data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given the set $\\{1,2,3,5,8,13,21,34,55\\}$, calculate the number of integers between 3 and 89 that cannot be written as the sum of exactly two elements of the set. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "400",
"split": "train"
} |
math | [
{
"content": "Given $F_{1}$ and $F_{2}$ are the foci of the hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, a regular triangle $M F_{1} F_{2}$ is constructed with $F_{1} F_{2}$ as one side. If the midpoint of the side $M F_{1}$ lies on the hyperbola, what is the eccentricity of the hyperbola?... | {
"ground_truth": "$\\sqrt{3}+1$",
"style": "rule"
} | {
"index": "401",
"split": "train"
} |
math | [
{
"content": "$A B C D$ is a parallelogram satisfying $A B=7, B C=2$, and $\\angle D A B=120^{\\circ}$. Parallelogram $E C F A$ is contained in $A B C D$ and is similar to it. Find the ratio of the area of $E C F A$ to the area of $A B C D$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{39}{67}",
"style": "rule"
} | {
"index": "402",
"split": "train"
} |
math | [
{
"content": "A four-digit number has the following properties:\r\n\r\n(a) It is a perfect square;\r\n\r\n(b) Its first two digits are equal\r\n\r\n(c) Its last two digits are equal.\r\n\r\nFind all such four-digit numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7744",
"style": "rule"
} | {
"index": "403",
"split": "train"
} |
math | [
{
"content": "For any real number $t$ , let $\\lfloor t \\rfloor$ denote the largest integer $\\le t$ . Suppose that $N$ is the greatest integer such that $$ \\left \\lfloor \\sqrt{\\left \\lfloor \\sqrt{\\left \\lfloor \\sqrt{N} \\right \\rfloor}\\right \\rfloor}\\right \\rfloor = 4 $$ Find the sum of... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "404",
"split": "train"
} |
math | [
{
"content": "If the graph of the function $f(x) = (1-x^2)(x^2+ax+b)$ is symmetric about the line $x = -2$, then the maximum value of $f(x)$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "405",
"split": "train"
} |
math | [
{
"content": "Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "406",
"split": "train"
} |
math | [
{
"content": "On the island of Misfortune with a population of 96 people, the government decided to implement five reforms. Each reform is disliked by exactly half of the citizens. A citizen protests if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the govern... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "407",
"split": "train"
} |
math | [
{
"content": "Given the areas of three squares in the diagram, find the area of the triangle formed. The triangle shares one side with each of two squares and the hypotenuse with the third square.\n[asy]\n/* Modified AMC8-like Problem */\ndraw((0,0)--(10,0)--(10,10)--cycle);\ndraw((10,0)--(20,0)--(20,10)--(10,1... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "408",
"split": "train"
} |
math | [
{
"content": "Given that $C_{n}^{4}$, $C_{n}^{5}$, and $C_{n}^{6}$ form an arithmetic sequence, find the value of $C_{n}^{12}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "91",
"style": "rule"
} | {
"index": "409",
"split": "train"
} |
math | [
{
"content": "A market survey shows that the price $f(x)$ (in yuan) and the sales volume $g(x)$ (in units) of a certain product in the past $20$ days are both functions of time $x$ (in days), and the price satisfies $f(x)=20-\\frac{1}{2}|x-10|$, and the sales volume satisfies $g(x)=80-2x$, where $0\\leqslant x\... | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "410",
"split": "train"
} |
math | [
{
"content": "Person A arrives between 7:00 and 8:00, while person B arrives between 7:20 and 7:50. The one who arrives first waits for the other for 10 minutes, after which they leave. Calculate the probability that the two people will meet. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "411",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\\leqslant 1-f\\left(\\ln a-\\ln x\\right)$ always holds, then the minimum value of $a$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{e}",
"style": "rule"
} | {
"index": "412",
"split": "train"
} |
math | [
{
"content": "In the sequence \\(\\left\\{a_{n}\\right\\}\\), \\(a_{1} = -1\\), \\(a_{2} = 1\\), \\(a_{3} = -2\\). Given that for all \\(n \\in \\mathbf{N}_{+}\\), \\(a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{n} + a_{n+1} + a_{n+2} + a_{n+3}\\), and \\(a_{n+1} a_{n+2} a_{n+3} \\neq 1\\), find the sum of the first 4321... | {
"ground_truth": "-4321",
"style": "rule"
} | {
"index": "413",
"split": "train"
} |
math | [
{
"content": "Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\\cdot 2 \\cdot 3\\cdot ... \\cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \\cdot 2^2 \\cdot 3^2\\cdot ... \\cdot 50^2$. Please output the final... | {
"ground_truth": "32768",
"style": "rule"
} | {
"index": "414",
"split": "train"
} |
math | [
{
"content": "Cookie Monster now finds a bigger cookie with the boundary described by the equation $x^2 + y^2 - 8 = 2x + 4y$. He wants to know both the radius and the area of this cookie to determine if it's enough for his dessert. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13\\pi",
"style": "rule"
} | {
"index": "415",
"split": "train"
} |
math | [
{
"content": "A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "416",
"split": "train"
} |
math | [
{
"content": "What integer value of $n$ will satisfy $n + 10 > 11$ and $-4n > -12$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "417",
"split": "train"
} |
math | [
{
"content": "Consider the 100th, 101st, and 102nd rows of Pascal's triangle, denoted as sequences $(p_i)$, $(q_i)$, and $(r_i)$ respectively. Calculate:\n\\[\n\\sum_{i = 0}^{100} \\frac{q_i}{r_i} - \\sum_{i = 0}^{99} \\frac{p_i}{q_i}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "418",
"split": "train"
} |
math | [
{
"content": "Given \\( x^{2} + y^{2} - 2x - 2y + 1 = 0 \\) where \\( x, y \\in \\mathbb{R} \\), find the minimum value of \\( F(x, y) = \\frac{x + 1}{y} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3/4",
"style": "rule"
} | {
"index": "419",
"split": "train"
} |
math | [
{
"content": "In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "420",
"split": "train"
} |
math | [
{
"content": "On the hypotenuse \\( AB \\) of a right triangle \\( ABC \\), square \\( ABDE \\) is constructed externally with \\( AC=2 \\) and \\( BC=5 \\). In what ratio does the angle bisector of angle \\( C \\) divide side \\( DE \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "2 : 5",
"style": "rule"
} | {
"index": "421",
"split": "train"
} |
math | [
{
"content": "Given that \\( x - \\frac{1}{x} = 5 \\), find the value of \\( x^4 - \\frac{1}{x^4} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "727",
"style": "rule"
} | {
"index": "422",
"split": "train"
} |
math | [
{
"content": "The numbers in the sequence $101$, $104$, $109$, $116$,$\\ldots$ are of the form $a_n=100+n^2$, where $n=1,2,3,\\ldots$ For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers. Please output the fina... | {
"ground_truth": "401",
"style": "rule"
} | {
"index": "423",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system \\(xOy\\), the equation of the ellipse \\(C\\) is given by the parametric form:\n\\[\n\\begin{cases}\nx=5\\cos\\varphi \\\\\ny=3\\sin\\varphi\n\\end{cases}\n\\]\nwhere \\(\\varphi\\) is the parameter.\n(I) Find the general equation of the straight line \\(l\\) th... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "424",
"split": "train"
} |
math | [
{
"content": "Determine how many prime dates occurred in 2008, a leap year. A \"prime date\" is when both the month and the day are prime numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "425",
"split": "train"
} |
math | [
{
"content": "A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a $\\textit{lune}$. Determine the area of this lune. Express your answer in terms of $\\pi$ and in simplest radical form.\n\... | {
"ground_truth": "\\frac{\\sqrt{3}}{4} - \\frac{1}{24}\\pi",
"style": "rule"
} | {
"index": "426",
"split": "train"
} |
math | [
{
"content": "In a right triangle $PQR$, medians are drawn from $P$ and $Q$ to divide segments $\\overline{QR}$ and $\\overline{PR}$ in half, respectively. If the length of the median from $P$ to the midpoint of $QR$ is $8$ units, and the median from $Q$ to the midpoint of $PR$ is $4\\sqrt{5}$ units, find the l... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "427",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\ln x+ \\frac {1}{2}ax^{2}-x-m$ ($m\\in\\mathbb{Z}$). \n(I) If $f(x)$ is an increasing function, find the range of values for $a$; \n(II) If $a < 0$, and $f(x) < 0$ always holds, find the minimum value of $m$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "428",
"split": "train"
} |
math | [
{
"content": "Find the limit, when $n$ tends to the infinity, of $$ \\frac{\\sum_{k=0}^{n} {{2n} \\choose {2k}} 3^k} {\\sum_{k=0}^{n-1} {{2n} \\choose {2k+1}} 3^k} $$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "429",
"split": "train"
} |
math | [
{
"content": "Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y = g(x) is symmetric about the line x = 2, g(2) = 4, determine the value of \\sum _{k=1}^{22}f(k). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-24",
"style": "rule"
} | {
"index": "430",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x)$ and its derivative $f'(x)$ have a domain of $R$, and $f(x+2)$ is an odd function, ${f'}(2-x)+{f'}(x)=2$, ${f'}(2)=2$, then $\\sum_{i=1}^{50}{{f'}}(i)=$____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "431",
"split": "train"
} |
math | [
{
"content": "Find the number of four-digit numbers with distinct digits, formed using the digits 0, 1, 2, ..., 9, such that the absolute difference between the units and hundreds digit is 8. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "154",
"style": "rule"
} | {
"index": "432",
"split": "train"
} |
math | [
{
"content": "A man chooses two positive integers \\( m \\) and \\( n \\). He defines a positive integer \\( k \\) to be good if a triangle with side lengths \\( \\log m \\), \\( \\log n \\), and \\( \\log k \\) exists. He finds that there are exactly 100 good numbers. Find the maximum possible value of \\( mn ... | {
"ground_truth": "134",
"style": "rule"
} | {
"index": "433",
"split": "train"
} |
math | [
{
"content": "What is the smallest natural number whose digits in decimal representation are either 0 or 1 and which is divisible by 225? (China Junior High School Mathematics League, 1989) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11111111100",
"style": "rule"
} | {
"index": "434",
"split": "train"
} |
math | [
{
"content": "Square the numbers \\( a = 101 \\) and \\( b = 10101 \\). Find the square root of the number \\( c = 102030405060504030201 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10101010101",
"style": "rule"
} | {
"index": "435",
"split": "train"
} |
math | [
{
"content": "In a physical education class, students line up in four rows to do exercises. One particular class has over 30 students, with three rows having the same number of students and one row having one more student than the other three rows. What is the smallest possible class size for this physical educ... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "436",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=a\\cos (x+\\frac{\\pi }{6})$, its graph passes through the point $(\\frac{\\pi }{2}, -\\frac{1}{2})$.\n\n(1) Find the value of $a$;\n\n(2) If $\\sin \\theta =\\frac{1}{3}, 0 < \\theta < \\frac{\\pi }{2}$, find $f(\\theta ).$ Please output the final answer within \\boxed{}.... | {
"ground_truth": "\\frac{2\\sqrt{6}-1}{6}",
"style": "rule"
} | {
"index": "437",
"split": "train"
} |
math | [
{
"content": "Draw a $2004 \\times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "561",
"style": "rule"
} | {
"index": "438",
"split": "train"
} |
math | [
{
"content": "Let $x$ be a positive real number. Find the minimum value of $4x^5 + 5x^{-4}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "439",
"split": "train"
} |
math | [
{
"content": "Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. \nA third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\\triangle APR$ is Please output the final an... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "440",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer with exactly 18 positive divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "288",
"style": "rule"
} | {
"index": "441",
"split": "train"
} |
math | [
{
"content": "Farmer James wishes to cover a circle with circumference $10 \\pi$ with six different types of colored arcs. Each type of arc has radius 5, has length either $\\pi$ or $2 \\pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to complet... | {
"ground_truth": "93",
"style": "rule"
} | {
"index": "442",
"split": "train"
} |
math | [
{
"content": "Given a circle \\\\(O: x^2 + y^2 = 2\\\\) and a line \\\\(l: y = kx - 2\\\\).\n\n\\\\((1)\\\\) If line \\\\(l\\\\) intersects circle \\\\(O\\\\) at two distinct points \\\\(A\\\\) and \\\\(B\\\\), and \\\\(\\angle AOB = \\frac{\\pi}{2}\\\\), find the value of \\\\(k\\\\).\n\n\\\\((2)\\\\) If \\\\(... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "443",
"split": "train"
} |
math | [
{
"content": "A tetrahedron \\(ABCD\\) has six edges with lengths \\(7, 13, 18, 27, 36, 41\\) units. If the length of \\(AB\\) is 41 units, then the length of \\(CD\\) is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "444",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a > c$. Given that $\\overrightarrow{BA} \\cdot \\overrightarrow{BC} = 2$ and $\\cos B = \\frac{1}{3}$, with $b=3$, find:\n\n$(1)$ The values of $a$ and $c$; $(2)$ The value of $\\cos(B - C)$. Plea... | {
"ground_truth": "\\frac{23}{27}",
"style": "rule"
} | {
"index": "445",
"split": "train"
} |
math | [
{
"content": "The sum of the ages of three people A, B, and C, denoted as \\(x, y, z\\), is 120, with \\(x, y, z \\in (20, 60)\\). How many ordered triples \\((x, y, z)\\) satisfy this condition? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1198",
"style": "rule"
} | {
"index": "446",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), side \\(AB = 6\\), \\(\\angle BAC = 30^\\circ\\), and the radius of the circumscribed circle is 5. Find side \\(AC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{3} + 4",
"style": "rule"
} | {
"index": "447",
"split": "train"
} |
math | [
{
"content": "How many numbers are in the list $-50, -44, -38, \\ldots, 68, 74$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "448",
"split": "train"
} |
math | [
{
"content": "Find the number of square units in the area of the triangle.\n\n[asy]size(125);\ndraw( (-10,-2) -- (2,10), Arrows);\ndraw( (0,-2)-- (0,10) ,Arrows);\ndraw( (5,0) -- (-10,0),Arrows);\n\nlabel(\"$l$\",(2,10), NE);\nlabel(\"$x$\", (5,0) , E);\nlabel(\"$y$\", (0,-2) , S);\nfilldraw( (-8,0) -- (0,8) --... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "449",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\angle A = 60^\\circ$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\\frac{MH + NH}{OH}$. Please output the final answe... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "450",
"split": "train"
} |
math | [
{
"content": "Let $M$ be the number of positive integers that are less than or equal to $2050$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "374",
"style": "rule"
} | {
"index": "451",
"split": "train"
} |
math | [
{
"content": "In the expansion of $(x-y)^{8}(x+y)$, the coefficient of $x^{7}y^{2}$ is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "452",
"split": "train"
} |
math | [
{
"content": "What is the number of centimeters in the length of $EF$ if $AB\\parallel CD\\parallel EF$?\n\n[asy]\n\nsize(4cm,4cm);\npair A,B,C,D,E,F,X;\n\nA=(0,1);\nB=(1,1);\nC=(1,0);\nX=(0,0);\nD=(1/3)*C+(2/3)*X;\n\ndraw (A--B--C--D);\ndraw(D--B);\ndraw(A--C);\n\nE=(0.6,0.4);\nF=(1,0.4);\n\ndraw(E--F);\n\nlab... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "453",
"split": "train"
} |
math | [
{
"content": "Suppose $x$ is in the interval $[0, \\pi/2]$ and $\\log_{24\\sin x} (24\\cos x)=\\frac{3}{2}$. Find $24\\cot^2 x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "192",
"style": "rule"
} | {
"index": "454",
"split": "train"
} |
math | [
{
"content": "Expand the product ${(x+2)(x+5)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x^2 + 7x + 10",
"style": "rule"
} | {
"index": "455",
"split": "train"
} |
math | [
{
"content": "Given that the monogram of a person's first, middle, and last initials is in alphabetical order with no letter repeated, and the last initial is 'M', calculate the number of possible monograms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "456",
"split": "train"
} |
math | [
{
"content": "A trainee cook took two buckets of unpeeled potatoes and peeled everything in an hour. Meanwhile, $25\\%$ of the potatoes went into peels. How much time did it take for him to collect exactly one bucket of peeled potatoes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "457",
"split": "train"
} |
math | [
{
"content": "Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on George's dice? Please o... | {
"ground_truth": "\\frac{19}{40}",
"style": "rule"
} | {
"index": "458",
"split": "train"
} |
math | [
{
"content": "Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He chooses a card at random. What is the probability that the number on the card he picks is a multiple of 4, 5 or 6? Express your answer as a reduced fraction. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{7}{15}",
"style": "rule"
} | {
"index": "459",
"split": "train"
} |
math | [
{
"content": "Two distinct positive integers $a$ and $b$ are factors of 48. If $a\\cdot b$ is not a factor of 48, what is the smallest possible value of $a\\cdot b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "460",
"split": "train"
} |
math | [
{
"content": "To factorize the quadratic trinomial $x^{2}+4x-5$, we can first add $4$ to $x^{2}+4x$ to make it a perfect square trinomial. Then, subtract $4$ so that the value of the entire expression remains unchanged. Therefore, we have: $x^{2}+4x-5=x^{2}+4x+4-4-5=\\left(x+2\\right)^{2}-9=\\left(x+2+3\\right)... | {
"ground_truth": "-24",
"style": "rule"
} | {
"index": "461",
"split": "train"
} |
math | [
{
"content": "Let $f(n)$ be the base-10 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. Express $\\frac{f(n)}{\\log_{10} 2}$ in terms of $n$. Recall that Pascal's triangle begins\n\n\\begin{tabular}{rccccccccc}\n$n=0$:& & & & & 1\\\\\\noalign{\\smallskip\\smallskip}\n$n=1$:& & & & 1 ... | {
"ground_truth": "n",
"style": "rule"
} | {
"index": "462",
"split": "train"
} |
math | [
{
"content": "A company offers its employees a salary increase, provided they increase their work productivity by 2% per week. If the company operates 5 days a week, by what percentage per day must employees increase their productivity to achieve the desired goal? Please output the final answer within \\boxed{}... | {
"ground_truth": "0.4",
"style": "rule"
} | {
"index": "463",
"split": "train"
} |
math | [
{
"content": "Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. How many minutes did Xiaoming spend on the bus t... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "464",
"split": "train"
} |
math | [
{
"content": "At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? Please o... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "465",
"split": "train"
} |
math | [
{
"content": "Given that $\\cos(x - \\frac{\\pi}{4}) = \\frac{\\sqrt{2}}{10}$, with $x \\in (\\frac{\\pi}{2}, \\frac{3\\pi}{4})$.\n(1) Find the value of $\\sin x$;\n(2) Find the value of $\\cos(2x - \\frac{\\pi}{3})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{7 + 24\\sqrt{3}}{50}",
"style": "rule"
} | {
"index": "466",
"split": "train"
} |
math | [
{
"content": "Let $x$ be inversely proportional to $y$. If $x = 4$ when $y = 2$, find the value of $x$ when $y = -3$ and when $y = 6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "467",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the curve $C_{1}$ is defined by $\\begin{cases} x=-2+\\cos \\alpha \\\\ y=-1+\\sin \\alpha \\end{cases}$ (where $\\alpha$ is a parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, t... | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "468",
"split": "train"
} |
math | [
{
"content": "Given that Joy has 40 thin rods, one each of every integer length from 1 cm through 40 cm, with rods of lengths 4 cm, 9 cm, and 18 cm already placed on a table, determine how many of the remaining rods can be chosen as the fourth rod to form a quadrilateral with positive area. Please output the fi... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "469",
"split": "train"
} |
math | [
{
"content": "Given set $A=\\{a-2, 12, 2a^2+5a\\}$, and $-3$ belongs to $A$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "470",
"split": "train"
} |
math | [
{
"content": "Two circles are centered at the origin, as shown. The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=3$, what is the value of $k$?\n\n[asy]\nunitsize(0.2 cm);\n\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\ndotfactor=4;\ndraw(Circle((0,0),7)); draw... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "471",
"split": "train"
} |
math | [
{
"content": "A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "472",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are positive integers with $3^{x} 5^{y} = 225$, what is the value of $x + y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "473",
"split": "train"
} |
math | [
{
"content": "A boy is riding a scooter from one bus stop to another and looking in the mirror to see if a bus appears behind him. As soon as the boy notices the bus, he can change the direction of his movement. What is the maximum distance between the bus stops so that the boy is guaranteed not to miss the bus... | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": "474",
"split": "train"
} |
math | [
{
"content": "The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1 is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4436",
"style": "rule"
} | {
"index": "475",
"split": "train"
} |
math | [
{
"content": "In recent years, China's scientific and technological achievements have been remarkable. The Beidou-3 global satellite navigation system has been operational for many years. The Beidou-3 global satellite navigation system consists of 24 medium Earth orbit satellites, 3 geostationary Earth orbit sa... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "476",
"split": "train"
} |
math | [
{
"content": "The sum of the coefficients of the expanded form of $(x+ \\frac{a}{x})(2x- \\frac{1}{x})^{5}$ is 2. Find the constant term in the expanded form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "477",
"split": "train"
} |
math | [
{
"content": "It is known that \\(4 \\operatorname{tg}^{2} Y + 4 \\operatorname{ctg}^{2} Y - \\frac{1}{\\sin ^{2} \\gamma} - \\frac{1}{\\cos ^{2} \\gamma} = 17\\). Find the value of the expression \\(\\cos ^{2} Y - \\cos ^{4} \\gamma\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{3}{25}",
"style": "rule"
} | {
"index": "478",
"split": "train"
} |
math | [
{
"content": "The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[17]{3}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\\cdots a_k$ is an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "479",
"split": "train"
} |
math | [
{
"content": "Consider a revised dataset given in the following stem-and-leaf plot, where $7|1$ represents 71:\n\n\\begin{tabular}{|c|c|}\\hline\n\\textbf{Tens} & \\textbf{Units} \\\\ \\hline\n2 & $0 \\hspace{2mm} 0 \\hspace{2mm} 1 \\hspace{2mm} 1 \\hspace{2mm} 2$ \\\\ \\hline\n3 & $3 \\hspace{2mm} 6 \\hspace{2... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "480",
"split": "train"
} |
math | [
{
"content": "A local government intends to encourage entrepreneurship by rewarding newly established small and micro enterprises with an annual output value between 500,000 and 5,000,000 RMB. The reward scheme follows these principles: The bonus amount $y$ (in ten thousand RMB) increases with the yearly output... | {
"ground_truth": "315",
"style": "rule"
} | {
"index": "481",
"split": "train"
} |
math | [
{
"content": "What integer is closest to the value of $\\sqrt[3]{6^3+8^3}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "482",
"split": "train"
} |
math | [
{
"content": "Given a pyramid-like structure with a rectangular base consisting of $4$ apples by $7$ apples, each apple above the first level resting in a pocket formed by four apples below, and the stack topped off with a single row of apples, determine the total number of apples in the stack. Please output th... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "483",
"split": "train"
} |
math | [
{
"content": "If $x$ is a number less than -2, which of the following expressions has the least value: $x$, $x+2$, $\\frac{1}{2}x$, $x-2$, or $2x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2x",
"style": "rule"
} | {
"index": "484",
"split": "train"
} |
math | [
{
"content": "How many distinct diagonals of a convex heptagon (7-sided polygon) can be drawn? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "485",
"split": "train"
} |
math | [
{
"content": "What is the probability that the arrow stops on a shaded region if a circular spinner is divided into six regions, four regions each have a central angle of $x^{\\circ}$, and the remaining regions have central angles of $20^{\\circ}$ and $140^{\\circ}$? Please output the final answer within \\boxe... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "486",
"split": "train"
} |
math | [
{
"content": "Given that point $P$ is on curve $C_1: y^2 = 8x$ and point $Q$ is on curve $C_2: (x-2)^2 + y^2 = 1$. If $O$ is the coordinate origin, find the maximum value of $\\frac{|OP|}{|PQ|}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4\\sqrt{7}}{7}",
"style": "rule"
} | {
"index": "487",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that is both a multiple of $7$ and a multiple of $4$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "488",
"split": "train"
} |
math | [
{
"content": "An eight-sided die is rolled seven times. Find the probability of rolling at least a seven at least six times. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{2048}",
"style": "rule"
} | {
"index": "489",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\frac{x^2}{1+x^2}$.\n\n$(1)$ Calculate the values of $f(2) + f\\left(\\frac{1}{2}\\right)$, $f(3) + f\\left(\\frac{1}{3}\\right)$, $f(4) + f\\left(\\frac{1}{4}\\right)$, and infer a general conclusion (proof not required);\n\n$(2)$ Calculate the value of $2f(2) + 2f(3) ... | {
"ground_truth": "4032",
"style": "rule"
} | {
"index": "490",
"split": "train"
} |
math | [
{
"content": "Suppose that $a$ and $b$ are positive integers for which $a$ has $3$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, then what is the least possible value of $b?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "491",
"split": "train"
} |
math | [
{
"content": "What integer $n$ satisfies $0 \\leq n < 151$ and $$100n \\equiv 93 \\pmod {151}~?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "492",
"split": "train"
} |
math | [
{
"content": "The point $O$ is the center of the circle circumscribed about $\\triangle ABC$, with $\\angle BOC = 120^{\\circ}$ and $\\angle AOB =\n140^{\\circ}$, as shown. What is the degree measure of $\\angle\nABC$?\n\n[asy]\npair A,B,C;\ndraw(Circle((0,0),20),linewidth(0.7));\nlabel(\"$O$\",(0,0),S);\nA=(-1... | {
"ground_truth": "50^{\\circ}",
"style": "rule"
} | {
"index": "493",
"split": "train"
} |
math | [
{
"content": "The quadratic $x^2-3x+9=x+41$ has two solutions. What is the positive difference between these solutions? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "494",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression:\n\\[4(1+4(1+4(1+4(1+4(1+4(1+4(1+4(1))))))))\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "87380",
"style": "rule"
} | {
"index": "495",
"split": "train"
} |
math | [
{
"content": "Let $s(n)$ be the number of 1's in the binary representation of $n$ . Find the number of ordered pairs of integers $(a,b)$ with $0 \\leq a < 64, 0 \\leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$ .\n\n*Author:Anderson Wang* Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "1458",
"style": "rule"
} | {
"index": "496",
"split": "train"
} |
math | [
{
"content": "Given a circle $C: (x-3)^{2}+(y-4)^{2}=1$, and points $A(-1,0)$, $B(1,0)$, let $P$ be a moving point on the circle, then the maximum and minimum values of $d=|PA|^{2}+|PB|^{2}$ are \\_\\_\\_\\_\\_\\_ and \\_\\_\\_\\_\\_\\_ respectively. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "497",
"split": "train"
} |
math | [
{
"content": "Solve the following equation and provide its root. If the equation has multiple roots, provide their product.\n\n\\[ \\sqrt{2 x^{2} + 8 x + 1} - x = 3 \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "498",
"split": "train"
} |
math | [
{
"content": "Compute $\\dbinom{5}{3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "499",
"split": "train"
} |
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