data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "The roots of $(x^{2}-3x+2)(x)(x-4)=0$ are: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0, 1, 2 and 4",
"style": "rule"
} | {
"index": "7200",
"split": "train"
} |
math | [
{
"content": "Define $\\phi_n(x)$ to be the number of integers $y$ less than or equal to $n$ such that $\\gcd(x,y)=1$ . Also, define $m=\\text{lcm}(2016,6102)$ . Compute $$ \\frac{\\phi_{m^m}(2016)}{\\phi_{m^m}(6102)}. $$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "339/392",
"style": "rule"
} | {
"index": "7201",
"split": "train"
} |
math | [
{
"content": "In our number system the base is ten. If the base were changed to four you would count as follows: \n$1,2,3,10,11,12,13,20,21,22,23,30,\\ldots$ The twentieth number would be: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "7202",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence with positive terms $\\{a_n\\}$, let $S_n$ be the sum of the first $n$ terms. If $S_3 + a_2 = 9a_3$, determine the common ratio. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7203",
"split": "train"
} |
math | [
{
"content": "A point $P$ is randomly placed inside the right triangle $\\triangle XYZ$ where $X$ is at $(0,6)$, $Y$ is at $(0,0)$, and $Z$ is at $(9,0)$. What is the probability that the area of triangle $PYZ$ is less than half of the area of triangle $XYZ$?\n[asy]\nsize(7cm);\ndefaultpen(linewidth(0.7));\npai... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "7204",
"split": "train"
} |
math | [
{
"content": "If $78$ is divided into three parts which are proportional to $1, \\frac{1}{3}, \\frac{1}{6},$ the middle part is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17\\frac{1}{3}",
"style": "rule"
} | {
"index": "7205",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are integers with $2x^{2}+8y=26$, what is a possible value of $x-y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "7206",
"split": "train"
} |
math | [
{
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both these circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fracti... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7207",
"split": "train"
} |
math | [
{
"content": "Jude repeatedly flips a coin. If he has already flipped $n$ heads, the coin lands heads with probability $\\frac{1}{n+2}$ and tails with probability $\\frac{n+1}{n+2}$. If Jude continues flipping forever, let $p$ be the probability that he flips 3 heads in a row at some point. Compute $\\lfloor 18... | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "7208",
"split": "train"
} |
math | [
{
"content": "Compute the exact value of the expression $\\left|\\pi - | \\pi - 7 | \\right|$. Write your answer using only integers and $\\pi$, without any absolute value signs. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7-2\\pi",
"style": "rule"
} | {
"index": "7209",
"split": "train"
} |
math | [
{
"content": "Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. What is the number of women in the luxury compartment if the train is carrying 300 passengers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "7210",
"split": "train"
} |
math | [
{
"content": "In the figure below, a 3-inch by 3-inch square adjoins a 10-inch by 10-inch square. What is the area of the shaded region? Express your answer in square inches as a common fraction. [asy]\nunitsize(2mm);\ndefaultpen(linewidth(0.7pt)+fontsize(12pt));\n\npair H=(0,0), A=(0,10), B=(10,10), C=(10,3), ... | {
"ground_truth": "\\frac{72}{13}",
"style": "rule"
} | {
"index": "7211",
"split": "train"
} |
math | [
{
"content": "What is the smallest number, \\( n \\), which is the product of 3 distinct primes where the mean of all its factors is not an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "7212",
"split": "train"
} |
math | [
{
"content": "How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "7213",
"split": "train"
} |
math | [
{
"content": "A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one digit 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\%",
"style": "rule"
} | {
"index": "7214",
"split": "train"
} |
math | [
{
"content": "In the $xy$-plane, the segment with endpoints $(-3,0)$ and $(27,0)$ is the diameter of a circle. A vertical line $x=k$ intersects the circle at two points, and one of the points has a $y$-coordinate of $12$. Find the value of $k$. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "7215",
"split": "train"
} |
math | [
{
"content": "A man has $10,000 to invest. He invests $4000 at 5% and $3500 at 4%. In order to have a yearly income of $500, he must invest the remainder at: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6.4\\%",
"style": "rule"
} | {
"index": "7216",
"split": "train"
} |
math | [
{
"content": "Joann ate a total of 100 lollipops in five days. Each day after the first day she ate six more than she had eaten on the previous day. How many lollipops did she eat on the third day? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7217",
"split": "train"
} |
math | [
{
"content": "Another trapezoid \\(ABCD\\) has \\(AD\\) parallel to \\(BC\\). \\(AC\\) and \\(BD\\) intersect at \\(P\\). If \\(\\frac{[ADP]}{[BCP]} = \\frac{1}{2}\\), find \\(\\frac{[ADP]}{[ABCD]}\\). (Here, the notation \\(\\left[P_1 \\cdots P_n\\right]\\) denotes the area of the polygon \\(P_1 \\cdots P_n\\)... | {
"ground_truth": "3 - 2\\sqrt{2}",
"style": "rule"
} | {
"index": "7218",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 1)(n - 5)(n - 9) \\dotsm (n - 101) < 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "7219",
"split": "train"
} |
math | [
{
"content": "The adjacent map is part of a city: the small rectangles are blocks, and the paths in between are streets. \nEach morning, a student walks from intersection $A$ to intersection $B$, always walking along streets shown, \nand always going east or south. For variety, at each intersection where he has... | {
"ground_truth": "\\frac{21}{32}",
"style": "rule"
} | {
"index": "7220",
"split": "train"
} |
math | [
{
"content": "Find $r$ such that $\\log_{81} (2r-1) = -1/2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": "7221",
"split": "train"
} |
math | [
{
"content": "Let the roots of the polynomial $f(x) = x^6 + 2x^3 + 1$ be denoted as $y_1, y_2, y_3, y_4, y_5, y_6$. Let $h(x) = x^3 - 3x$. Find the product $\\prod_{i=1}^6 h(y_i)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "676",
"style": "rule"
} | {
"index": "7222",
"split": "train"
} |
math | [
{
"content": "Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0),(2,0),(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. Compute the number of ways to choose one or more of the seven edges such that the resulting figure is traceab... | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "7223",
"split": "train"
} |
math | [
{
"content": "What is the tens digit of the smallest six-digit positive integer that is divisible by each of $10,11,12,13,14$, and 15? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7224",
"split": "train"
} |
math | [
{
"content": "What is the result of the correct calculation for the product $0.08 \\times 3.25$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.26",
"style": "rule"
} | {
"index": "7225",
"split": "train"
} |
math | [
{
"content": "Let \\( x \\) and \\( y \\) be positive integers, with \\( x < y \\). The leading digit of \\( \\lg x \\) is \\( a \\), and the trailing digit is \\( \\alpha \\); the leading digit of \\( \\lg y \\) is \\( b \\), and the trailing digit is \\( \\beta \\). They satisfy the conditions \\( a^{2} + b^{... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "7226",
"split": "train"
} |
math | [
{
"content": "An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "6.67",
"style": "rule"
} | {
"index": "7227",
"split": "train"
} |
math | [
{
"content": "Find a four-digit number such that the square of the sum of the two-digit number formed by its first two digits and the two-digit number formed by its last two digits is exactly equal to the four-digit number itself. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2025",
"style": "rule"
} | {
"index": "7228",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be real numbers such that $x + y = 3.$ Find the maximum value of\n\\[x^4 y + x^3 y + x^2 y + xy + xy^2 + xy^3 + xy^4.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{400}{11}",
"style": "rule"
} | {
"index": "7229",
"split": "train"
} |
math | [
{
"content": "Given a box contains a total of 180 marbles, 25% are silver, 20% are gold, 15% are bronze, 10% are sapphire, and 10% are ruby, and the remainder are diamond marbles. If 10% of the gold marbles are removed, calculate the number of marbles left in the box. Please output the final answer within \\box... | {
"ground_truth": "176",
"style": "rule"
} | {
"index": "7230",
"split": "train"
} |
math | [
{
"content": "A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After a total of $n... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "7231",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\frac{1}{2}\\cos 2A = \\cos^2 A - \\cos A$.\n(I) Find the measure of angle $A$;\n(II) If $a=3$, $\\sin B = 2\\sin C$, find the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "7232",
"split": "train"
} |
math | [
{
"content": "If $a \\lt 0$, the graph of the function $f\\left(x\\right)=a^{2}\\sin 2x+\\left(a-2\\right)\\cos 2x$ is symmetric with respect to the line $x=-\\frac{π}{8}$. Find the maximum value of $f\\left(x\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": "7233",
"split": "train"
} |
math | [
{
"content": "Determine the remainder when $$2^{\\frac{1 \\cdot 2}{2}}+2^{\\frac{2 \\cdot 3}{2}}+\\cdots+2^{\\frac{2011 \\cdot 2012}{2}}$$ is divided by 7. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7234",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin \\alpha = \\frac{4}{5}$ and $\\alpha$ is an angle in the second quadrant, find the value of $\\cot (\\frac{\\pi}{4} - \\frac{\\alpha}{2})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "7235",
"split": "train"
} |
math | [
{
"content": "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 $S_{i+1},$ ... | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "7236",
"split": "train"
} |
math | [
{
"content": "Find the largest constant $c>0$ such that for every positive integer $n\\ge 2$ , there always exist a positive divisor $d$ of $n$ such that $$ d\\le \\sqrt{n}\\hspace{0.5cm} \\text{and} \\hspace{0.5cm} \\tau(d)\\ge c\\sqrt{\\tau(n)} $$ where $\\tau(n)$ is the number of divisors of $n... | {
"ground_truth": "\\frac{1}{\\sqrt{2}}",
"style": "rule"
} | {
"index": "7237",
"split": "train"
} |
math | [
{
"content": "Given that point P is on the curve $y=\\frac{1}{2}{{e}^{x}}$ and point Q is on the curve $y=x$, determine the minimum value of the distance $|PQ|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2}(1 - \\ln 2)",
"style": "rule"
} | {
"index": "7238",
"split": "train"
} |
math | [
{
"content": "The sum of all positive integers $m$ such that $\\frac{13!}{m}$ is a perfect square can be written as $2^a3^b5^c7^d11^e13^f,$ where $a,b,c,d,e,$ and $f$ are positive integers. Find $a+b+c+d+e+f.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "012",
"style": "rule"
} | {
"index": "7239",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of\n\\[4x + \\frac{1}{x^4}\\]for $x > 0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "7240",
"split": "train"
} |
math | [
{
"content": "Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be real numbers satisfying the following equations: $$\\frac{a_{1}}{k^{2}+1}+\\frac{a_{2}}{k^{2}+2}+\\frac{a_{3}}{k^{2}+3}+\\frac{a_{4}}{k^{2}+4}+\\frac{a_{5}}{k^{2}+5}=\\frac{1}{k^{2}} \\text { for } k=1,2,3,4,5$$ Find the value of $\\frac{a_{1}}{37}+\\frac... | {
"ground_truth": "\\frac{187465}{6744582}",
"style": "rule"
} | {
"index": "7241",
"split": "train"
} |
math | [
{
"content": "For how many positive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "212",
"style": "rule"
} | {
"index": "7242",
"split": "train"
} |
math | [
{
"content": "How many of the natural numbers from 1 to 600, inclusive, contain the digit 5 at least once? (The numbers 152 and 553 are two natural numbers that contain the digit 5 at least once, but 430 is not.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "195",
"style": "rule"
} | {
"index": "7243",
"split": "train"
} |
math | [
{
"content": "There is a four-digit number \\( A \\). By rearranging the digits (none of which are 0), the largest possible number is 7668 greater than \\( A \\), and the smallest possible number is 594 less than \\( A \\). What is \\( A \\)? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "1963",
"style": "rule"
} | {
"index": "7244",
"split": "train"
} |
math | [
{
"content": "1. The converse of the proposition \"If $x > 1$, then ${x}^{2} > 1$\" is ________.\n2. Let $P$ be a point on the parabola ${{y}^{2}=4x}$ such that the distance from $P$ to the line $x+2=0$ is $6$. The distance from $P$ to the focus $F$ of the parabola is ________.\n3. In a geometric sequence $\\\\... | {
"ground_truth": "\\frac{32}{5}",
"style": "rule"
} | {
"index": "7245",
"split": "train"
} |
math | [
{
"content": "In the diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $6.$ What is the area of the shaded region?\n\n[asy]\nsize(100);\nimport graph;\nfilldraw(Circle((0,0),2),mediumgray);\nfilldraw(Circle((-1,0),1),... | {
"ground_truth": "18\\pi",
"style": "rule"
} | {
"index": "7246",
"split": "train"
} |
math | [
{
"content": "A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{65}",
"style": "rule"
} | {
"index": "7247",
"split": "train"
} |
math | [
{
"content": "Calculate the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x=0.4$ using Horner's method, and then determine the value of $v_1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5.2",
"style": "rule"
} | {
"index": "7248",
"split": "train"
} |
math | [
{
"content": "Positive integers $a$, $b$, and $c$ are chosen so that $a<b<c$, and the system of equations \\[\n2x + y = 2003\\quad\\text{and}\\quad y = |x-a| + |x-b| + |x-c|\n\\]has exactly one solution. What is the minimum value of $c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "1002",
"style": "rule"
} | {
"index": "7249",
"split": "train"
} |
math | [
{
"content": "Given $(b_1, b_2, ... b_7)$ be a list of the first 7 even positive integers such that for each $2 \\le i \\le 7$, either $b_i + 2$ or $b_i - 2$ or both appear somewhere before $b_i$ in the list, determine the number of such lists. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "7250",
"split": "train"
} |
math | [
{
"content": "A rectangle has a length of $\\frac{3}{5}$ and an area of $\\frac{1}{3}$. What is the width of the rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\\\frac{5}{9}",
"style": "rule"
} | {
"index": "7251",
"split": "train"
} |
math | [
{
"content": "What is the number halfway between $\\frac{1}{12}$ and $\\frac{1}{10}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{120}",
"style": "rule"
} | {
"index": "7252",
"split": "train"
} |
math | [
{
"content": "A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle? Ple... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7253",
"split": "train"
} |
math | [
{
"content": "Triangle \\( ABC \\) has \\( AB=24 \\), \\( AC=26 \\), and \\( BC=22 \\). Points \\( D \\) and \\( E \\) are located on \\( \\overline{AB} \\) and \\( \\overline{AC} \\), respectively, so that \\( \\overline{DE} \\) is parallel to \\( \\overline{BC} \\) and contains the center of the inscribed cir... | {
"ground_truth": "\\frac{275}{18}",
"style": "rule"
} | {
"index": "7254",
"split": "train"
} |
math | [
{
"content": "Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "968",
"style": "rule"
} | {
"index": "7255",
"split": "train"
} |
math | [
{
"content": "Place the numbers 1, 2, 3, 4, 2, 6, 7, and 8 at the eight vertices of a cube such that the sum of the numbers on any given face is at least 10. Determine the minimum possible sum of the numbers on each face. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7256",
"split": "train"
} |
math | [
{
"content": "At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain ... | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "7257",
"split": "train"
} |
math | [
{
"content": "The sides $x$ and $y$ of a scalene triangle satisfy $x + \\frac{2\\Delta }{x}=y+ \\frac{2\\Delta }{y}$ , where $\\Delta$ is the area of the triangle. If $x = 60, y = 63$ , what is the length of the largest side of the triangle? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "87",
"style": "rule"
} | {
"index": "7258",
"split": "train"
} |
math | [
{
"content": "Given an angle $a$ and a point $P(-4,3)$ on its terminal side, find the value of $\\dfrac{\\cos \\left( \\dfrac{\\pi}{2}+a\\right)\\sin \\left(-\\pi-a\\right)}{\\cos \\left( \\dfrac{11\\pi}{2}-a\\right)\\sin \\left( \\dfrac{9\\pi}{2}+a\\right)}$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "- \\dfrac{3}{4}",
"style": "rule"
} | {
"index": "7259",
"split": "train"
} |
math | [
{
"content": "How many factors of 8000 are perfect squares? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "7260",
"split": "train"
} |
math | [
{
"content": "Convert the point $(-2,-2)$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( 2 \\sqrt{2}, \\frac{5 \\pi}{4} \\right)",
"style": "rule"
} | {
"index": "7261",
"split": "train"
} |
math | [
{
"content": "Find the smallest composite number that has no prime factors less than 15. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "289",
"style": "rule"
} | {
"index": "7262",
"split": "train"
} |
math | [
{
"content": "A checker can move in one direction on a divided strip into cells, moving either to the adjacent cell or skipping one cell in one move. In how many ways can it move 10 cells? 11 cells? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "7263",
"split": "train"
} |
math | [
{
"content": "Quarter circles of radius 1' form a pattern as shown below. What is the area, in square feet, of the shaded region in a 3-foot length of this pattern? The quarter circles' flat sides face outward, alternating top and bottom along the length. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\frac{3}{4}\\pi",
"style": "rule"
} | {
"index": "7264",
"split": "train"
} |
math | [
{
"content": "Given that the product of two positive integers $a$ and $b$ is $161$ after reversing the digits of the two-digit number $a$, find the correct value of the product of $a$ and $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "224",
"style": "rule"
} | {
"index": "7265",
"split": "train"
} |
math | [
{
"content": "A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\clubsuit$) are black... | {
"ground_truth": "\\dfrac{13}{51}",
"style": "rule"
} | {
"index": "7266",
"split": "train"
} |
math | [
{
"content": "If the function $f(x)=\\frac{1}{2}(2-m)x^{2}+(n-8)x+1$ $(m > 2)$ is monotonically decreasing in the interval $[-2,-1]$, find the maximum value of $mn$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7267",
"split": "train"
} |
math | [
{
"content": "In triangle \\( \\triangle ABC \\), \\( AB = \\sqrt{2} \\), \\( AC = \\sqrt{3} \\), and \\( \\angle BAC = 30^\\circ \\). Let \\( P \\) be an arbitrary point in the plane containing \\( \\triangle ABC \\). Find the minimum value of \\( \\mu = \\overrightarrow{PA} \\cdot \\overrightarrow{PB} + \\ove... | {
"ground_truth": "\\frac{\\sqrt{2}}{2} - \\frac{5}{3}",
"style": "rule"
} | {
"index": "7268",
"split": "train"
} |
math | [
{
"content": "In our daily life, for a pair of new bicycle tires, the rear tire wears out faster than the front tire. Through testing, it is found that the front tire of a general bicycle is scrapped after traveling 11,000 kilometers, while the rear tire is scrapped after traveling 9,000 kilometers. It is evide... | {
"ground_truth": "4950",
"style": "rule"
} | {
"index": "7269",
"split": "train"
} |
math | [
{
"content": "What is the remainder when the integer equal to \\( QT^2 \\) is divided by 100, given that \\( QU = 9 \\sqrt{33} \\) and \\( UT = 40 \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "7270",
"split": "train"
} |
math | [
{
"content": "Find the pattern and fill in the blanks:\n\n1. 12, 16, 20, \\_\\_\\_\\_\\_\\_, \\_\\_\\_\\_\\_\\_\n2. 2, 4, 8, \\_\\_\\_\\_\\_\\_, \\_\\_\\_\\_\\_\\_ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "7271",
"split": "train"
} |
math | [
{
"content": "When $\\frac{3}{1250}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7272",
"split": "train"
} |
math | [
{
"content": "Circle $O$ has chord $A B$. A circle is tangent to $O$ at $T$ and tangent to $A B$ at $X$ such that $A X=2 X B$. What is \\frac{A T}{B T}? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7273",
"split": "train"
} |
math | [
{
"content": "Determine the maximum number of different sets consisting of three terms that form arithmetic progressions and can be chosen from a sequence of real numbers \\( a_1, a_2, \\ldots, a_{101} \\), where\n\\[ \na_1 < a_2 < a_3 < \\cdots < a_{101} .\n\\] Please output the final answer within \\boxed{}."... | {
"ground_truth": "2500",
"style": "rule"
} | {
"index": "7274",
"split": "train"
} |
math | [
{
"content": "Find $A$ and $B$ such that\n\\[\\frac{3x+5}{x^2-x-42}=\\frac{A}{x-7}+\\frac{B}{x+6}.\\]Write your answer in the form $(A,B)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2,1)",
"style": "rule"
} | {
"index": "7275",
"split": "train"
} |
math | [
{
"content": "The quadratic polynomial \\( f(x) = a x^{2} + b x + c \\) has exactly one root, and the quadratic polynomial \\( f(3x + 2) - 2f(2x - 1) \\) also has exactly one root. Find the root of the polynomial \\( f(x) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "7276",
"split": "train"
} |
math | [
{
"content": "Positive real numbers $a$ , $b$ , $c$ satisfy $a+b+c=1$ . Find the smallest possible value of $$ E(a,b,c)=\\frac{a^3}{1-a^2}+\\frac{b^3}{1-b^2}+\\frac{c^3}{1-c^2}. $$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "7277",
"split": "train"
} |
math | [
{
"content": "Katie has a fair 2019-sided die with sides labeled $1,2, \\ldots, 2019$. After each roll, she replaces her $n$-sided die with an $(n+1)$-sided die having the $n$ sides of her previous die and an additional side with the number she just rolled. What is the probability that Katie's $2019^{\\text {th... | {
"ground_truth": "\\frac{1}{2019}",
"style": "rule"
} | {
"index": "7278",
"split": "train"
} |
math | [
{
"content": "Given points $a$ and $b$ in the plane, let $a \\oplus b$ be the unique point $c$ such that $a b c$ is an equilateral triangle with $a, b, c$ in the clockwise orientation. Solve $(x \\oplus(0,0)) \\oplus(1,1)=(1,-1)$ for $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\left(\\frac{1-\\sqrt{3}}{2}, \\frac{3-\\sqrt{3}}{2}\\right)",
"style": "rule"
} | {
"index": "7279",
"split": "train"
} |
math | [
{
"content": "Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follo... | {
"ground_truth": "99\\%",
"style": "rule"
} | {
"index": "7280",
"split": "train"
} |
math | [
{
"content": "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 $\\... | {
"ground_truth": "546",
"style": "rule"
} | {
"index": "7281",
"split": "train"
} |
math | [
{
"content": "Find $0.4 \\cdot 0.6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.24",
"style": "rule"
} | {
"index": "7282",
"split": "train"
} |
math | [
{
"content": "Two cubes with the faces numbered 1 through 6 are tossed and the numbers shown on the top faces are added. What is the probability that the sum is even? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac12",
"style": "rule"
} | {
"index": "7283",
"split": "train"
} |
math | [
{
"content": "Express $537_8 + 1C2E_{16}$ as a base 10 integer, where $C$ and $E$ denote the hexadecimal digits with values 12 and 14 respectively. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7565",
"style": "rule"
} | {
"index": "7284",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $b\\sin A=(3b-c)\\sin B$\n(1) If $2\\sin A=3\\sin B$, and the perimeter of $\\triangle ABC$ is $8$, find $c$\n(2) If $\\triangle ABC$ is an isosceles triangle, find $\\cos 2B$. Please output the fin... | {
"ground_truth": "\\frac{17}{81}",
"style": "rule"
} | {
"index": "7285",
"split": "train"
} |
math | [
{
"content": "A line in the plane is called strange if it passes through \\((a, 0)\\) and \\((0, 10-a)\\) for some \\(a\\) in the interval \\([0,10]\\). A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming poi... | {
"ground_truth": "50/3",
"style": "rule"
} | {
"index": "7286",
"split": "train"
} |
math | [
{
"content": "Each of given $100$ numbers was increased by $1$. Then each number was increased by $1$ once more. Given that the fi\frst time the sum of the squares of the numbers was not changed \ffind how this sum was changed the second time. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "7287",
"split": "train"
} |
math | [
{
"content": "A certain middle school assigns numbers to each student, where the last digit indicates the gender of the student: 1 for male and 2 for female. If 028432 represents \"a female student who is number 43 in class 8 and enrolled in the year 2002,\" then the number for a male student who is number 23 i... | {
"ground_truth": "086231",
"style": "rule"
} | {
"index": "7288",
"split": "train"
} |
math | [
{
"content": "A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were i... | {
"ground_truth": "998",
"style": "rule"
} | {
"index": "7289",
"split": "train"
} |
math | [
{
"content": "How many of the natural numbers from 1 to 700, inclusive, contain the digit 3 at least once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "214",
"style": "rule"
} | {
"index": "7290",
"split": "train"
} |
math | [
{
"content": "The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "7291",
"split": "train"
} |
math | [
{
"content": "The points \\( M \\) and \\( N \\) are chosen on the angle bisector \\( AL \\) of a triangle \\( ABC \\) such that \\( \\angle ABM = \\angle ACN = 23^\\circ \\). \\( X \\) is a point inside the triangle such that \\( BX = CX \\) and \\( \\angle BXC = 2\\angle BML \\). Find \\( \\angle MXN \\). Ple... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "7292",
"split": "train"
} |
math | [
{
"content": "To solve the problem, we need to find the value of $\\log_{4}{\\frac{1}{8}}$.\nA) $-\\frac{1}{2}$\nB) $-\\frac{3}{2}$\nC) $\\frac{1}{2}$\nD) $\\frac{3}{2}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "7293",
"split": "train"
} |
math | [
{
"content": "Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7294",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive angle \\( x \\) for which\n\n\\[ \n2^{\\sin^2 x} \\cdot 4^{\\cos^2 x} \\cdot 2^{\\tan x} = 8 \n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "7295",
"split": "train"
} |
math | [
{
"content": "If I have four boxes arranged in a $2$ x $2$ grid, in how many distinct ways can I place the digits $1$, $2$, and $3$ in the boxes such that each box contains at most one digit? (I only have one of each digit, so one box will remain blank.) Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "7296",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "7297",
"split": "train"
} |
math | [
{
"content": "Through points \\(A(0, 14)\\) and \\(B(0, 4)\\), two parallel lines are drawn. The first line, passing through point \\(A\\), intersects the hyperbola \\(y = \\frac{1}{x}\\) at points \\(K\\) and \\(L\\). The second line, passing through point \\(B\\), intersects the hyperbola \\(y = \\frac{1}{x}\... | {
"ground_truth": "3.5",
"style": "rule"
} | {
"index": "7298",
"split": "train"
} |
math | [
{
"content": "Let $a_n$ be the closest to $\\sqrt n$ integer. \nFind the sum $1/a_1 + 1/a_2 + ... + 1/a_{1980}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "7299",
"split": "train"
} |
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