mathrm{(A) \\ }4 \\qquad \\mathrm{(B) \\ }\\frac {58}{13} \\qquad \\mathrm{(C) \\ }\\frac{60}{13} \\qquad \\mathrm{(D) \\ }5 \\qquad \\mathrm{(E) \\ }6$" + } + }, + { + "question": "Return your final response within \\boxed{}. A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 6$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2676", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 \\qquad \\textbf{(E)}\\ 6$" + } + }, + { + "question": "Return your final response within \\boxed{}. Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 5 \\qquad\\textbf{(C) } 9 \\qquad\\textbf{(D) } 12 \\qquad\\textbf{(E) } 15$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2677", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$\n$\\textbf{(A) } 3 \\qquad\\textbf{(B) } 5 \\qquad\\textbf{(C) } 9 \\qquad\\textbf{(D) } 12 \\qquad\\textbf{(E) } 15$" + } + }, + { + "question": "Return your final response within \\boxed{}. For any three real numbers $a$, $b$, and $c$, with $b\\neq c$, the operation $\\otimes$ is defined by:\n\\[\\otimes(a,b,c)=\\frac{a}{b-c}\\]\nWhat is $\\otimes(\\otimes(1,2,3),\\otimes(2,3,1),\\otimes(3,1,2))$?\n$\\mathrm{(A) \\ } -\\frac{1}{2}\\qquad \\mathrm{(B) \\ } -\\frac{1}{4} \\qquad \\mathrm{(C) \\ } 0 \\qquad \\mathrm{(D) \\ } \\frac{1}{4} \\qquad \\mathrm{(E) \\ } \\frac{1}{2}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "B", + "index": "Sky-T1_10k_2678", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. For any three real numbers $a$, $b$, and $c$, with $b\\neq c$, the operation $\\otimes$ is defined by:\n\\[\\otimes(a,b,c)=\\frac{a}{b-c}\\]\nWhat is $\\otimes(\\otimes(1,2,3),\\otimes(2,3,1),\\otimes(3,1,2))$?\n$\\mathrm{(A) \\ } -\\frac{1}{2}\\qquad \\mathrm{(B) \\ } -\\frac{1}{4} \\qquad \\mathrm{(C) \\ } 0 \\qquad \\mathrm{(D) \\ } \\frac{1}{4} \\qquad \\mathrm{(E) \\ } \\frac{1}{2}$" + } + }, + { + "question": "Return your final response within \\boxed{}. If $m>0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m=$\n$\\text{(A) } 1\\quad \\text{(B) } \\sqrt{2}\\quad \\text{(C) } \\sqrt{3}\\quad \\text{(D) } 2\\quad \\text{(E) } \\sqrt{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2679", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. If $m>0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m=$\n$\\text{(A) } 1\\quad \\text{(B) } \\sqrt{2}\\quad \\text{(C) } \\sqrt{3}\\quad \\text{(D) } 2\\quad \\text{(E) } \\sqrt{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?\n$\\textbf{(A) }1979\\qquad\\textbf{(B) }1980\\qquad\\textbf{(C) }1981\\qquad\\textbf{(D) }1982\\qquad \\textbf{(E) }1983$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "A", + "index": "Sky-T1_10k_2680", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?\n$\\textbf{(A) }1979\\qquad\\textbf{(B) }1980\\qquad\\textbf{(C) }1981\\qquad\\textbf{(D) }1982\\qquad \\textbf{(E) }1983$" + } + }, + { + "question": "Return your final response within \\boxed{}. Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB 5\\sqrt2$? \n\n\n$\\textbf{(A)}\\ \\frac{2-\\sqrt2}{2}\\qquad\\textbf{(B)}\\ \\frac{1}{3}\\qquad\\textbf{(C)}\\ \\frac{3-\\sqrt3}{3}\\qquad\\textbf{(D)}\\ \\frac{1}{2}\\qquad\\textbf{(E)}\\ \\frac{5-\\sqrt5}{5}$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "C", + "index": "Sky-T1_10k_2769", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Triangle $ABC$ is a right triangle with $\\angle ACB$ as its right angle, $m\\angle ABC = 60^\\circ$ , and $AB = 10$. Let $P$ be randomly chosen inside $ABC$ , and extend $\\overline{BP}$ to meet $\\overline{AC}$ at $D$. What is the probability that $BD > 5\\sqrt2$? \n\n\n$\\textbf{(A)}\\ \\frac{2-\\sqrt2}{2}\\qquad\\textbf{(B)}\\ \\frac{1}{3}\\qquad\\textbf{(C)}\\ \\frac{3-\\sqrt3}{3}\\qquad\\textbf{(D)}\\ \\frac{1}{2}\\qquad\\textbf{(E)}\\ \\frac{5-\\sqrt5}{5}$" + } + }, + { + "question": "Return your final response within \\boxed{}. Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes $5$ hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass in the highway (not in the station)?\n$\\text{(A)}\\ 5\\qquad\\text{(B)}\\ 6\\qquad\\text{(C)}\\ 9\\qquad\\text{(D)}\\ 10\\qquad\\text{(E)}\\ 11$", + "source": "Sky-T1_10k", + "category": "math", + "ground_truth": "D", + "index": "Sky-T1_10k_2770", + "multimodal_info": { + "image": [], + "audio": [] + }, + "meta_info": { + "solution": "Return your final response within \\boxed{}. Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes $5$ hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass in the highway (not in the station)?\n$\\text{(A)}\\ 5\\qquad\\text{(B)}\\ 6\\qquad\\text{(C)}\\ 9\\qquad\\text{(D)}\\ 10\\qquad\\text{(E)}\\ 11$" + } + }, + { + "question": "Return your final response within \\boxed{}. Which statement is correct?\n$\\mathrm{(A)\\ } \\text{If } x<0, \\text{then } x^2>x. \\qquad \\mathrm{(B) \\ } \\text{If } x^2>0, \\text{then } x>0.$\n$\\qquad \\mathrm{(C) \\ } \\text{If } x^2>x, \\text{then } x>0. \\qquad \\mathrm{(D) \\ } \\text{If } x^2>x, \\text{then } x<0.$\n$\\qquad \\mathrm{(E) \\ }\\text{If } x<1, \\text{then } x^2x. \\qquad \\mathrm{(B) \\ } \\text{If } x^2>0, \\text{then } x>0.$\n$\\qquad \\mathrm{(C) \\ } \\text{If } x^2>x, \\text{then } x>0. \\qquad \\mathrm{(D) \\ } \\text{If } x^2>x, \\text{then } x<0.$\n$\\qquad \\mathrm{(E) \\ }\\text{If } x<1, \\text{then } x^2