data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given that $α$ is an angle in the second quadrant and $\\cos (α+π)= \\frac {3}{13}$.\n(1) Find the value of $\\tan α$;\n(2) Find the value of $\\sin (α- \\frac {π}{2}) \\cdot \\sin (-α-π)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{12\\sqrt{10}}{169}"
} | {
"split": "train",
"index": 18249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"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. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac12"
} | {
"split": "train",
"index": 18250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two types of steel plates need to be cut into three sizes $A$, $B$, and $C$. The number of each size that can be obtained from each type of steel plate is shown in the table:\n\n\\begin{tabular}{|l|c|c|c|}\n\\hline & Size $A$ & Size $B$ & Size $C$ \\\\\n\\hline First type of st... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f\\left(x\\right)=\\ln |a+\\frac{1}{{1-x}}|+b$ is an odd function, then $a=$____, $b=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\ln 2"
} | {
"split": "train",
"index": 18252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people are playing \"Easter egg battle.\" In front of them is a large basket of eggs. They randomly pick one egg each and hit them against each other. One of the eggs breaks, the defeated player takes a new egg, and the winner keeps their egg for the next round (the outcome... | math | {
"style": "rule",
"ground_truth": "11/12"
} | {
"split": "train",
"index": 18253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "749"
} | {
"split": "train",
"index": 18254,
"solution": "All integers $a$ will have factorization $2^a3^b5^c7^d...$. Therefore, the number of factors in $a^7$ is $(7a+1)(7b+1)...$, and for $a^8$ is $(8a+1)(8b+1)...$. The most salient step afterwards is to realize that all numbers $N$ not $1 \\pmod{7}$ and also not $1 \\p... |
deepscaler | [
{
"role": "user",
"content": "Given the nine-sided regular polygon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\\{A_1, A_2, \\ldots A_9\\}$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 18255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair 6-sided die is rolled. If I roll $n$, then I win $n^2$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$15.17"
} | {
"split": "train",
"index": 18256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts? Let's think s... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 18257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence\n$$\n\\begin{array}{l}\na_{0}=134, a_{1}=150, \\\\\na_{k+1}=a_{k-1}-\\frac{k}{a_{k}} \\quad (k=1,2, \\cdots, n-1).\n\\end{array}\n$$\n\nIf \\( a_n = 0 \\), then \\( n \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 18258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111, R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=R_{24}/R_4$ is an integer whose base-ten representation is a sequence containing only ones ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 18259,
"solution": "1. **Expression for $R_k$:** \n The number $R_k$ is defined as an integer consisting of $k$ ones in its decimal representation. We can express $R_k$ as:\n \\[\n R_k = 111\\ldots1 \\quad (\\text{$k$ times})\n \\]\n This can be written as a sum of powers o... |
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there, given that it contains repeating letters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 18260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the six solutions of $x^6=-64$ are written in the form $a+bi$, where $a$ and $b$ are real, find the product of those solutions with $a>0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 18261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive integer $n$ such that\n\n$$\\arctan\\frac {1}{3} + \\arctan\\frac {1}{4} + \\arctan\\frac {1}{5} + \\arctan\\frac {1}{n} = \\frac {\\pi}{4}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 18262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Choose one of the three conditions in $①$ $ac=\\sqrt{3}$, $②$ $c\\sin A=3$, $③$ $c=\\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.<br/>Qu... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 18263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 20 students in a class. In total, 10 of them have black hair, 5 of them wear glasses, and 3 of them both have black hair and wear glasses. How many of the students have black hair but do not wear glasses? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 18264,
"solution": "Since 10 students have black hair and 3 students have black hair and wear glasses, then a total of $10-3=7$ students have black hair but do not wear glasses."
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x,$ if \\[|x-20| + |x-18| = |2x-36|.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 18265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Along a road, there are 25 poles in a single row. Sometimes a siskin lands on one of the poles, and immediately a siskin flies off from one of the neighboring poles (if there was at least one siskin sitting on the neighboring poles at that moment). Also, each pole can hold no m... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 18266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\(PQRS\\), the lengths of the bases \\(PQ\\) and \\(RS\\) are 10 and 23 respectively. The legs of the trapezoid are extended beyond \\(P\\) and \\(Q\\) to meet at point \\(T\\). What is the ratio of the area of triangle \\(TPQ\\) to the area of trapezoid \\(PQRS\\... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{429}"
} | {
"split": "train",
"index": 18267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in the $24$-sided polygon shown i... | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 18268,
"solution": "1. **Understanding the Problem:**\n We have three identical square sheets of paper, each with side length $6$. The middle sheet is rotated $30^\\circ$ clockwise, and the top sheet is rotated $60^\\circ$ clockwise. We need to find the area of the resulting $24$-si... |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a$ and $b$ satisfy $a^2+b^2-2a+4b=-5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 18269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street? Let's think st... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 18270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral \\(ABCD\\), we have \\(AB=5\\), \\(BC=6\\), \\(CD=5\\), \\(DA=4\\), and \\(\\angle ABC=90^\\circ\\). Let \\(AC\\) and \\(BD\\) meet at \\(E\\). Compute \\(\\frac{BE}{ED}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 18271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$O$ is the center of square $A B C D$, and $M$ and $N$ are the midpoints of $\\overline{B C}$ and $\\overline{A D}$, respectively. Points $A^{\\prime}, B^{\\prime}, C^{\\prime}, D^{\\prime}$ are chosen on $\\overline{A O}, \\overline{B O}, \\overline{C O}, \\overline{D O}$, res... | math | {
"style": "rule",
"ground_truth": "8634"
} | {
"split": "train",
"index": 18272,
"solution": "Assume without loss of generality that the side length of $A B C D$ is 1 so that the area of the square is also 1 . This also means that $O M=O N=\\frac{1}{2}$. As $A^{\\prime} B^{\\prime} M C^{\\prime} D^{\\prime} N$ is equiangular, it can be seen that $\\angle A^... |
deepscaler | [
{
"role": "user",
"content": "John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.\nJohn looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this ... | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 18273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a trapezoid with $AB \\parallel CD$, $AB=6$, and $CD=15$. If the area of $\\triangle AED=30$, what is the area of $\\triangle AEB?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a, b \\in \\mathbb{R}$, $m = ab + 1$, $n = a + b$.\n- $(1)$ If $a > 1$, $b < 1$, compare the sizes of $m$ and $n$.\n- $(2)$ If $a > 1$, $b > 1$, and $m - n = 49$, find the minimum value of $a + b$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 18275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jenna is at a festival with six friends, making a total of seven people. They all want to participate in various group activities requiring groups of four or three people. How many different groups of four can be formed, and how many different groups of three can be formed from... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 18276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the sum of all the real coefficients of the expansion of ${(1+ix)}^{2011}$. What is $\\log_{2}(T)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1005"
} | {
"split": "train",
"index": 18277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers in Grace's sequence, starting from 43 and each number being 4 less than the previous one, are positive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 18278,
"solution": "We write out the numbers in the sequence until we obtain a negative number: $43,39,35,31,27,23,19,15,11,7,3,-1$. Since each number is 4 less than the number before it, then once a negative number is reached, every following number will be negative. Thus, Grace writ... |
deepscaler | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is an ace? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{52}"
} | {
"split": "train",
"index": 18279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x^{5}+ax^{3}+bx-8$, if $f(-2)=10$, find the value of $f(2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"split": "train",
"index": 18280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the differences between the scores of 14 students in a group and the class average score of 85 are 2, 3, -3, -5, 12, 12, 8, 2, -1, 4, -10, -2, 5, 5, find the average score of this group. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "87.29"
} | {
"split": "train",
"index": 18281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=4\\sin x\\cos \\left(x- \\frac {\\pi}{3}\\right)- \\sqrt {3}$. \n(I) Find the smallest positive period and zeros of $f(x)$. \n(II) Find the maximum and minimum values of $f(x)$ in the interval $\\left[ \\frac {\\pi}{24}, \\frac {3\\pi}{4}\\right]$. Le... | math | {
"style": "rule",
"ground_truth": "- \\sqrt {2}"
} | {
"split": "train",
"index": 18282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the different prime factors of $210630$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 18283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The difference when the product of 4 and a number is subtracted from 16 is greater than 10. How many positive integers satisfy the given conditions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 18284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $\\frac{3x - 1}{4x-4} = \\frac23$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 18285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Miki's father is saving money in a piggy bank for the family's vacation, adding to it once a week. Miki counts and notes how much money has accumulated every week and looks for patterns in the growth. Let $P_{n}$ denote the amount in the piggy bank in the $n$-th week (in forint... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 2x^3 - 3ax^2 + 3a - 2$ ($a \\in \\mathbb{R}$).\n\n$(1)$ If $a=1$, determine the intervals of monotonicity for the function $f(x)$.\n\n$(2)$ If the maximum value of $f(x)$ is $0$, find the value of the real number $a$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{3}"
} | {
"split": "train",
"index": 18287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The prime factorization of 2160 is $2^4 \\times 3^3 \\times 5$. How many of its positive integer factors are perfect squares? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 18288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1, f(2)=-4, f(-3)=-9$, and $f(4)=-16$, find $f(1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 18289,
"solution": "The given data tells us that the roots of $f(x)+x^{2}$ are $-1,2,-3$, and 4. Combining with the fact that $f$ is monic and quartic we get $f(x)+x^{2}=(x+1)(x-2)(x+3)(x-4)$. Hence $f(1)=(2)(-1)(4)(-3)-1=\\mathbf{23}$."
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\alpha, \\beta, \\gamma\\) satisfy \\(0<\\alpha<\\beta<\\gamma<2 \\pi\\), and for any \\(x \\in \\mathbf{R}\\), \\(\\cos (x+\\alpha) + \\cos (x+\\beta) + \\cos (x+\\gamma) = 0\\), determine the value of \\(\\gamma - \\alpha\\). Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"split": "train",
"index": 18290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $5^{5^{5^5}}$ is divided by 500. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 18291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( M \\) is the midpoint of side \\( BC \\) of triangle \\( ABC \\), where \\( AB = 17 \\), \\( AC = 30 \\), and \\( BC = 19 \\). A circle is constructed with diameter \\( AB \\). A point \\( X \\) is chosen arbitrarily on this circle. What is the minimum possible value ... | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"split": "train",
"index": 18292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions to\n\\[\\left| x^2 - x + \\frac{1}{2010} \\right| = \\frac{1}{2010}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2008}{1005}"
} | {
"split": "train",
"index": 18293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, $1000+1000+2$ and $1000+2+1000$ are considered to be the same way)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "334000"
} | {
"split": "train",
"index": 18294,
"solution": "Call the three numbers that sum to $2002 A, B$, and $C$. In order to prevent redundancy, we will consider only cases where $A \\leq B \\leq C$. Then $A$ can range from 1 to 667, inclusive. For odd $A$, there are $1000-\\frac{3(A-1)}{2}$ possible values for $B$. For... |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha \\in \\left(0, \\frac{\\pi}{3}\\right)$, satisfying $\\sqrt{3}\\sin\\alpha + \\cos\\alpha = \\frac{\\sqrt{6}}{2}$.\n\n$(1)$ Find the value of $\\cos\\left(\\alpha + \\frac{\\pi}{6}\\right)$; \n$(2)$ Find the value of $\\cos\\left(2\\alpha + \\frac{7\\pi}{12}\\righ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2} - \\sqrt{30}}{8}"
} | {
"split": "train",
"index": 18295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a sequence of polynomials as follows: let $a_{1}=3 x^{2}-x$, let $a_{2}=3 x^{2}-7 x+3$, and for $n \\geq 1$, let $a_{n+2}=\\frac{5}{2} a_{n+1}-a_{n}$. As $n$ tends to infinity, what is the limit of the sum of the roots of $a_{n}$ ? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{3}"
} | {
"split": "train",
"index": 18296,
"solution": "By using standard methods for solving linear recurrences, we see that this recurrence has a characteristic polynomial of $x^{2}-\\frac{5}{2} x+1=\\left(x-\\frac{1}{2}\\right)(x-2)$, hence $a_{n}(x)=c(x) \\cdot 2^{n}+d(x) \\cdot 2^{-n}$ for some polynomials $c$ and ... |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that\n1. each of $T_1, T_2$ , and $T_3$ is a subset of $\\{1, 2, 3, 4\\}$ ,\n2. $T_1 \\subseteq T_2 \\cup T_3$ ,\n3. $T_2 \\subseteq T_1 \\cup T_3$ , and\n4. $T_3\\subseteq T_1 \\cup T_2$ . Let's think s... | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 18297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To welcome the 2008 Olympic Games, a craft factory plans to produce the Olympic logo \"China Seal\" and the Olympic mascot \"Fuwa\". The factory mainly uses two types of materials, A and B. It is known that producing a set of the Olympic logo requires 4 boxes of material A and ... | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"split": "train",
"index": 18298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of the infinite series \\[\n\\sum_{n=2}^{\\infty} \\frac{n^4+3n^2+10n+10}{2^n \\cdot \\left(n^4+4\\right)}\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{10}"
} | {
"split": "train",
"index": 18299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the 95 numbers $1^2, 2^2, 3^2, \\ldots, 95^2$, how many of them have an odd digit in the tens place? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 18300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Anna Quick needs to pick up her friend from the airport every Friday. The airport is a certain distance from her home. If she drives at an average speed of 50 miles per hour, she arrives 5 minutes late, and if she drives at an average speed of 75 miles per hour, she arrive... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 18301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the 6-digit number beginning and ending in the digit 2 that is the product of three consecutive even integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "287232"
} | {
"split": "train",
"index": 18302,
"solution": "Because the last digit of the product is 2, none of the three consecutive even integers end in 0. Thus they must end in $2,4,6$ or $4,6,8$, so they must end in $4,6,8$ since $2 \\cdot 4 \\cdot 6$ does not end in 2. Call the middle integer $n$. Then the product is $... |
deepscaler | [
{
"role": "user",
"content": "The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (Assume that the graph has the standard unit scale.)\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nrea... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 18303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its ... | math | {
"style": "rule",
"ground_truth": "683"
} | {
"split": "train",
"index": 18304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $p$ if $12^3=\\frac{9^2}3\\cdot2^{12p}$. 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": 18305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.) Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 18306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the difference between the sum of the first 1000 even counting numbers including 0, and the sum of the first 1000 odd counting numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1000"
} | {
"split": "train",
"index": 18307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\\circ$, $60^\\circ$, and $60.001^\\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 18308,
"solution": "1. **Define the angles and setup the problem:** Let $\\triangle A_0B_0C_0$ be a triangle with angles $\\angle C_0A_0B_0 = x_0 = 59.999^\\circ$, $\\angle A_0B_0C_0 = y_0 = 60^\\circ$, and $\\angle B_0C_0A_0 = z_0 = 60.001^\\circ$. For each positive integer $n$, defi... |
deepscaler | [
{
"role": "user",
"content": "Observation: Given $\\sqrt{5}≈2.236$, $\\sqrt{50}≈7.071$, $\\sqrt[3]{6.137}≈1.8308$, $\\sqrt[3]{6137}≈18.308$; fill in the blanks:<br/>① If $\\sqrt{0.5}\\approx \\_\\_\\_\\_\\_\\_.$<br/>② If $\\sqrt[3]{x}≈-0.18308$, then $x\\approx \\_\\_\\_\\_\\_\\_$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "-0.006137"
} | {
"split": "train",
"index": 18309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer $n$ is chosen uniformly at random from the set $\\{1,2,3, \\ldots, 2023!\\}$. Compute the probability that $$\\operatorname{gcd}\\left(n^{n}+50, n+1\\right)=1$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{265}{357}"
} | {
"split": "train",
"index": 18310,
"solution": "If $n$ is even, we need $\\operatorname{gcd}(n+1,51)=1$. If $n$ is odd, we need $\\operatorname{gcd}(n+1,49)=1$. Thus, the answer is $$\\frac{1}{2}\\left(\\frac{\\varphi(49)}{49}+\\frac{\\varphi(51)}{51}\\right)=\\frac{265}{357}$$"
} |
deepscaler | [
{
"role": "user",
"content": "From the sixteen natural numbers $1, 2, 3, 4, \\cdots, 15, 16$, if $n$ numbers are randomly selected, there will always be such two numbers: one is three times the other. What is the smallest $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two hourglasses - one for 7 minutes and one for 11 minutes. An egg needs to be boiled for 15 minutes. How can you measure this amount of time using the hourglasses? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 18312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of triangle $ABC$ given below:\n\n[asy]\nunitsize(1inch);\npair A,B,C;\nA = (0,0);\nB = (1,0);\nC = (0,1);\ndraw (A--B--C--A,linewidth(0.9));\ndraw(rightanglemark(B,A,C,3));\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,N);\nlabel(\"$1$\",(B+C)/2,NE);... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 18313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer \\( n \\) is said to be increasing if, by reversing the digits of \\( n \\), we get an integer larger than \\( n \\). For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four-digit positiv... | math | {
"style": "rule",
"ground_truth": "4005"
} | {
"split": "train",
"index": 18314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An escalator has \\( n \\) visible steps and descends at a constant speed. Two boys, \\( A \\) and \\( Z \\), walk down the moving escalator at a steady pace. Boy \\( A \\) walks twice as many steps per minute as boy \\( Z \\). \\( A \\) reaches the bottom after walking 27 step... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 18315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\"Sun\", \"Xing\", and \"Zhe\" each represent a digit. The product of the two three-digit numbers 'SunXingZhe' and 'ZheXingSun' is 78445. Can you find the sum of these two three-digit numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "686"
} | {
"split": "train",
"index": 18316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy: $|\\overrightarrow{a}| = \\sqrt{2}$, $|\\overrightarrow{b}| = 2$, and $(\\overrightarrow{a} - \\overrightarrow{b}) \\perp \\overrightarrow{a}$, find the angle between $\\overrightarrow{a}$ and $\\overri... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 18317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three workers are digging a pit. They work in shifts, with each worker working as long as it takes for the other two to dig half of the pit. Working in this manner, they dug the pit. How many times faster would the three workers dig the same pit if they worked simultaneously? L... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 18318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered four-tuples of integers $(a,b,c,d)\\,$ with $0 < a < b < c < d < 500\\,$ satisfy $a + d = b + c\\,$ and $bc - ad = 93\\,$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "870"
} | {
"split": "train",
"index": 18319,
"solution": "Let $b = a + m$ and $c = a + m + n$. From $a + d = b + c$, $d = b + c - a = a + 2m + n$.\n Substituting $b = a + m$, $c = a + m + n$, and $d = b + c - a = a + 2m + n$ into $bc - ad = 93$, \\[bc - ad = (a + m)(a + m + n) - a(a + 2m + n) = m(m + n). = 93 = 3(31)\\] H... |
deepscaler | [
{
"role": "user",
"content": "Arrange 6 volunteers for 3 different tasks, each task requires 2 people. Due to the work requirements, A and B must work on the same task, and C and D cannot work on the same task. How many different arrangements are there? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the mean of one set consisting of seven numbers is 18, and the mean of another set consisting of eight numbers is 16. What is the mean of all fifteen numbers combined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{254}{15}"
} | {
"split": "train",
"index": 18321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of the greatest common divisor and the least common multiple of $18$ and $42.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "756"
} | {
"split": "train",
"index": 18322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Last year Mr. Jon Q. Public received an inheritance. He paid $20\\%$ in federal taxes on the inheritance, and paid $10\\%$ of what he had left in state taxes. He paid a total of $\\textdollar10500$ for both taxes. How many dollars was his inheritance? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "37500"
} | {
"split": "train",
"index": 18323,
"solution": "1. **Define the variable**: Let $x$ be the total amount of Mr. Jon Q. Public's inheritance.\n\n2. **Calculate federal taxes**: He pays $20\\%$ in federal taxes, which is $0.2x$.\n\n3. **Calculate remaining amount after federal taxes**: After paying federal taxes, h... |
deepscaler | [
{
"role": "user",
"content": "How many positive perfect square integers are factors of the product $\\left(2^{10}\\right)\\left(3^{12}\\right)\\left(5^{15}\\right)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 18324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 18325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a trapezoid with bases 3 and 4, find the length of the segment parallel to the bases that divides the area of the trapezoid in the ratio $5:2$, counting from the shorter base. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"split": "train",
"index": 18326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$ . Find the maximum value of $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 18327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two points are chosen inside the square $\\{(x, y) \\mid 0 \\leq x, y \\leq 1\\}$ uniformly at random, and a unit square is drawn centered at each point with edges parallel to the coordinate axes. The expected area of the union of the two squares can be expressed as $\\frac{a}{... | math | {
"style": "rule",
"ground_truth": "1409"
} | {
"split": "train",
"index": 18328,
"solution": "Let $\\Delta x$ and $\\Delta y$ be the positive differences between the $x$ coordinates and $y$ coordinates of the centers of the squares, respectively. Then, the length of the intersection of the squares along the $x$ dimension is $1-\\Delta x$, and likewise the l... |
deepscaler | [
{
"role": "user",
"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. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 18329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each letter represents a non-zero digit. What is the value of $t?$ \\begin{align*}\nc + o &= u \\\\\nu + n &= t \\\\\nt + c &= s \\\\\no + n + s &= 12\n\\end{align*} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 18330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the second quadrant, and $\\tan 2\\theta = -2\\sqrt{2}$.\n(1) Find the value of $\\tan \\theta$.\n(2) Calculate the value of $\\frac {2\\cos^{2} \\frac {\\theta}{2}-\\sin\\theta-\\tan \\frac {5\\pi}{4}}{\\sqrt {2}\\sin(\\theta + \\frac {\\pi}... | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 18331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of one square is $3$ times the perimeter of another square. The area of the larger square is how many times the area of the smaller square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 18332,
"solution": "1. **Define Variables:**\n Let $a$ be the side length of the larger square and $b$ be the side length of the smaller square.\n\n2. **Relate Perimeters:**\n The perimeter of a square is given by $4$ times the side length. If the perimeter of the larger square is... |
deepscaler | [
{
"role": "user",
"content": "In Zuminglish, all words consist only of the letters $M, O,$ and $P$. As in English, $O$ is said to be a vowel and $M$ and $P$ are consonants. A string of $M's, O's,$ and $P's$ is a word in Zuminglish if and only if between any two $O's$ there appear at least two consonants. Le... | math | {
"style": "rule",
"ground_truth": "936"
} | {
"split": "train",
"index": 18333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles triangle \\(ABC \\) (\\(AB = BC\\)), a point \\(D\\) is taken on the side \\(BC\\) such that \\(BD : DC = 1 : 4\\).\n\nIn what ratio does the line \\(AD\\) divide the height \\(BE\\) of the triangle \\(ABC\\), counted from the vertex \\(B\\)? Let's think step by... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 18334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum value of the product $\\prod_{i=1}^{6} \\frac{a_{i}-a_{i+1}}{a_{i+2}-a_{i+3}}$ given that $\\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\\right)$ is a permutation of $(1,2,3,4,5,6)$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 18335,
"solution": "The product always evaluates to 1."
} |
deepscaler | [
{
"role": "user",
"content": "\nA batch of disaster relief supplies is loaded into 26 trucks. The trucks travel at a constant speed of \\( v \\) kilometers per hour directly to the disaster area. If the distance between the two locations is 400 kilometers and the distance between every two trucks must be at... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The operation $\\nabla$ is defined by $g \\nabla h=g^{2}-h^{2}$. If $g>0$ and $g \\nabla 6=45$, what is the value of $g$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 18337,
"solution": "Using the definition of the operation, $g \\nabla 6=45$ gives $g^{2}-6^{2}=45$. Thus, $g^{2}=45+36=81$. Since $g>0$, then $g=\\sqrt{81}=9$."
} |
deepscaler | [
{
"role": "user",
"content": "How many times do the graphs $r = 4 \\cos \\theta$ and $r = 8 \\sin \\theta$ intersect? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 18338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Maria needs to build a circular fence around a garden. Based on city regulations, the garden's diameter needs to be close to 30 meters, with an allowable error of up to $10\\%$. After building, the fence turned out to have a diameter of 33 meters. Calculate the area she thought... | math | {
"style": "rule",
"ground_truth": "21\\%"
} | {
"split": "train",
"index": 18339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin x + \\cos x = \\frac{1}{2}$, where $x \\in [0, \\pi]$, find the value of $\\sin x - \\cos x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"split": "train",
"index": 18340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ Find the prime factorization of the minimum possible value of $x$ and determine the sum of the exponents and the prime factors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 18341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{1 \\le a < b < c} \\frac{1}{2^a 3^b 5^c}.\\](The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \\le a < b < c.$) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1624}"
} | {
"split": "train",
"index": 18342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle PQRS and right triangle SRT share side SR and have the same area. Rectangle PQRS has dimensions PQ = 4 and PS = 8. Find the length of side RT. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 18343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, if for all $n \\in \\mathbb{N}^*$, it holds that $a_n = -3a_{n+1}$, and $$\\lim_{n \\to \\infty}(a_{2}+a_{4}+a_{6}+\\ldots+a_{2n}) = \\frac{9}{2},$$ then the value of $a_1$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 18344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the points $(7, -9)$ and $(1, 7)$ as the endpoints of a diameter of a circle, calculate the sum of the coordinates of the center of the circle, and also determine the radius of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{73}"
} | {
"split": "train",
"index": 18345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tram ticket is called \"lucky in Leningrad style\" if the sum of its first three digits is equal to the sum of its last three digits. A tram ticket is called \"lucky in Moscow style\" if the sum of its digits in even positions is equal to the sum of its digits in odd position... | math | {
"style": "rule",
"ground_truth": "6700"
} | {
"split": "train",
"index": 18346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(0^{\\circ} < \\alpha < 30^{\\circ}\\), and \\(\\sin^6 \\alpha + \\cos^6 \\alpha = \\frac{7}{12}\\), then \\(1998 \\cos \\alpha = ?\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "333 \\sqrt{30}"
} | {
"split": "train",
"index": 18347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\frac {x}{1+x}$, then $f(1)+f(2)+f(3)+\\ldots+f(2017)+f( \\frac {1}{2})+f( \\frac {1}{3})+\\ldots+f( \\frac {1}{2017})=$ \\_\\_\\_\\_\\_\\_ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4033}{2}"
} | {
"split": "train",
"index": 18348,
"solution": ""
} |
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