data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "How many of the 200 students surveyed said that their favourite food was sandwiches, given the circle graph results? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26949,
"solution": "Since the angle in the sector representing cookies is $90^{\\circ}$, then this sector represents $\\frac{1}{4}$ of the total circle. Therefore, 25% of the students chose cookies as their favourite food. Thus, the percentage of students who chose sandwiches was $100... |
deepscaler | [
{
"role": "user",
"content": "In a 6 by 6 grid of points, what fraction of the larger rectangle's area is inside the shaded right triangle? The vertices of the shaded triangle correspond to grid points and are located at (1,1), (1,5), and (4,1). Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 26950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $0.5\\overline{10}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{101}{198}"
} | {
"split": "train",
"index": 26951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt{125}\\cdot\\sqrt{45}\\cdot \\sqrt{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75\\sqrt{10}"
} | {
"split": "train",
"index": 26952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(2x +3y)^{20}$ , find the number of coefficients divisible by $144$ .\n\n*Proposed by Hannah Shen* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $B$ and $D$ coincide, forming the pentagon $ABEFC$. What is the length of segment $EF$? Express your answer in simplest radical form. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 26954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \\cdot y = -2$. What is the value of \n\\[x + \\frac{x^3}{y^2} + \\frac{y^3}{x^2} + y?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 26955,
"solution": "\nGiven the equations:\n1. \\(x + y = 4\\)\n2. \\(x \\cdot y = -2\\)\n\nWe need to find the value of:\n\\[x + \\frac{x^3}{y^2} + \\frac{y^3}{x^2} + y\\]\n\n#### Step-by-step calculation:\n\n1. **Expression Simplification**:\n \\[x + \\frac{x^3}{y^2} + \\frac{y^3}... |
deepscaler | [
{
"role": "user",
"content": "Let $p(x)$ be a monic polynomial of degree 7 such that $p(0) = 0,$ $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ $p(6) = 6,$ and $p(7) = 7.$ Find $p(8).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40328"
} | {
"split": "train",
"index": 26956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the inequality $ax^{2}+bx+c \\gt 0$ with the solution set $\\{x\\left|\\right.1 \\lt x \\lt 2\\}$, find the solution set of the inequality $cx^{2}+bx+a \\gt 0$ in terms of $x$. When studying the above problem, Xiaoming and Xiaoning respectively came up with the following ... | math | {
"style": "rule",
"ground_truth": "-490"
} | {
"split": "train",
"index": 26957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the function $g$ on the set of integers such that \\[g(n)= \\begin{cases} n-4 & \\mbox{if } n \\geq 2000 \\\\ g(g(n+6)) & \\mbox{if } n < 2000. \\end{cases}\\] Determine $g(172)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 26958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the line $x+ay+6=0$ is parallel to the line $(a-2)x+3y+2a=0$, determine the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose there are 3 counterfeit coins of equal weight mixed with 12 genuine coins. All counterfeit coins weigh differently from the genuine coins. A pair of coins is selected at random without replacement from the 15 coins, followed by selecting a second pair from the remaining... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{19}"
} | {
"split": "train",
"index": 26960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\begin{cases} 2^{x}, & x < 0 \\\\ f(x-1)+1, & x\\geqslant 0 \\end{cases}$, calculate the value of $f(2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{2}"
} | {
"split": "train",
"index": 26961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different ways can 6 different books be distributed according to the following requirements? \n(1) Among three people, A, B, and C, one person gets 1 book, another gets 2 books, and the last one gets 3 books; \n(2) The books are evenly distributed to A, B, and C, wit... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 26962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of one side of the square \\(ABCD\\) is 4 units. A circle is drawn tangent to \\(\\overline{BC}\\) and passing through the vertices \\(A\\) and \\(D\\). Find the area of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25 \\pi}{4}"
} | {
"split": "train",
"index": 26963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shi... | math | {
"style": "rule",
"ground_truth": "7,15"
} | {
"split": "train",
"index": 26964,
"solution": "Let Alice's tower be of a height $a$, and Bob's tower a height $b$. Reflect the diagram over the ice to obtain an isosceles trapezoid. Then we get that by Ptolemy's Theorem, $4 a b=26^{2}-16^{2}=4 \\cdot 105$, thus $a b=105$. Hence $a \\in\\{1,3,5,7,15,21,35,105\\}... |
deepscaler | [
{
"role": "user",
"content": "Solve the equation: $x^{2}+5x-1=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-5-\\sqrt{29}}{2}"
} | {
"split": "train",
"index": 26965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $P$ and $R$ are located at (2, 1) and (12, 15) respectively. Point $M$ is the midpoint of segment $\\overline{PR}$. Segment $\\overline{PR}$ is reflected over the $x$-axis. What is the sum of the coordinates of the image of point $M$ (the midpoint of the reflected segmen... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest possible area, in square units, of a right triangle with side lengths $7$ units and $10$ units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 26967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a plane, five points are given such that no three points are collinear. The points are pairwise connected by lines. What is the number of intersection points of these lines (excluding the original five points), assuming that none of the drawn lines are parallel to each other... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 26968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be a subset of $\\{1,2,3,...,100\\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 26969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle can be divided into \\( n \\) equal squares. The same rectangle can also be divided into \\( n + 76 \\) equal squares. Find all possible values of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 26970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the value of $3^2 \\times 4 \\times 6^3 \\times 7!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39191040"
} | {
"split": "train",
"index": 26971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $f(2017)$ for a function $f(x)$ defined on $\\mathbb{R}$ that satisfies $f(x) \\cdot f(x+2) = 13$ given that $f(3) = 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{4}"
} | {
"split": "train",
"index": 26972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 26973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers aft... | math | {
"style": "rule",
"ground_truth": "314"
} | {
"split": "train",
"index": 26974,
"solution": "Let the biking rate be $b$, swimming rate be $s$, jogging rate be $j$, all in km/h.\n We have $2b + 3j + 4s = 74,2j + 3s + 4b = 91$. Subtracting the second from twice the first gives $4j + 5s = 57$. Mod 4, we need $s\\equiv1\\pmod{4}$. Thus, $(j,s) = (13,1),(8,5),(... |
deepscaler | [
{
"role": "user",
"content": "\nIt is known that the numbers \\( x, y, z \\) form an arithmetic progression in the given order with a common difference \\( \\alpha = \\arccos \\left(-\\frac{3}{7}\\right) \\), and the numbers \\( \\frac{1}{\\cos x}, \\frac{7}{\\cos y}, \\frac{1}{\\cos z} \\) also form an ari... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{13}"
} | {
"split": "train",
"index": 26975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a square of side length 1. $X$ and $Y$ are points on $BC$ and $CD$ respectively such that $CX = CY = m$. When extended, $AB$ meets $DX$ at $P$, $AD$ meets $BY$ at $Q$, $AX$ meets $DC$ at $R$, and $AY$ meets $BC$ at $S$. If $P, Q, R, S$ are collinear, find $m$. Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{3 - \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 26976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the following complex numbers $z$, which one has the property that $z^5$ has the greatest real part?\n- $z = -3$\n- $z = -2 + \\frac{1}{2}i$\n- $z = -\\frac{3}{2} + \\frac{3}{2}i$\n- $z = -1 + 2i$\n- $z = 3i$\nA) $-243$\nB) $-12.3125$\nC) $-168.75$\nD) $39$\nE) $0$ Let's thi... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 26977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a tennis club, each member has exactly $k > 0$ friends, and a tournament is organized in rounds such that each pair of friends faces each other in matches exactly once. Rounds are played in simultaneous matches, choosing pairs until they cannot choose any more (that is, amon... | math | {
"style": "rule",
"ground_truth": "2k - 1"
} | {
"split": "train",
"index": 26978,
"solution": "\nConsider a tennis club where each member has exactly \\( k > 0 \\) friends. The task is to determine the maximum number of rounds in a tournament where each pair of friends plays exactly once. Matches in each round are simultaneous, meaning no participant can pla... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest composite number that has no prime factors less than 15. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "289"
} | {
"split": "train",
"index": 26979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure shown, the ratio of $BD$ to $DC$ is $4$ to $3$. The area of $\\triangle ABD$ is $24$ square centimeters. What is the area of $\\triangle ADC$? [asy] size(85); defaultpen(linewidth(1)+fontsize(10));\n\npair A = (0,5.5), B=(0,0), D = (2,0), C = (3,0);\n\ndraw(A--B--... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 26980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( D \\) and \\( E \\) are located on the diagonals \\( A B_{1} \\) and \\( C A_{1} \\) of the lateral faces of the prism \\( A B C A_{1} B_{1} C_{1} \\) such that the lines \\( D E \\) and \\( B C_{1} \\) are parallel. Find the ratio of the segments \\( D E \\) and \\(... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 26981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a 10-ring target, the probabilities of hitting scores 10, 9, 8, 7, and 6 are $\\frac{1}{5}, \\frac{1}{4}, \\frac{1}{6}, \\frac{1}{8},$ and $\\frac{1}{10}$ respectively. The probability of hitting any other score (from 5 to 1) is $\\frac{1}{12}$. $A$ pays $B$ the score amount... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 26982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer is called jubilant if the number of 1 's in its binary representation is even. For example, $6=110_{2}$ is a jubilant number. What is the 2009 th smallest jubilant number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4018"
} | {
"split": "train",
"index": 26983,
"solution": "Notice that for each pair of consecutive positive integers $2 k$ and $2 k+1$, their binary representation differs by exactly one 1 (in the units digit), so exactly one of 2 and 3 is jubilant, exactly one of 4 and 5 is jubilant, etc. It follows that there are exactl... |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator: $\\sqrt{\\frac{3}{8}}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{4}"
} | {
"split": "train",
"index": 26984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n>0$ be an integer. Each face of a regular tetrahedron is painted in one of $n$ colors (the faces are not necessarily painted different colors.) Suppose there are $n^{3}$ possible colorings, where rotations, but not reflections, of the same coloring are considered the same... | math | {
"style": "rule",
"ground_truth": "1,11"
} | {
"split": "train",
"index": 26985,
"solution": "We count the possible number of colorings. If four colors are used, there are two different colorings that are mirror images of each other, for a total of $2\\binom{n}{4}$ colorings. If three colors are used, we choose one color to use twice (which determines the c... |
deepscaler | [
{
"role": "user",
"content": "Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 26986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $x$ and $y$ are positive real numbers such that $x^2 - 2xy + 3y^2 = 9$. Find the maximum possible value of $x^2 + 2xy + 3y^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18 + 9\\sqrt{3}"
} | {
"split": "train",
"index": 26987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand the following expression: $(13x+15)\\cdot 2x$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26x^2+30x"
} | {
"split": "train",
"index": 26988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many rectangles can be formed when the vertices are chosen from points on a 4x4 grid (having 16 points)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 26989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the measure, in units, of the hypotenuse of a right triangle with leg lengths of 75 and 100 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 26990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of the binomial ${({(\\frac{1}{x}}^{\\frac{1}{4}}+{{x}^{2}}^{\\frac{1}{3}})}^{n})$, the coefficient of the third last term is $45$. Find the coefficient of the term containing $x^{3}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 26991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that\n\n(a) the numbers are all different,\n(b) they sum to $13$, and\n(c) they are in increasing order, left to right.\nFirst, Casey looks at the number on... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 26992,
"solution": "1. **Initial Possibilities**: Given the conditions (a) all numbers are different, (b) they sum to $13$, and (c) they are in increasing order, we list all possible sets of numbers:\n - $(1,2,10)$\n - $(1,3,9)$\n - $(1,4,8)$\n - $(1,5,7)$\n - $(2,3,8)$\n ... |
deepscaler | [
{
"role": "user",
"content": "The perimeter of a rectangle is 24 inches. What is the number of square inches in the maximum possible area for this rectangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 26993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to choose 6 different numbers from $1, 2, \\cdots, 49$, where at least two numbers are consecutive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\binom{49}{6} - \\binom{44}{6}"
} | {
"split": "train",
"index": 26994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the Cesaro sum of a sequence with 99 terms is 1000, calculate the Cesaro sum of the sequence with 100 terms consisting of the numbers 1 and the first 99 terms of the original sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "991"
} | {
"split": "train",
"index": 26995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the square roots of a positive number are $x+1$ and $4-2x$, then the positive number is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 26996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C$: $\\frac{x^{2}}{4}-y^{2}=1$ with left and right foci $F\\_1$ and $F\\_2$ respectively, find the area of the quadrilateral $P\\_1P\\_2P\\_3P\\_4$ for a point $P$ on the hyperbola that satisfies $\\overrightarrow{PF_{1}}\\cdot \\overrightarrow{PF_{2}}=0$. ... | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{6}}{5}"
} | {
"split": "train",
"index": 26997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Diagonals \\( AC \\) and \\( BD \\) of the cyclic quadrilateral \\( ABCD \\) are perpendicular and intersect at point \\( M \\). It is known that \\( AM = 3 \\), \\( BM = 4 \\), and \\( CM = 6 \\). Find \\( CD \\). Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"split": "train",
"index": 26998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The shortest distance from a point on the curve $y=\\ln x$ to the line $y=x+2$ is what value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 26999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $0.\\overline{3}+0.\\overline{2}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac 59"
} | {
"split": "train",
"index": 27000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), \\(AB = 20\\), \\(BC = 21\\), and \\(CA = 29\\). Point \\(M\\) is on side \\(AB\\) with \\(\\frac{AM}{MB}=\\frac{3}{2}\\), while point \\(N\\) is on side \\(BC\\) with \\(\\frac{CN}{NB}=2\\). Points \\(P\\) and \\(Q\\) are on side \\(AC\\) such that line ... | math | {
"style": "rule",
"ground_truth": "224/15"
} | {
"split": "train",
"index": 27001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a certain meeting, a total of \\(12k\\) people attended. Each person greeted exactly \\(3k+6\\) other people. For any two individuals, the number of people who greeted both of them is the same. How many people attended the meeting? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 27002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a village, a plot of land is shaped as a right triangle, where one of the legs measures 5 units and another measures 12 units. Farmer Euclid decides to leave a small unplanted square at the vertex of this right angle, and the shortest distance from this unplanted square to t... | math | {
"style": "rule",
"ground_truth": "\\frac{412}{1000}"
} | {
"split": "train",
"index": 27003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A herd of elephants. Springs are bubbling at the bottom of the lake. A herd of 183 elephants could drink it dry in one day, and a herd of 37 elephants could do so in 5 days. How many days will it take for 1 elephant to drink the lake dry? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "365"
} | {
"split": "train",
"index": 27004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all integral solutions of \\[ a^2\\plus{}b^2\\plus{}c^2\\equal{}a^2b^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 0)"
} | {
"split": "train",
"index": 27005,
"solution": "\nWe are tasked with finding all integral solutions to the equation:\n\n\\[\na^2 + b^2 + c^2 = a^2b^2.\n\\]\n\nFirst, let's rewrite the equation and rearrange the terms:\n\n\\[\na^2b^2 - a^2 - b^2 = c^2.\n\\]\n\nThis suggests that \\( c^2 \\) must be non-negative, ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{\\pi}{4} < \\alpha < \\frac{3\\pi}{4}$, $0 < \\beta < \\frac{\\pi}{4}$, $\\sin(\\alpha + \\frac{\\pi}{4}) = \\frac{3}{5}$, and $\\cos(\\frac{\\pi}{4} + \\beta) = \\frac{5}{13}$, find the value of $\\sin(\\alpha + \\beta)$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"split": "train",
"index": 27006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} 1 & \\cos (a - b) & \\cos a \\\\ \\cos(a - b) & 1 & \\cos b \\\\ \\cos a & \\cos b & 1 \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 27007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of real solutions to\n\\[(x^{2006} + 1)(x^{2004} + x^{2002} + x^{2000} + \\dots + x^2 + 1) = 2006x^{2005}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 27008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dormitory of a certain high school senior class has 8 people. In a health check, the weights of 7 people were measured to be 60, 55, 60, 55, 65, 50, 50 (in kilograms), respectively. One person was not measured due to some reasons, and it is known that the weight of this stude... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the infinite geometric series: $$\\frac{5}{3} - \\frac{3}{5} + \\frac{9}{25} - \\frac{27}{125} + \\dots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{102}"
} | {
"split": "train",
"index": 27010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain academic knowledge competition, where the total score is 100 points, if the scores (ξ) of the competitors follow a normal distribution (N(80,σ^2) where σ > 0), and the probability that ξ falls within the interval (70,90) is 0.8, then calculate the probability that ... | math | {
"style": "rule",
"ground_truth": "0.1"
} | {
"split": "train",
"index": 27011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six bags of marbles contain $18, 19, 21, 23, 25$ and $34$ marbles, respectively. One bag contains chipped marbles only. The other $5$ bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Ja... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 27012,
"solution": "1. **Identify the total number of marbles:** \n The total number of marbles in all bags is $18 + 19 + 21 + 23 + 25 + 34 = 140$.\n\n2. **Set up the relationship between Jane's and George's marbles:** \n Let the number of marbles George has be $x$. Then, Jane h... |
deepscaler | [
{
"role": "user",
"content": "Given a woman was x years old in the year $x^2$, determine her birth year. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1980"
} | {
"split": "train",
"index": 27013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integer values of $x$ satisfy $|x|<3\\pi$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 27014,
"solution": "1. **Translate the inequality involving absolute value**: The inequality $|x| < 3\\pi$ can be rewritten as $-3\\pi < x < 3\\pi$.\n\n2. **Approximate $\\pi$ and calculate $3\\pi$**: Since $\\pi \\approx 3.14$, we calculate $3\\pi \\approx 3 \\times 3.14 = 9.42$.\n\n... |
deepscaler | [
{
"role": "user",
"content": "Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$ , find the sum of digits of $E$ Let's ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 27015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the product $(12)(15)(16)$, each factor written in base $b$, equals $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 27016,
"solution": "1. **Convert the given numbers to base 10 expressions**: \n - The numbers $12$, $15$, and $16$ in base $b$ are represented as $(b+2)$, $(b+5)$, and $(b+6)$ respectively.\n - The product $(12)(15)(16)$ in base $b$ translates to $(b+2)(b+5)(b+6)$ in base 10.\n\n2... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\frac{1}{3}x^{3}+ax^{2}+bx$ $(a,b\\in \\mathbb{R})$ attains a maximum value of $9$ at $x=-3$.\n\n$(I)$ Find the values of $a$ and $b$;\n\n$(II)$ Find the maximum and minimum values of the function $f(x)$ in the interval $[-3,3]$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "- \\frac{5}{3}"
} | {
"split": "train",
"index": 27017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, point $E$ is on $AB$, point $F$ is on $AC$, and $BF$ intersects $CE$ at point $P$. If the areas of quadrilateral $AEPF$ and triangles $BEP$ and $CFP$ are all equal to 4, what is the area of $\\triangle BPC$? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers \\underline{a} \\underline{b} \\underline{c}, if \\underline{a} \\underline{b} \\underline{c} is a multiple of $x$, then the three-digit (base 10) number \\underline... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 27019,
"solution": "Note that $\\overline{a b c 0}-\\overline{b c a}=a\\left(10^{4}-1\\right)$ must also be a multiple of $x$. Choosing $a=1$ means that $x$ divides $10^{3}-1$, and this is clearly a necessary and sufficient condition. The only two-digit factors of $10^{3}-1$ are 27 an... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the odd integers between $300$ and $500$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40000"
} | {
"split": "train",
"index": 27020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Investment funds A, B, and C claim that they can earn profits of 200%, 300%, and 500% respectively in one year. Tommy has $90,000 and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has... | math | {
"style": "rule",
"ground_truth": "30000"
} | {
"split": "train",
"index": 27021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean (average) of four numbers is $85$. If the largest of these numbers is $97$, then the mean of the remaining three numbers is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81.0"
} | {
"split": "train",
"index": 27022,
"solution": "1. Let the four numbers be $a, b, c,$ and $97$. Given that the arithmetic mean of these numbers is $85$, we can set up the equation:\n \\[\n \\frac{a+b+c+97}{4} = 85\n \\]\n\n2. Multiply both sides of the equation by $4$ to eliminate the fraction:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "The six edges of a tetrahedron $ABCD$ measure $7, 13, 18, 27, 36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is\n$\\textbf{(A)}\\ 7\\qquad \\textbf{(B)}\\ 13\\qquad \\textbf{(C)}\\ 18\\qquad \\textbf{(D)}\\ 27\\qquad \\textbf{(E)}\\ 36$\n L... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 27023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a 5-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $200$. For how many values of $n$ is $q+r$ divisible by $7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13050"
} | {
"split": "train",
"index": 27024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nWest, Non-West, Russia:\n1st place - Russia: 302790.13 cubic meters/person\n2nd place - Non-West: 26848.55 cubic meters/person\n3rd place - West: 21428 cubic meters/person Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "302790.13"
} | {
"split": "train",
"index": 27025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is \\( t \\) and the positive difference between the squares of these two numbers is 208. What is the larger of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 27026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular octagon is inscribed in a circle of radius 2 units. What is the area of the octagon? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16 \\sqrt{2} - 8(2)"
} | {
"split": "train",
"index": 27027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\[f(x) =\n\\begin{cases}\nx^2-4 &\\quad \\text{if } x \\ge -4, \\\\\nx + 3 &\\quad \\text{otherwise},\n\\end{cases}\n\\]then for how many values of $x$ is $f(f(x)) = 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 27028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the vertices of a regular 300-gon, the numbers from 1 to 300 are arranged in some order, each number appearing exactly once. It turns out that for each number \\(a\\), there are as many numbers smaller than \\(a\\) among the 15 closest numbers to it clockwise as there are am... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $3x^7-x^6-7x^5+2x^3+4x^2-11$ is divided by $2x-4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 27030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate\n\\[\\sum_{m = 1}^\\infty \\sum_{n = 1}^\\infty \\frac{1}{mn(m + n + 1)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 27031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid. What is $DE$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 27032,
"solution": "1. **Identify the areas of the shapes**: \n - The area of rectangle $ABCD$ is given by the formula for the area of a rectangle, $AB \\times AD$. Given $AB = 5$ and $AD = 6$, the area of rectangle $ABCD$ is:\n \\[\n AB \\cdot AD = 5 \\cdot 6 = 30\n \\]... |
deepscaler | [
{
"role": "user",
"content": "Cut a 3-meter-long rope into 7 equal segments. Each segment accounts for \\_\\_\\_\\_\\_\\_ of the total length, and each segment is \\_\\_\\_\\_\\_\\_ meters long. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 27033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) < 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 27034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the four vertices of the triangular pyramid $P-ABC$ lie on the surface of the sphere $O$, and $PA = PB = PC$. The triangle $ABC$ is an equilateral triangle with side length 2. Points $E$ and $F$ are the midpoints of $AC$ and $BC$ respectively, and $\\angle EPF = 60^\... | math | {
"style": "rule",
"ground_truth": "6 \\pi"
} | {
"split": "train",
"index": 27035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four circles, each with a radius of 7 inches, are arranged such that they touch each other and the sides of a square as shown. Determine the area, in square inches, of the square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 27036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nWhat is the maximum area that a rectangle can have if the coordinates of its vertices satisfy the equation \\( |y-x| = (y+x+1)(5-x-y) \\), and its sides are parallel to the lines \\( y = x \\) and \\( y = -x \\)? Give the square of the value of the maximum area found as the a... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 27037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\sqrt{2 \\cdot 4! \\cdot 4!}$ expressed as a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24\\sqrt{2}"
} | {
"split": "train",
"index": 27038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the focus of the parabola $y^{2}=ax$ coincides with the left focus of the ellipse $\\frac{x^{2}}{6}+ \\frac{y^{2}}{2}=1$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 27039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All three vertices of $\\triangle ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 27040,
"solution": "1. **Identify the Coordinates of Points on the Parabola**: \n Since $\\triangle ABC$ has all vertices on the parabola $y = x^2$ and $A$ is at the origin, the coordinates of $A$ are $(0,0)$. Let the coordinates of $B$ and $C$ be $(-a, a^2)$ and $(a, a^2)$ respecti... |
deepscaler | [
{
"role": "user",
"content": "When finding the sum $\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{7}$, the least common denominator used is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 27041,
"solution": "To find the least common denominator (LCD) for the sum $\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{7}$, we need to determine the least common multiple (LCM) of the denominators: $2, 3, 4, 5, 6, 7$.\n\n1. **Prime Factorization**:\n ... |
deepscaler | [
{
"role": "user",
"content": "Find the number of permutations \\( a_1, a_2, \\ldots, a_{10} \\) of the numbers \\( 1, 2, \\ldots, 10 \\), such that \\( a_{i+1} \\) is not less than \\( a_i - 1 \\) for \\( i = 1, 2, \\ldots, 9 \\). Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 27042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be an integer of the form $a^{2}+b^{2}$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \\leq \\sqrt{n}$, then $p$ divides $a b$. Determine all such $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n = 2, 5, 13"
} | {
"split": "train",
"index": 27043,
"solution": "A prime $p$ divides $a b$ if and only if divides either $a$ or $b$. If $n=a^{2}+b^{2}$ is a composite then it has a prime divisor $p \\leq \\sqrt{n}$, and if $p$ divides $a$ it divides $b$ and vice-versa, which is not possible because $a$ and $b$ are coprime. There... |
deepscaler | [
{
"role": "user",
"content": "Find $x$ such that $\\log_x 4=\\log_{27} 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 27044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all positive integer divisors of $129,600$. Calculate the number of numbers that are the product of two distinct elements of $S$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "488"
} | {
"split": "train",
"index": 27045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nSvetlana takes a triplet of numbers and transforms it by the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and smallest numbers in the triplet on the 1580th step of applying this rule, if the initial triple... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 27046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two right triangles, $ABC$ and $ACD$, are joined at side $AC$. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. Determine the number of square units in the area of the fourth square. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "138"
} | {
"split": "train",
"index": 27047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the sum \\[ \\sum_{n=0}^{332} (-1)^{n} {1008 \\choose 3n} \\] and find the remainder when the sum is divided by $500$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 27048,
"solution": ""
} |
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