data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let triangle $ABC$ with incenter $I$ and circumcircle $\\Gamma$ satisfy $AB = 6\\sqrt{3}, BC = 14,$ and $CA = 22$ . Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$ . Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 2900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given algebraic expressions $A=2m^{2}+3my+2y-1$ and $B=m^{2}-my$. Find:<br/>\n$(1)$ Simplify $3A-2\\left(A+B\\right)$.<br/>\n$(2)$ If $\\left(m-1\\right)^{2}+|y+2|=0$, find the value of $3A-2\\left(A+B\\right)$.<br/>\n$(3)$ If the value of $3A-2\\left(A+B\\right)$ is independen... | math | {
"style": "rule",
"ground_truth": "-0.4"
} | {
"split": "train",
"index": 2901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(12,16)$ that does not go inside the circle $(x-6)^{2}+(y-8)^{2}= 25$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}+\\frac{5\\pi}{3}"
} | {
"split": "train",
"index": 2902,
"solution": "1. **Identify the Points and Circle**: \n - Let $A(0,0)$ and $D(12,16)$ be the start and end points, respectively.\n - The circle is centered at $O(6,8)$ with radius $5$, given by the equation $(x-6)^2 + (y-8)^2 = 25$.\n\n2. **Calculate Distance $OA$**:\n - Us... |
deepscaler | [
{
"role": "user",
"content": "Let $g$ be a function taking the positive integers to the positive integers, such that:\n\n(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)\n(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$,\n(iii) if $m \\neq n$ and $m^n = n^m$, then ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 2903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$ , and also that $ AX \\equal{} ... | math | {
"style": "rule",
"ground_truth": "5/2"
} | {
"split": "train",
"index": 2904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3, and Buddy wins otherwise. If the probability that Ainsley wins is $\\frac{a}{b}... | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 2905,
"solution": "We let $X$ be the event of a multiple of 3 being rolled and $Y$ be the event of a nonmultiple of 3 being rolled. In order for Ainsley to win, she needs event $X$ to happen consecutively; meanwhile, Buddy just needs $Y$ then $X$ to occur. Thus, if $Y$ occurs in the f... |
deepscaler | [
{
"role": "user",
"content": "Find the largest natural number \\( n \\) with the following property: for any odd prime number \\( p \\) less than \\( n \\), the difference \\( n - p \\) is also a prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the students are numbered from 01 to 70, determine the 7th individual selected by reading rightward starting from the number in the 9th row and the 9th column of the random number table. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 2907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest value of the parameter \\(a\\) for which the equation:\n\n\\[ (|x-2| + 2a)^2 - 3(|x-2| + 2a) + 4a(3-4a) = 0 \\]\n\nhas exactly three solutions. Indicate the largest value in the answer. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 2908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five monkeys share a pile of peanuts. The first monkey divides the peanuts into five piles, leaving one peanut which it eats, and takes away one pile. The second monkey then divides the remaining peanuts into five piles, leaving exactly one peanut, eats it, and takes away one p... | math | {
"style": "rule",
"ground_truth": "3121"
} | {
"split": "train",
"index": 2909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2, \\cdots$ where the number of 2s between consecutive 1s increases by 1 each time, what is the sum of the first 1234 terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2419"
} | {
"split": "train",
"index": 2910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have created a new game where for each day in May, if the date is a prime number, I walk three steps forward; if the date is composite, I walk one step backward. If I stop on May 31st, how many steps long is my walk back to the starting point? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 2911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A six-digit number begins with digit 1 and ends with digit 7. If the digit in the units place is decreased by 1 and moved to the first place, the resulting number is five times the original number. Find this number. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"split": "train",
"index": 2912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The midpoints of the sides of a parallelogram with area $P$ are joined to form a smaller parallelogram inside it. What is the ratio of the area of the smaller parallelogram to the area of the original parallelogram? Express your answer as a common fraction. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 2913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $$\\frac {x^{2}}{a^{2}}- \\frac {y^{2}}{b^{2}}=1$$ ($a>0$, $b>0$) with its right focus at $F(c, 0)$.\n(1) If one of the asymptotes of the hyperbola is $y=x$ and $c=2$, find the equation of the hyperbola;\n(2) With the origin $O$ as the center and $c$ as the ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 2914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square $EFGH$ is inscribed in the region bounded by the parabola $y = x^2 - 6x + 5$ and the $x$-axis. Find the area of square $EFGH$.\n\\[\n\\text{[asy]}\nunitsize(0.8 cm);\n\nreal parab (real x) {\n return(x^2 - 6*x + 5);\n}\n\npair E, F, G, H;\nreal x = -1 + sqrt(3);\n\nE ... | math | {
"style": "rule",
"ground_truth": "24 - 8\\sqrt{5}"
} | {
"split": "train",
"index": 2915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyram... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 2916,
"solution": "Let $B$ and $C$ be the vertices adjacent to $A$ on the same base as $A$, and let $D$ be the last vertex of the triangular pyramid. Notice that we can already find some lengths. We have $AB=AC=12$ (given) and $BC=BD=\\sqrt{144+h^2}$ by the Pythagorean Theorem. Let $M... |
deepscaler | [
{
"role": "user",
"content": "In a box, there are 22 kg of cranberries. How, using a single 2-kilogram weight and a two-pan scale, can you measure out 17 kg of cranberries in two weighings? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 2917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ defined by $a_{0} = \\frac{1}{2}$ and $a_{n+1} = a_{n} + \\frac{a_{n}^{2}}{2023}$ for $n=0,1,2,\\ldots$, find the integer $k$ such that $a_{k} < 1 < a_{k+1}$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"split": "train",
"index": 2918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $IJKL$ has one vertex on each side of square $WXYZ$. Point $I$ is on $WZ$ such that $WI = 9 \\cdot IZ$. Determine the ratio of the area of square $IJKL$ to the area of square $WXYZ$.\nA) $\\frac{1}{20}$ \nB) $\\frac{1}{50}$ \nC) $\\frac{1}{40}$ \nD) $\\frac{1}{64}$ \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{50}"
} | {
"split": "train",
"index": 2919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 2920,
"solution": "1. **Determine individual rates**: \n - Cagney's rate of frosting cupcakes is 1 cupcake per 20 seconds.\n - Lacey's rate of frosting cupcakes is 1 cupcake per 30 seconds.\n\n2. **Calculate the combined rate**:\n - The combined rate \\( R \\) when two people ar... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the distance $|AB|$ where point $A$ is the intersection of the line $y=a$ and the line $y=2x+2$, and point $B$ is the intersection of the line $y=a$ and the curve $y=x+\\ln x$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 2921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a four-digit natural number with the following property: if we swap its first two digits with the second two digits, we get a four-digit number that is 99 less.\n\nHow many such numbers are there in total, and how many of them are divisible by 9? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\displaystyle \\frac{2+4-8+16+32-64}{4+8-16+32+64-128}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two identical smaller cubes are stacked next to a larger cube. Each of the two smaller cubes has a volume of 8. What is the volume of the larger cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 2924,
"solution": "Since $2 \\times 2 \\times 2=8$, a cube with edge length 2 has volume 8. Therefore, each of the cubes with volume 8 have a height of 2. This means that the larger cube has a height of $2+2=4$, which means that its volume is $4^{3}=4 \\times 4 \\times 4=64$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of a quadrilateral with vertices at $(0,0)$, $(4,3)$, $(7,0)$, and $(4,4)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 2925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each continuous function $f: [0,1] \\to \\mathbb{R}$, let $I(f) = \\int_0^1 x^2 f(x)\\,dx$ and $J(x) = \\int_0^1 x \\left(f(x)\\right)^2\\,dx$. Find the maximum value of $I(f) - J(f)$ over all such functions $f$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "1/16"
} | {
"split": "train",
"index": 2926,
"solution": "The answer is $1/16$. We have \\begin{align*} &\\int_0^1 x^2 f (x)\\,dx - \\int_0^1 x f(x)^2\\,dx \\\\ &= \\int_0^1 (x^3/4 - x ( f(x)-x/2)^2)\\,dx \\\\ &\\leq \\int_0^1 x^3/4\\,dx = 1/16, \\end{align*} with equality when $f(x) = x/2$."
} |
deepscaler | [
{
"role": "user",
"content": "In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midsegment is 5. Find the height of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.8"
} | {
"split": "train",
"index": 2927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(3 \\times 3 \\times 5 \\times 5 \\times 7 \\times 9 = 3 \\times 3 \\times 7 \\times n \\times n\\), what is a possible value of \\(n\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2928,
"solution": "Since \\(3 \\times 3 \\times 5 \\times 5 \\times 7 \\times 9 = 3 \\times 3 \\times 7 \\times n \\times n\\), then \\(n \\times n = \\frac{3 \\times 3 \\times 5 \\times 5 \\times 7 \\times 9}{3 \\times 3 \\times 7} = 5 \\times 5 \\times 9 = 5 \\times 5 \\times 3 \\ti... |
deepscaler | [
{
"role": "user",
"content": "Given a cube with a side length of \\(4\\), if a solid cube of side length \\(1\\) is removed from each corner, calculate the total number of edges of the resulting structure. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 2929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of the vertices of the polygon such that they form the vertices of a **right triangle**. What is the value of $n$ ? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 2930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the smallest positive integer such that the remainder of $3n+45$ , when divided by $1060$ , is $16$ . Find the remainder of $18n+17$ upon division by $1920$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1043"
} | {
"split": "train",
"index": 2931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric series \\(\\left\\{a_{n}\\right\\}\\) with the sum of its first \\(n\\) terms denoted by \\(S_{n}\\), and satisfying the equation \\(S_{n}=\\frac{\\left(a_{n}+1\\right)^{2}}{4}\\), find the value of \\(S_{20}\\). Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 2932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What expression describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a>\\frac{1}{2}"
} | {
"split": "train",
"index": 2933,
"solution": "To solve this problem, we need to analyze the conditions under which the circle $x^2 + y^2 = a^2$ and the parabola $y = x^2 - a$ intersect at exactly three points.\n\n1. **Equation Setup**: Substitute $y = x^2 - a$ into the circle equation:\n \\[\n x^2 + (x^2 - ... |
deepscaler | [
{
"role": "user",
"content": "Divers extracted a certain number of pearls, not exceeding 1000. The distribution of the pearls happens as follows: each diver in turn approaches the heap of pearls and takes either exactly half or exactly one-third of the remaining pearls. After all divers have taken their sha... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $-2(-x^2y+xy^2)-[-3x^2y^2+3x^2y+(3x^2y^2-3xy^2)]$, where $x=-1$, $y=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 2935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 2936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x)=\\frac{\\sin (\\pi x)-\\cos (\\pi x)+2}{\\sqrt{x}} \\) for \\( \\frac{1}{4} \\leqslant x \\leqslant \\frac{5}{4} \\), find the minimum value of \\( f(x) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{5}}{5} - \\frac{2\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 2937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each row of a seating arrangement seats either 9 or 10 people. A total of 100 people are to be seated. How many rows seat exactly 10 people if every seat is occupied? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\( 2^{2021} \\) and \\( 5^{2021} \\) are written out one after the other. How many digits are written in total? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 2939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\\text{eight}}.$ Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "585"
} | {
"split": "train",
"index": 2940,
"solution": "Assuming that such palindrome is greater than $777_8 = 511,$ we conclude that the palindrome has four digits when written in base $8.$ Let such palindrome be \\[(\\underline{ABBA})_8 = 512A + 64B + 8B + A = 513A + 72B.\\]\n It is clear that $A=1,$ so we repeatedly a... |
deepscaler | [
{
"role": "user",
"content": "Four cars $A, B, C,$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ travel clockwise, while $C$ and $D$ travel counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meet... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 2941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the constant \\(\\lambda\\) such that for all positive integers \\(n\\) and any positive real numbers \\(x_{k}\\) \\((1 \\leq k \\leq n)\\) summing to 1, the following inequality holds:\n$$\n\\lambda \\prod_{k=1}^{n}\\left(1-x_{k}\\right) \\geq 1-\\sum... | math | {
"style": "rule",
"ground_truth": "e"
} | {
"split": "train",
"index": 2942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function \\( f(x) = x^2 + ax + b \\), determine the minimum value of \\( F = \\max_{|x| \\leq 1} \\mid f(x) \\), where \\( a \\) and \\( b \\) are real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square was cut into 25 smaller squares, of which exactly one has a side length different from 1 (each of the others has a side length of 1).\n\nFind the area of the original square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 2944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of:\n\\[\\frac{abc(a + b + c)}{(a + b)^3 (b + c)^3}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 2945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age.... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 2946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 2947,
"solution": "1. **Define Variables:**\n Let $p$ be the rate at which Paula paints (in house/hours), $h$ be the combined rate of the two helpers (in house/hours), and $L$ be the lunch break duration (in hours).\n\n2. **Set Up Equations:**\n From the problem, we can set up the... |
deepscaler | [
{
"role": "user",
"content": "Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a $3$-digit number with $a\\ge1$ and $a+b+c\\le7$. At the end of the trip, the odometer showed ... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 2948,
"solution": "1. **Set up the equation based on the problem statement:**\n Danica drove for $x$ hours at a speed of $55$ miles per hour. The odometer reading at the start was $abc$ and at the end was $cba$. In terms of digits, this translates to:\n \\[\n 100a + 10b + c \\te... |
deepscaler | [
{
"role": "user",
"content": "In a regional frisbee league, teams have 7 members and each of the 5 teams takes turns hosting matches. At each match, each team selects three members of that team to be on the match committee, except the host team, which selects four members. How many possible 13-member match ... | math | {
"style": "rule",
"ground_truth": "262,609,375"
} | {
"split": "train",
"index": 2949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In-Young generates a string of $B$ zeroes and ones using the following method:\n- First, she flips a fair coin. If it lands heads, her first digit will be a 0, and if it lands tails, her first digit will be a 1.\n- For each subsequent bit, she flips an unfair coin, which lands ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 2950,
"solution": "Since each digit is dependent on the previous, and the first digit is random, we note that the probability that In Young obtains a particular string is the same probability as that she obtains the inverse string (i.e. that where the positions of the 0 s and 1 s are ... |
deepscaler | [
{
"role": "user",
"content": "If the polynomial $x^2+x^{10}=a+a_{1}(x+1)+\\ldots+a_{9}(x+1)^{9}+a_{10}(x+1)^{10}$, then find the value of $a_{9}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 2951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an $8 \\times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes unti... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 2952,
"solution": "Let the expected number of minutes it will take the rook to reach the upper right corner from the top or right edges be $E_{e}$, and let the expected number of minutes it will take the rook to reach the upper right corner from any other square be $E_{c}$. Note that ... |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"split": "train",
"index": 2953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square sheet of paper with sides of length $10$ cm is initially folded in half horizontally. The folded paper is then folded diagonally corner to corner, forming a triangular shape. If this shape is then cut along the diagonal fold, what is the ratio of the perimeter of one o... | math | {
"style": "rule",
"ground_truth": "\\frac{15 + \\sqrt{125}}{40}"
} | {
"split": "train",
"index": 2954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions... | math | {
"style": "rule",
"ground_truth": "3030"
} | {
"split": "train",
"index": 2955,
"solution": "Take one vertex of the cube as origin and establish 3D coordinates along the cube's edges.\nDefine a beam as $x-dir$ if its long edge is parallel to x-axis. Similarly for $y-dir$ and $z-dir$ .\nDefine a beam's location as (direction, ( $1 \\times 1$ face's location ... |
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"basic'' are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 2956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many factors of 2 are in the prime factorization of 1984!? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"split": "train",
"index": 2957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin A+\\sin B=1$ and $\\cos A+\\cos B= \\frac{3}{2}$, what is the value of $\\cos(A-B)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 2958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer that is divisible by three distinct primes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 2959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many even divisors does $9!$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 2960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest odd number with four different prime factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1155"
} | {
"split": "train",
"index": 2961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "James writes down three integers. Alex picks some two of those integers, takes the average of them, and adds the result to the third integer. If the possible final results Alex could get are 42, 13, and 37, what are the three integers James originally chose? Let's think step by... | math | {
"style": "rule",
"ground_truth": "-20, 28, 38"
} | {
"split": "train",
"index": 2962,
"solution": "Let $x, y, z$ be the integers. We have $$\\begin{aligned} & \\frac{x+y}{2}+z=42 \\\\ & \\frac{y+z}{2}+x=13 \\\\ & \\frac{x+z}{2}+y=37 \\end{aligned}$$ Adding these three equations yields $2(x+y+z)=92$, so $\\frac{x+y}{2}+z=23+\\frac{z}{2}=42$ so $z=38$. Similarly, $... |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, the medians $\\overline{XT}$ and $\\overline{YS}$ are perpendicular. If $XT = 15$ and $YS = 20$, find the length of side $XZ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{3}"
} | {
"split": "train",
"index": 2963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral \\(A B C D\\) is inscribed in a circle with radius 6 and center at point \\(O\\). Its diagonals \\(A C\\) and \\(B D\\) are mutually perpendicular and intersect at point \\(K\\). Points \\(E\\) and \\(F\\) are the midpoints of \\(A C\\) and \\(B D\\), respective... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 2964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the region enclosed by the graph of the equation $x^2-14x+3y+70=21+11y-y^2$ that lies below the line $y=x-3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8 \\pi"
} | {
"split": "train",
"index": 2965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least common multiple of the numbers 1056 and 792, and then add 100 to your result? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3268"
} | {
"split": "train",
"index": 2966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $(x_n)$ is defined by $x_1 = 115$ and $x_k = x_{k - 1}^2 + x_{k - 1}$ for all $k \\ge 2.$ Compute\n\\[\\frac{1}{x_1 + 1} + \\frac{1}{x_2 + 1} + \\frac{1}{x_3 + 1} + \\dotsb.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{115}"
} | {
"split": "train",
"index": 2967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a\\_n\\}$, $a\\_1=1$, $a\\_{n+1}=3a\\_n (n∈N^{})$，then $a\\_3=$ _______ , $S\\_5=$ _______ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 2968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse with its foci on the $y$-axis, an eccentricity of $\\frac{2\\sqrt{2}}{3}$, and one focus at $(0, 2\\sqrt{2})$.\n(1) Find the standard equation of the ellipse;\n(2) A moving line $l$ passes through point $P(-1,0)$, intersecting a circle $O$ centered at the origi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{62}}{2}"
} | {
"split": "train",
"index": 2969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in $\\{(x, y) \\mid 1 \\leq x, y \\leq 5\\}$ blue such that the distance between any two blue points is not an integer. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 2970,
"solution": "We can see that no two blue points can have the same $x$ or $y$ coordinate. The blue points then must make a permutation of $1,2,3,4,5$ that avoid the pattern of $3-4-5$ triangles. It is not hard to use complementary counting to get the answer from here. There are 8... |
deepscaler | [
{
"role": "user",
"content": "Roger has exactly one of each of the first 22 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. The graph below shows the number of states that joined the union in each decade. What fraction of Roger's 22 coins repr... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{11}"
} | {
"split": "train",
"index": 2971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Misha is the 50th best as well as the 50th worst student in her grade. How many students are in Misha's grade? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 2972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two non-collinear vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $|\\overrightarrow{a}|=|\\overrightarrow{b}|$, and $\\overrightarrow{a}\\perp(\\overrightarrow{a}-2\\overrightarrow{b})$, the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 2973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\tan \\theta = \\frac{5}{12}\\), where \\(180^{\\circ} \\leq \\theta \\leq 270^{\\circ}\\). If \\(A = \\cos \\theta + \\sin \\theta\\), find the value of \\(A\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{17}{13}"
} | {
"split": "train",
"index": 2974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 2975,
"solution": "1. **Total Outcomes**: When three fair dice are tossed, each die has 6 faces, and each face is equally likely to come up. Therefore, the total number of outcomes when three dice are tossed is $6 \\times 6 \\times 6 = 6^3 = 216$.\n\n2. **Favorable Outcomes**: We need... |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 81 be written as the sum of three positive perfect squares if the order of the three perfect squares does not matter? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $10 \\times 10$ table consists of 100 unit cells. A block is a $2 \\times 2$ square consisting of 4 unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n-1$ blocks of $C$ cover the table. Fin... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 2977,
"solution": "Consider an infinite table divided into unit cells. Any $2 \\times 2$ square consisting of 4 unit cells of the table we also call a block. Fix arbitrary finite set $M$ of blocks lying on the table. Now we will consider arbitrary finite sets of unit cells of the tabl... |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer that must divide the product of any $5$ consecutive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 2978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=ax^{3}+bx+1$, and $f(-2)=3$, find $f(2)=$ _____ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 2979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $\\sqrt{3}b\\cos A - a\\sin B = 0$.\n\n$(1)$ Find the measure of angle $A$; \n$(2)$ Given that $c=4$ and the area of $\\triangle ABC$ is $6\\sqrt... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 2980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $A B C$ is a triangle with circumcenter $O$ and orthocenter $H$ such that $A, B, C, O$, and $H$ are all on distinct points with integer coordinates. What is the second smallest possible value of the circumradius of $A B C$ ? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 2981,
"solution": "Assume without loss of generality that the circumcenter is at the origin. By well known properties of the Euler line, the centroid $G$ is such that $O, G$, and $H$ are collinear, with $G$ in between $O$ and $H$, such that $G H=2 G O$. Thus, since $G=\\frac{1}{3}(A+B... |
deepscaler | [
{
"role": "user",
"content": "A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\\frac {a\\sqrt {b}}{c}$, where $a$, $b$, and $c$ are posi... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 2982,
"solution": "\nTo solve this problem, we need to understand the geometry of the octahedron and how the plane intersects it. An octahedron can be thought of as two square pyramids base-to-base. Each face of the octahedron is an equilateral triangle.\n\n#### Step 1: Understanding ... |
deepscaler | [
{
"role": "user",
"content": "A wealthy king has his blacksmith fashion him a large cup, whose inside is a cone of height 9 inches and base diameter 6 inches. At one of his many feasts, he orders the mug to be filled to the brim with cranberry juice. For each positive integer $n$, the king stirs his drink v... | math | {
"style": "rule",
"ground_truth": "\\frac{216 \\pi^{3}-2187 \\sqrt{3}}{8 \\pi^{2}}"
} | {
"split": "train",
"index": 2983,
"solution": "First, we find the total amount of juice consumed. We can simply subtract the amount of juice remaining at infinity from the initial amount of juice in the cup, which of course is simply the volume of the cup; we'll denote this value by $V$. Since volume in the cup ... |
deepscaler | [
{
"role": "user",
"content": "Select 4 individuals from a group of 6 to visit Paris, London, Sydney, and Moscow, with the requirement that each city is visited by one person, each individual visits only one city, and among these 6 individuals, individuals A and B shall not visit Paris. The total number of d... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 2984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Mendez awards extra credit on quizzes to his students with quiz grades that exceed the class mean. Given that 107 students take the same quiz, what is the largest number of students who can be awarded extra credit? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 2985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The chess club has 20 members: 12 boys and 8 girls. A 4-person team is chosen at random. What is the probability that the team has at least 2 boys and at least 1 girl? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4103}{4845}"
} | {
"split": "train",
"index": 2986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of functions $f:\\{1,2,3,4,5,6,7,8,9,10\\} \\rightarrow \\{1,2,3,4,5\\}$ that have the property that for $1\\leq x\\leq 5$ it is true that $f(f(x))=x$ . Given that $N$ can be written in the form $5^a\\cdot b$ for positive integers $a$ and $b$... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 2987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the closed region formed by the line $y = nx$ and the curve $y = x^2$ is \\_\\_\\_\\_\\_\\_ when the binomial coefficients of the third and fourth terms in the expansion of $(x - \\frac{2}{x})^n$ are equal. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{125}{6}"
} | {
"split": "train",
"index": 2988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{15}{4}, 0 \\right)"
} | {
"split": "train",
"index": 2989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all natural numbers whose own divisors can be paired such that the numbers in each pair differ by 545. An own divisor of a natural number is a natural divisor different from one and the number itself. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "1094"
} | {
"split": "train",
"index": 2990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^3 + 3(a-1)x^2 - 12ax + b$ has a local maximum $M$ at $x=x_1$ and a local minimum $N$ at $x=x_2$,\n\n(1) If the tangent line of the graph of $f(x)$ at its intersection with the y-axis is $24x - y - 10 = 0$, find the values of $x_1$, $x_2$, $M$, and $... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 2991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of $a$ is it true that the line $y=x+a$ passes through the vertex of parabola $y=x^2+a^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 2992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve \\[\\frac{3x^2+2x+1}{x-1}=3x+1\\]for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}2"
} | {
"split": "train",
"index": 2993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four points in the order \\( A, B, C, D \\) lie on a circle with the extension of \\( AB \\) meeting the extension of \\( DC \\) at \\( E \\) and the extension of \\( AD \\) meeting the extension of \\( BC \\) at \\( F \\). Let \\( EP \\) and \\( FQ \\) be tangents to this circ... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"split": "train",
"index": 2994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1, a_2, a_3,\\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-275"
} | {
"split": "train",
"index": 2995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "By permuting the digits of 20130518, how many different eight-digit positive odd numbers can be formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3600"
} | {
"split": "train",
"index": 2996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, what is the value of $x$? [asy]\nsize(120);\ndraw(Circle((0,0),1));\ndraw((0,0)--(.5,sqrt(3)/2));\ndraw((0,0)--(sqrt(3)/2,.5));\ndraw((0,0)--(sqrt(3)/2,-.5));\ndraw((0,0)--(-1,0));\nlabel(\"$4x^\\circ$\",(0,0),NNW); label(\"$5x^\\circ$\",(0,0),SSW);\nlabel(\"$2x... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 2997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\\sum^{2014}_{k=0} a_k {x \\choose k}$ for some real numbers $a_0,... , a_{2014}$ . Find the largest integer $m$ such that $2^m$ divides $a_{2013}$ . Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 2998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $y=kx^{\\frac{1}{4}}$ and $y=3\\sqrt{2}$ at $x=81$, what is the value of $y$ at $x=4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 2999,
"solution": ""
} |
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