data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "A said: \"I am 10 years old, 2 years younger than B, and 1 year older than C.\"\nB said: \"I am not the youngest, C and I have a 3-year difference, and C is 13 years old.\"\nC said: \"I am younger than A, A is 11 years old, and B is 3 years older than A.\"\nAmong the three statements made by each ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "1900",
"split": "train"
} |
math | [
{
"content": "Three numbers $a, b,$ and $c$ were written on a board. They were erased and replaced with the numbers $a-1, b+1,$ and $c^2$. After this, it turned out that the same numbers were on the board as initially (possibly in a different order). What values can the number $a$ take, given that the sum of th... | {
"ground_truth": "1004",
"style": "rule"
} | {
"index": "1901",
"split": "train"
} |
math | [
{
"content": "The secret object is a rectangle measuring $200 \\times 300$ meters. Outside the object, there is one guard stationed at each of its four corners. An intruder approached the perimeter of the secret object from the outside, and all the guards ran towards the intruder using the shortest paths along ... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "1902",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = \\cos(x + \\theta) + \\sqrt{2}\\sin(x + \\phi)$ be an even function, where $\\theta$ and $\\phi$ are acute angles, and $\\cos\\theta = \\frac{\\sqrt{6}}{3}\\sin\\phi$. Then, evaluate the value of $\\theta + \\phi$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{7\\pi}{12}",
"style": "rule"
} | {
"index": "1903",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=x^{3}+x+1$. Suppose $g$ is a cubic polynomial such that $g(0)=-1$, and the roots of $g$ are the squares of the roots of $f$. Find $g(9)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "899",
"style": "rule"
} | {
"index": "1904",
"split": "train"
} |
math | [
{
"content": "Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12.5\\%",
"style": "rule"
} | {
"index": "1905",
"split": "train"
} |
math | [
{
"content": "The diagram shows an arc \\( PQ \\) of a circle with center \\( O \\) and radius 8. Angle \\( QOP \\) is a right angle, the point \\( M \\) is the midpoint of \\( OP \\), and \\( N \\) lies on the arc \\( PQ \\) so that \\( MN \\) is perpendicular to \\( OP \\). Which of the following is closest t... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "1906",
"split": "train"
} |
math | [
{
"content": "Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24, q(0)=30$, and $p(q(x))=q(p(x))$ for all real numbers $x$. Find the ordered pair $(p(3), q(6))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(3,-24)",
"style": "rule"
} | {
"index": "1907",
"split": "train"
} |
math | [
{
"content": "Determine the number of ordered pairs $(a,b)$ of integers such that $\\log_a b + 6\\log_b a=5, 2 \\leq a \\leq 2005,$ and $2 \\leq b \\leq 2005.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "1908",
"split": "train"
} |
math | [
{
"content": "Given that Chelsea is leading by 60 points halfway through a 120-shot archery competition, each shot can score 10, 7, 3, or 0 points, and Chelsea always scores at least 3 points. If Chelsea's next \\(n\\) shots are all for 10 points, she will secure her victory regardless of her opponent's scoring... | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "1909",
"split": "train"
} |
math | [
{
"content": "The numbers $1000^{2}, 1001^{2}, 1002^{2}, \\ldots$ have their last two digits discarded. How many of the first terms of the resulting sequence form an arithmetic progression? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "1910",
"split": "train"
} |
math | [
{
"content": "The graph of the function $f(x)=\\sin({ωx-\\frac{π}{6}})$, where $0<ω<6$, is shifted to the right by $\\frac{π}{6}$ units to obtain the graph of the function $g(x)$. If $\\left(0,\\frac{π}{ω}\\right)$ is a monotone interval of $g(x)$, and $F(x)=f(x)+g(x)$, determine the maximum value of $F(x)$. Pl... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "1911",
"split": "train"
} |
math | [
{
"content": "How many ways are there to choose 4 cards from a standard deck of 52 cards, where two cards come from one suit and the other two each come from different suits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "158184",
"style": "rule"
} | {
"index": "1912",
"split": "train"
} |
math | [
{
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 2, \\pi, \\frac{\\pi}{4} \\right)$ in spherical coordinates to rectangular coordinates. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-\\sqrt{2}, 0, \\sqrt{2})",
"style": "rule"
} | {
"index": "1913",
"split": "train"
} |
math | [
{
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $C$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, and point $A(1, \\frac{3}{2})$ on ellipse $C$ has a sum of distances to these two points equal to $4$.\n(I) Find the equation of the ellipse $C$ and the coordinates... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "1914",
"split": "train"
} |
math | [
{
"content": "The inclination angle of the line $\\sqrt{3}x + y - 1 = 0$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "1915",
"split": "train"
} |
math | [
{
"content": "Given that Ben spent some amount of money and David spent $0.5 less for each dollar Ben spent, and Ben paid $16.00 more than David, determine the total amount they spent together in the bagel store. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48.00",
"style": "rule"
} | {
"index": "1916",
"split": "train"
} |
math | [
{
"content": "When the product of $\\underbrace{\\frac{2}{3} \\times \\frac{2}{3} \\times \\cdots \\times \\frac{2}{3}}_{10}$ is written as a decimal, what are the first two decimal places? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.01",
"style": "rule"
} | {
"index": "1917",
"split": "train"
} |
math | [
{
"content": "The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{3}<x<\\frac{1}{2}",
"style": "rule"
} | {
"index": "1918",
"split": "train"
} |
math | [
{
"content": "Given \\( 0 \\leq m-n \\leq 1 \\) and \\( 2 \\leq m+n \\leq 4 \\), when \\( m - 2n \\) reaches its maximum value, what is the value of \\( 2019m + 2020n \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2019",
"style": "rule"
} | {
"index": "1919",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $(xOy)$, the equation of circle $C$ is $((x-4)^{2}+y^{2}=1)$. If there exists at least one point on the line $y=kx-3$ such that a circle with this point as the center and $2$ as the radius intersects with circle $C$, then the maximum value of $k$ is _____. Please... | {
"ground_truth": "\\frac{24}{7}",
"style": "rule"
} | {
"index": "1920",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row t... | {
"ground_truth": "2005!",
"style": "rule"
} | {
"index": "1921",
"split": "train"
} |
math | [
{
"content": "Two right triangles $\\triangle ABC$ and $\\triangle ABD$ share a side $AB$ in such way that $AB=8$, $BC=12$, and $BD=10$. Let $E$ be a point on $BC$ such that $BE=9$. Determine the area of $\\triangle ABE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "1922",
"split": "train"
} |
math | [
{
"content": "If $$1+22+333+4444+55555+666666+7777777+88888888$$is congruent to $n$ modulo 9, where $0\\le n<9$, then what is the value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "1923",
"split": "train"
} |
math | [
{
"content": "Evaluate the ratio $\\frac{10^{3000} + 10^{3004}}{10^{3001} + 10^{3003}}$ and determine which whole number it is closest to. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "1924",
"split": "train"
} |
math | [
{
"content": "A church rings its bells every 15 minutes, the school rings its bells every 20 minutes and the day care center rings its bells every 25 minutes. If they all ring their bells at noon on the same day, at what time will they next all ring their bells together? (Answer in the form AB:CD without am or... | {
"ground_truth": "05\\!:\\!00",
"style": "rule"
} | {
"index": "1925",
"split": "train"
} |
math | [
{
"content": "How many integers between $1000$ and $9999$ have four distinct digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4536",
"style": "rule"
} | {
"index": "1926",
"split": "train"
} |
math | [
{
"content": "An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, find the ratio of the shorter side to the longer side of this frame... | {
"ground_truth": "2:3",
"style": "rule"
} | {
"index": "1927",
"split": "train"
} |
math | [
{
"content": "In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ What is the area of $\\triangle ABD?$ \n\n[asy]\ndraw((0,0)--(2,5)--(8,5)--(15,0)--cycle);\ndraw((8,5)--(0,0));\nlabel(\"$D$\",(0,0),W);\nlabel(\"$A$\",(2,5),NW);\nlabel(\"$B$\",(8,5),NE);\nlabel(\"... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "1928",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that satisfies the congruence $4x \\equiv 13 \\pmod{27}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "1929",
"split": "train"
} |
math | [
{
"content": "Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to color the dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.) Please output the fi... | {
"ground_truth": "7983360",
"style": "rule"
} | {
"index": "1930",
"split": "train"
} |
math | [
{
"content": "Find the remainder when $x^3$ is divided by $x^2 + 5x + 1.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24x + 5",
"style": "rule"
} | {
"index": "1931",
"split": "train"
} |
math | [
{
"content": "Approximate the reading indicated by the arrow in the diagram of a measuring device. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42.3",
"style": "rule"
} | {
"index": "1932",
"split": "train"
} |
math | [
{
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(B\\) and \\(C\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \\(5 \\sqrt{17}\\). The center of a third sphere with a radius of 8 is at point \\... | {
"ground_truth": "2\\sqrt{19}",
"style": "rule"
} | {
"index": "1933",
"split": "train"
} |
math | [
{
"content": "If $\\log_2(\\log_2(\\log_2(x)))=2$, then how many digits are in the base-ten representation for x? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "1934",
"split": "train"
} |
math | [
{
"content": "Let $x$ be a multiple of $7200$. Determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+5)(4x+7)$ and $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "1935",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,$ the midpoint of $\\overline{BC}$ is $(1,5,-1),$ the midpoint of $\\overline{AC}$ is $(0,4,-2),$ and the midpoint of $\\overline{AB}$ is $(2,3,4).$ Find the coordinates of vertex $A.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1, 2, 3)",
"style": "rule"
} | {
"index": "1936",
"split": "train"
} |
math | [
{
"content": "Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1. When the smallest number is removed, the average is increased by 1. When both the largest and the smallest numbers are removed, the average of the remaining four numbers is 20. Find the... | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "1937",
"split": "train"
} |
math | [
{
"content": "A two-digit positive integer is said to be $cuddly$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "1938",
"split": "train"
} |
math | [
{
"content": "Consider a string of $n$ $7$s, $7777\\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. How many values of $n$ are possible if the inserted $+$ signs create a sum of $7350$ using groups of $7$s, $77$s, $777$s, and possibly $7777$s? Please output the final answer with... | {
"ground_truth": "117",
"style": "rule"
} | {
"index": "1939",
"split": "train"
} |
math | [
{
"content": "The train arrives at a station randomly between 1:00 PM and 3:00 PM and waits for 30 minutes before departing. If John also arrives randomly at the station within the same time period, what is the probability that he will find the train at the station? Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{7}{32}",
"style": "rule"
} | {
"index": "1940",
"split": "train"
} |
math | [
{
"content": "If $a$ and $b$ are positive integers for which $ab - 6a + 5b = 373$, what is the minimal possible value of $|a - b|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "1941",
"split": "train"
} |
math | [
{
"content": "A shape was cut out from a regular hexagon as shown in the picture. The marked points on both the perimeter and inside the hexagon divide the respective line segments into quarters.\n\nWhat is the ratio of the areas of the original hexagon to the cut-out shape? Please output the final answer withi... | {
"ground_truth": "4:1",
"style": "rule"
} | {
"index": "1942",
"split": "train"
} |
math | [
{
"content": "If a four-digit number $\\overline{a b c d}$ meets the condition $a + b = c + d$, it is called a \"good number.\" For instance, 2011 is a \"good number.\" How many \"good numbers\" are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "615",
"style": "rule"
} | {
"index": "1943",
"split": "train"
} |
math | [
{
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2028",
"style": "rule"
} | {
"index": "1944",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\\begin{cases} x=t \\\\ y= \\sqrt {2}+2t \\end{cases}$ (where $t$ is the parameter), with point $O$ as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve $C$ is $\\rho=4\\cos... | {
"ground_truth": "\\frac {3 \\sqrt {10}}{5}",
"style": "rule"
} | {
"index": "1945",
"split": "train"
} |
math | [
{
"content": "Knowing the horizontal component of the tension force, find the minimum force acting on a buoy from the current.\n\nConsider the forces acting on the buoy along the horizontal axis: the force from the current acts in one direction, and the horizontal component of the tension force acts in the oppo... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "1946",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $B= \\frac {\\pi}{3}$ and $(a-b+c)(a+b-c)= \\frac {3}{7}bc$. \n(Ⅰ) Find the value of $\\cos C$; \n(Ⅱ) If $a=5$, find the area of $\\triangle ABC$. Please output the final answer wit... | {
"ground_truth": "10 \\sqrt {3}",
"style": "rule"
} | {
"index": "1947",
"split": "train"
} |
math | [
{
"content": "Find the product of all positive integral values of $n$ such that $n^2 - 21n + 110 = p$ for some prime number $p$. Assume there is at least one such $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "1948",
"split": "train"
} |
math | [
{
"content": "The picture shows the same die in three different positions. When the die is rolled, what is the probability of rolling a 'YES'?\n\nA) \\(\\frac{1}{3}\\)\nB) \\(\\frac{1}{2}\\)\nC) \\(\\frac{5}{9}\\)\nD) \\(\\frac{2}{3}\\)\nE) \\(\\frac{5}{6}\\) Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "1949",
"split": "train"
} |
math | [
{
"content": "In chess tournaments, each victory is worth 1 point, each draw is worth 0.5 points, and each loss is worth zero points. In the \"Magistral Championship\" of chess, only International Masters (IMs) and Grand Masters (GMs) participated. The number of GMs was ten times the number of IMs. Each player ... | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "1950",
"split": "train"
} |
math | [
{
"content": "Six small circles, each of radius 4 units, are tangent to a large circle. Each small circle is also tangent to its two neighboring small circles. Additionally, all small circles are tangent to a horizontal line that bisects the large circle. What is the diameter of the large circle in units? Pleas... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1951",
"split": "train"
} |
math | [
{
"content": "In the expansion of $(x^2+ \\frac{4}{x^2}-4)^3(x+3)$, find the constant term. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-240",
"style": "rule"
} | {
"index": "1952",
"split": "train"
} |
math | [
{
"content": "If the radius of a circle is increased by $1$ unit, the ratio of the new circumference to the new diameter is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\pi",
"style": "rule"
} | {
"index": "1953",
"split": "train"
} |
math | [
{
"content": "The cells of a $100 \\times 100$ table are painted white. In one move, you are allowed to choose any 99 cells from one row or one column and repaint each of them in the opposite color – from white to black or from black to white. What is the minimum number of moves needed to obtain a table with a ... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "1954",
"split": "train"
} |
math | [
{
"content": "Let $T$ be a subset of $\\{1,2,3,...,100\\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "1955",
"split": "train"
} |
math | [
{
"content": "What is the largest divisor of 342 that is also a factor of 285? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "1956",
"split": "train"
} |
math | [
{
"content": "The volume of a box is 360 cubic units where $a, b,$ and $c$ are integers such that $1 < c < b < a$ with $c$ being a prime number. Determine the largest possible value of $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "1957",
"split": "train"
} |
math | [
{
"content": "Given that $min\\{ a,b\\}$ represents the smaller value between the real numbers $a$ and $b$, and the vectors $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ satisfy $(\\vert\\overrightarrow{a}\\vert=1,\\vert\\overrightarrow{b}\\vert=2,\\overrightarrow{a}\\cdot\\overrightarrow{b}=0... | {
"ground_truth": "\\frac{2\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "1958",
"split": "train"
} |
math | [
{
"content": "In the rectangular prism \\(A B C D - A_{1} B_{1} C_{1} D_{1}\\), \\(A B = A A_{1} = 2\\), \\(A D = 2 \\sqrt{3}\\). Point \\(M\\) lies within plane \\(B A_{1} C_{1}\\). Find the minimum value of \\(\\overrightarrow{M A} \\cdot \\overrightarrow{M C}\\). Please output the final answer within \\boxed... | {
"ground_truth": "-\\frac{16}{7}",
"style": "rule"
} | {
"index": "1959",
"split": "train"
} |
math | [
{
"content": "The average of the numbers $1, 2, 3, \\dots, 50, y$ is $51y$. What is $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{51}{104}",
"style": "rule"
} | {
"index": "1960",
"split": "train"
} |
math | [
{
"content": "Given a function $f(x)$ defined on $R$ such that $f(2x+2)=-f(2x)$. If $f(x)=4x+3$ when $x\\in(\\frac{1}{2},\\frac{5}{4})$, then $f(2023)=\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "1961",
"split": "train"
} |
math | [
{
"content": "In the center of a circular field stands the geologists' house. From it, 6 straight roads extend, dividing the field into 6 equal sectors. Two geologists set off on a journey from their house at a speed of 5 km/h along randomly chosen roads. Determine the probability that the distance between them... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "1962",
"split": "train"
} |
math | [
{
"content": "From the set $\\{1,2,3, \\cdots, 20\\}$, choose 4 different numbers such that these 4 numbers form an arithmetic sequence. Determine the number of such arithmetic sequences. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "114",
"style": "rule"
} | {
"index": "1963",
"split": "train"
} |
math | [
{
"content": "A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\\frac{a\\pi+b\\sqrt{c}}{d\\pi-e\\sqrt{f}},$ where $a, b, c, d, e,$ and $f$ ar... | {
"ground_truth": "592",
"style": "rule"
} | {
"index": "1964",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\(AB\\) is a diameter of a circle with center \\(O\\). \\(C\\) and \\(D\\) are points on the circle. \\(OD\\) intersects \\(AC\\) at \\(P\\), \\(OC\\) intersects \\(BD\\) at \\(Q\\), and \\(AC\\) intersects \\(BD\\) at \\(R\\). If \\(\\angle BOQ = 60^{\\circ}\\) and \\(\\angle APO... | {
"ground_truth": "95",
"style": "rule"
} | {
"index": "1965",
"split": "train"
} |
math | [
{
"content": "How many squares are shown in the drawing? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "1966",
"split": "train"
} |
math | [
{
"content": "Arrange all positive integers whose digits sum to 8 in ascending order to form a sequence $\\{a_n\\}$, called the $P$ sequence. Then identify the position of 2015 within this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "83",
"style": "rule"
} | {
"index": "1967",
"split": "train"
} |
math | [
{
"content": "For how many positive integers $n$ , $1\\leq n\\leq 2008$ , can the set \\[\\{1,2,3,\\ldots,4n\\}\\] be divided into $n$ disjoint $4$ -element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset? Please output t... | {
"ground_truth": "1004",
"style": "rule"
} | {
"index": "1968",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers \\( x \\), where \\( x \\neq 9 \\), such that\n\\[\n\\log _{\\frac{x}{9}}\\left(\\frac{x^{2}}{3}\\right)<6+\\log _{3}\\left(\\frac{9}{x}\\right) .\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "223",
"style": "rule"
} | {
"index": "1969",
"split": "train"
} |
math | [
{
"content": "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? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "1970",
"split": "train"
} |
math | [
{
"content": "Given a cube \\( ABCD A_1 B_1 C_1 D_1 \\) with an edge length of 1. A sphere passes through vertices \\( A \\) and \\( C \\) and the midpoints \\( F \\) and \\( E \\) of edges \\( B_1 C_1 \\) and \\( C_1 D_1 \\) respectively. Find the radius \\( R \\) of this sphere. Please output the final answer... | {
"ground_truth": "\\frac{\\sqrt{41}}{8}",
"style": "rule"
} | {
"index": "1971",
"split": "train"
} |
math | [
{
"content": "If $f\\left(x\\right)=\\ln |a+\\frac{1}{{1-x}}|+b$ is an odd function, then $a=$____, $b=$____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\ln 2",
"style": "rule"
} | {
"index": "1972",
"split": "train"
} |
math | [
{
"content": "A regular triangular prism has a triangle $ABC$ with side $a$ as its base. Points $A_{1}, B_{1}$, and $C_{1}$ are taken on the lateral edges and are located at distances of $a / 2, a, 3a / 2$ from the base plane, respectively. Find the angle between the planes $ABC$ and $A_{1}B_{1}C_{1}$. Please o... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "1973",
"split": "train"
} |
math | [
{
"content": "On a rectangular sheet of paper, a picture in the shape of a \"cross\" is drawn from two rectangles \\(ABCD\\) and \\(EFGH\\), whose sides are parallel to the edges of the sheet. It is known that \\(AB = 9\\), \\(BC = 5\\), \\(EF = 3\\), and \\(FG = 10\\). Find the area of the quadrilateral \\(AFC... | {
"ground_truth": "52.5",
"style": "rule"
} | {
"index": "1974",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $EFGH$, $\\angle F$ is a right angle, diagonal $\\overline{EG}$ is perpendicular to $\\overline{GH}$, $EF=20$, $FG=24$, and $GH=16$. Find the perimeter of $EFGH$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60 + 8\\sqrt{19}",
"style": "rule"
} | {
"index": "1975",
"split": "train"
} |
math | [
{
"content": "The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9 and -7",
"style": "rule"
} | {
"index": "1976",
"split": "train"
} |
math | [
{
"content": "Suppose the function $f(x)$ is defined on the domain $\\{x_1,x_2,x_3\\}$, so that the graph of $y=f(x)$ consists of just three points. Suppose those three points form a triangle of area $32$.\n\nThe graph of $y = 2f(2x)$ also consists of just three points. What is the area of the triangle formed b... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "1977",
"split": "train"
} |
math | [
{
"content": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and t... | {
"ground_truth": "181",
"style": "rule"
} | {
"index": "1978",
"split": "train"
} |
math | [
{
"content": "Five numbers, $a_1$, $a_2$, $a_3$, $a_4$, are drawn randomly and without replacement from the set $\\{1, 2, 3, \\dots, 50\\}$. Four other numbers, $b_1$, $b_2$, $b_3$, $b_4$, are then drawn randomly and without replacement from the remaining set of 46 numbers. Let $p$ be the probability that, afte... | {
"ground_truth": "71",
"style": "rule"
} | {
"index": "1979",
"split": "train"
} |
math | [
{
"content": "Arrange the 9 numbers 12, 13, ..., 20 in a row such that the sum of every three consecutive numbers is a multiple of 3. How many such arrangements are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "1980",
"split": "train"
} |
math | [
{
"content": "The bases \\( AB \\) and \\( CD \\) of the trapezoid \\( ABCD \\) are 41 and 24 respectively, and its diagonals are mutually perpendicular. Find the scalar (dot) product of the vectors \\( \\overrightarrow{AD} \\) and \\( \\overrightarrow{BC} \\). Please output the final answer within \\boxed{}.",... | {
"ground_truth": "984",
"style": "rule"
} | {
"index": "1981",
"split": "train"
} |
math | [
{
"content": "Morgan uses a spreadsheet to create a table of values. In the first column, she lists the positive integers from 1 to 400. She then puts integers in the second column in the following way: if the integer in the first column of a given row is $n$, the number in the second column of that row is $3 n... | {
"ground_truth": "131",
"style": "rule"
} | {
"index": "1982",
"split": "train"
} |
math | [
{
"content": "Consider a regular polygon with $2^n$ sides, for $n \\ge 2$ , inscribed in a circle of radius $1$ . Denote the area of this polygon by $A_n$ . Compute $\\prod_{i=2}^{\\infty}\\frac{A_i}{A_{i+1}}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{\\pi}",
"style": "rule"
} | {
"index": "1983",
"split": "train"
} |
math | [
{
"content": "There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without looking at the screen, a... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "1984",
"split": "train"
} |
math | [
{
"content": "Square A has a perimeter of $24$ cm. Square B has an area equal to one-fourth the area of square A. What is the perimeter of square B?\n\n[asy]\ndraw((0,0)--(7,0));\ndraw((7,0)--(7,7));\ndraw((7,7)--(0,7));\ndraw((0,7)--(0,0));\ndraw((11,2)--(11,5));\ndraw((11,5)--(14,5));\ndraw((14,5)--(14,2));... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "1985",
"split": "train"
} |
math | [
{
"content": "Compute $\\sin 270^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "1986",
"split": "train"
} |
math | [
{
"content": "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? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "1987",
"split": "train"
} |
math | [
{
"content": "What is the greatest common factor of 252 and 96? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "1988",
"split": "train"
} |
math | [
{
"content": "The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $6$. How many two-digit numbers have this property? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "1989",
"split": "train"
} |
math | [
{
"content": "Given 6 people $A$, $B$, $C$, $D$, $E$, $F$, they are randomly arranged in a line. The probability of the event \"$A$ is adjacent to $B$ and $A$ is not adjacent to $C$\" is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{15}",
"style": "rule"
} | {
"index": "1990",
"split": "train"
} |
math | [
{
"content": "Two distinct numbers a and b are chosen randomly from the set $\\{3, 3^2, 3^3, ..., 3^{20}\\}$. What is the probability that $\\mathrm{log}_a b$ is an integer?\nA) $\\frac{12}{19}$\nB) $\\frac{1}{4}$\nC) $\\frac{24}{95}$\nD) $\\frac{1}{5}$ Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "\\frac{24}{95}",
"style": "rule"
} | {
"index": "1991",
"split": "train"
} |
math | [
{
"content": "A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of $34$ points, and the Cougars won by a margin of $14$ points. How many points did the Panthers score? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "1992",
"split": "train"
} |
math | [
{
"content": "If $\\angle A=20^\\circ$ and $\\angle AFG=\\angle AGF,$ then how many degrees is $\\angle B+\\angle D?$ [asy]\n/* AMC8 2000 #24 Problem */\npair A=(0,80), B=(46,108), C=(100,80), D=(54,18), E=(19,0);\ndraw(A--C--E--B--D--cycle);\nlabel(\"$A$\", A, W);\nlabel(\"$B$ \", B, N);\nlabel(\"$C$\", shift(... | {
"ground_truth": "80^\\circ",
"style": "rule"
} | {
"index": "1993",
"split": "train"
} |
math | [
{
"content": "Find the maximum number of Permutation of set { $1,2,3,...,2014$ } such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1007",
"style": "rule"
} | {
"index": "1994",
"split": "train"
} |
math | [
{
"content": "A polygon \\(\\mathcal{P}\\) is drawn on the 2D coordinate plane. Each side of \\(\\mathcal{P}\\) is either parallel to the \\(x\\) axis or the \\(y\\) axis (the vertices of \\(\\mathcal{P}\\) do not have to be lattice points). Given that the interior of \\(\\mathcal{P}\\) includes the interior of... | {
"ground_truth": "8 \\sqrt{2022}",
"style": "rule"
} | {
"index": "1995",
"split": "train"
} |
math | [
{
"content": "P is a polynomial. When P is divided by $x-1$, the remainder is -4 . When P is divided by $x-2$, the remainder is -1 . When $P$ is divided by $x-3$, the remainder is 4 . Determine the remainder when $P$ is divided by $x^{3}-6 x^{2}+11 x-6$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "x^{2}-5",
"style": "rule"
} | {
"index": "1996",
"split": "train"
} |
math | [
{
"content": "Which of the cones below can be formed from a $252^{\\circ}$ sector of a circle of radius 10 by aligning the two straight sides?\n\n[asy]\ndraw((5.8,8.1)..(-10,0)--(0,0)--(3.1,-9.5)..cycle);\nlabel(\"10\",(-5,0),S);\nlabel(\"$252^{\\circ}$\",(0,0),NE);\n[/asy]\n\nA. base radius = 6, slant =10\n\nB... | {
"ground_truth": "C",
"style": "rule"
} | {
"index": "1997",
"split": "train"
} |
math | [
{
"content": "Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$ , write the square and the cube of a positive integer. Determine what that number can be. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "1998",
"split": "train"
} |
math | [
{
"content": "Given an ellipse C: $$\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$$ (where $a>b>0$) that passes through the point $E(\\sqrt 3, 1)$, with an eccentricity of $\\frac{\\sqrt{6}}{3}$, and O as the origin of the coordinate system.\n(Ⅰ) Find the equation of the ellipse C.\n(Ⅱ) If point P is a moving point o... | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "1999",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.