data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, side lengths are $XY = 30$, $YZ = 45$, and $XZ = 51$. Points P and Q are on $XY$ and $XZ$ respectively, such that $XP = 18$ and $XQ = 15$. Determine the ratio of the area of triangle $XPQ$ to the area of quadrilateral $PQZY$.\nA) $\\frac{459}{625}$\nB) $\\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{459}{625}"
} | {
"split": "train",
"index": 23449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work wa... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 23450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ is inscribed in a semicircle with diameter $\\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 23451,
"solution": "1. **Identify the Geometry and Given Values**: We are given a rectangle $ABCD$ inscribed in a semicircle with diameter $\\overline{FE}$. The lengths $DA = 16$ and $FD = AE = 9$ are provided.\n\n2. **Complete the Circle**: To utilize symmetry and the properties of a... |
deepscaler | [
{
"role": "user",
"content": "The graph of $xy = 1$ is a hyperbola. Find the distance between the foci of this hyperbola. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 23452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 23453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, and $\\frac{1}{a} + \\frac{2}{b} = 2$.\n(1) Find the minimum value of $ab$;\n(2) Find the minimum value of $a + 2b$, and find the corresponding values of $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 23454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 3-element subsets of the set $\\{1,2,3, \\ldots, 19\\}$ have sum of elements divisible by 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "244"
} | {
"split": "train",
"index": 23455,
"solution": "Consider the elements of the sets mod 4. Then we would need to have sets of the form $\\{0,0,0\\}$, $\\{0,2,2\\},\\{0,1,3\\},\\{1,1,2\\}$, or $\\{2,3,3\\}$. In the set $\\{1,2, \\ldots, 19\\}$ there are four elements divisible by 4 and 5 elements congruent to each ... |
deepscaler | [
{
"role": "user",
"content": "Snow White entered a room with a round table surrounded by 30 chairs. Some of the chairs were occupied by dwarfs. It turned out that Snow White couldn't sit in a way such that no one was sitting next to her. What is the minimum number of dwarfs that could have been at the table... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 23457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a$, $b$, and $c$ are the three sides of $\\triangle ABC$, and $3a^2+3b^2-3c^2+2ab=0$, then $\\tan C= \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2 \\sqrt {2}"
} | {
"split": "train",
"index": 23458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train passenger knows that the speed of their train is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noted that the passing train took 3 seconds to pass completely. Determine the speed of the passing train, given that i... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 23459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A region is bounded by semicircular arcs created on each side of an equilateral triangle, where each side measures \\(1/\\pi\\). Calculate the total perimeter of this shaped region. 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": 23460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)(1\\frac{3}{4}-\\frac{3}{8}+\\frac{5}{6})÷(-\\frac{1}{24})$;<br/>$(2)-2^2+(-4)÷2×\\frac{1}{2}+|-3|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of $x$ when $5(2x - 3) + 7(5 - 3x^2 + 4x) - 6(3x - 2)$ is simplified. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 23462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some integers $m$ and $n$, the expression $(x+m)(x+n)$ is equal to a quadratic expression in $x$ with a constant term of -12. Which of the following cannot be a value of $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 23463,
"solution": "Expanding, $(x+m)(x+n)=x^{2}+n x+m x+m n=x^{2}+(m+n) x+m n$. The constant term of this quadratic expression is $m n$, and so $m n=-12$. Since $m$ and $n$ are integers, they are each divisors of -12 and thus of 12. Of the given possibilities, only 5 is not a divisor... |
deepscaler | [
{
"role": "user",
"content": "Crystal decides to alter her running routine slightly. She heads due north for 2 miles, then goes northwest for 2 miles, followed by a southwest direction for 2 miles, and she finishes with a final segment directly back to her starting point. Calculate the distance of this fina... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 23464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $y$: $$\\log_4 \\frac{2y+8}{3y-2} + \\log_4 \\frac{3y-2}{2y-5}=2$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{44}{15}"
} | {
"split": "train",
"index": 23465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$ have the sums of their first n terms denoted by $S_n$ and $T_n$, respectively. If $\\frac{a_n}{b_n} = \\frac{2n-1}{n+1}$, then calculate the value of $\\frac{S_{11}}{T_{11}}$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{7}"
} | {
"split": "train",
"index": 23466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two positive integers \\(m\\) and \\(n\\) are chosen such that \\(m\\) is the smallest positive integer with only two positive divisors, and \\(n\\) is the largest integer less than 200 that has exactly four positive divisors. What is \\(m+n\\)? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 23467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Number $a$ is such that $\\forall a_1, a_2, a_3, a_4 \\in \\mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\\sum_{1 \\leq i < j \\leq 4} ((a_i - k_i) - (a_j - k_j))^2 \\leq a$. Find the minimum of $a$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"split": "train",
"index": 23468,
"solution": "\nLet \\( a \\) be such that for all \\( a_1, a_2, a_3, a_4 \\in \\mathbb{R} \\), there exist integers \\( k_1, k_2, k_3, k_4 \\) such that\n\\[\n\\sum_{1 \\leq i < j \\leq 4} ((a_i - k_i) - (a_j - k_j))^2 \\leq a.\n\\]\nWe aim to find the minimum value of \\( a \\... |
deepscaler | [
{
"role": "user",
"content": "Let $A,B,C,D$ , be four different points on a line $\\ell$ , so that $AB=BC=CD$ . In one of the semiplanes determined by the line $\\ell$ , the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $16^{\\frac{1}{2}}-625^{\\frac{1}{2}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-21"
} | {
"split": "train",
"index": 23470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kimberly borrows $1000$ dollars from Lucy, who charged interest of $5\\%$ per month (which compounds monthly). What is the least integer number of months after which Kimberly will owe more than twice as much as she borrowed? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 23471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Liang went to the science museum. The science museum is located 400 meters northeast of Xiao Liang's home. When Xiao Liang left home, he walked 600 meters in the northwest direction by mistake. At this time, he did not see the science museum. He asked a lady, and she didn'... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 23472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least odd prime factor of $2019^8+1$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 23473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A novice economist-cryptographer received a cryptogram from a ruler which contained a secret decree about implementing an itemized tax on a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that a greater amount could no... | math | {
"style": "rule",
"ground_truth": "6480"
} | {
"split": "train",
"index": 23474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle with center $O$, the measure of $\\angle BAC$ is $45^\\circ$, and the radius of the circle $OA=15$ cm. Also, $\\angle BAC$ subtends another arc $BC$ which does not include point $A$. Compute the length of arc $BC$ in terms of $\\pi$. [asy]\ndraw(circle((0,0),1));\nd... | math | {
"style": "rule",
"ground_truth": "22.5\\pi"
} | {
"split": "train",
"index": 23475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( S = \\{1, 2, 3, \\cdots, 50\\} \\). Find the smallest positive integer \\( n \\) such that every subset of \\( S \\) with \\( n \\) elements contains three numbers that can be the side lengths of a right triangle. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 23476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $\\frac{\\sin^{2}B+\\sin^{2}C-\\sin^{2}A}{\\sin B \\sin C}$ given that $\\frac{\\sin B}{\\sin C}=\\frac{AC}{AB}$, $\\frac{\\sin C}{\\sin B}=\\frac{AB}{AC}$, and $\\frac{\\sin A}{\\sin B \\sin C}=\\frac{BC}{AC \\cdot AB}$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{83}{80}"
} | {
"split": "train",
"index": 23477,
"solution": "Using the Law of Sines, we have $$\\frac{\\sin^{2}B+\\sin^{2}C-\\sin^{2}A}{\\sin B \\sin C}=\\frac{\\sin B}{\\sin C}+\\frac{\\sin C}{\\sin B}-\\frac{\\sin A}{\\sin B} \\frac{\\sin A}{\\sin C}=\\frac{AC}{AB}+\\frac{AB}{AC}-\\frac{BC}{AC} \\frac{BC}{AB}=\\frac{83}{80... |
deepscaler | [
{
"role": "user",
"content": "In base $R_1$ the expanded fraction $F_1$ becomes $.373737\\cdots$, and the expanded fraction $F_2$ becomes $.737373\\cdots$. In base $R_2$ fraction $F_1$, when expanded, becomes $.252525\\cdots$, while the fraction $F_2$ becomes $.525252\\cdots$. The sum of $R_1$ and $R_2$, ea... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 23478,
"solution": "1. **Express $F_1$ and $F_2$ in base $R_1$ and $R_2$:**\n - For $F_1$ in base $R_1$, we have:\n \\[\n F_1 = 0.373737\\ldots_{R_1}\n \\]\n Multiplying by $R_1^2$:\n \\[\n R_1^2 F_1 = 37.373737\\ldots_{R_1}\n \\]\n Subtracting the or... |
deepscaler | [
{
"role": "user",
"content": "Determine the periodicity of the following functions. If the function is periodic, find its smallest positive period:\n(1) \\( y = \\tan x - \\cot x \\);\n(2) \\( y = \\sin (\\cos x) \\);\n(3) \\( y = \\sin x^{2} \\). Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 23479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two rectangles have integer dimensions, and both have a perimeter of 144 cm. What is the greatest possible difference between the areas of two such rectangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1225"
} | {
"split": "train",
"index": 23480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all numbers of the form $\\overline{13 x y 45 z}$ that are divisible by 792. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1380456"
} | {
"split": "train",
"index": 23481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many kings can be placed on an $8 \\times 8$ chessboard without any of them being in check? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes 15 men working steadily 4 days to dig the foundation for a new apartment. How many days would it have taken 25 men working at the same rate to dig the foundation? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"split": "train",
"index": 23483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two identical cylindrical containers are connected at the bottom by a small tube with a tap. While the tap was closed, water was poured into the first container, and oil was poured into the second one, so that the liquid levels were the same and equal to $h = 40$ cm. At what le... | math | {
"style": "rule",
"ground_truth": "16.47"
} | {
"split": "train",
"index": 23484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six students participate in an apple eating contest. The graph shows the number of apples eaten by each participating student. Aaron ate the most apples and Zeb ate the fewest. How many more apples than Zeb did Aaron eat?\n\n[asy]\ndefaultpen(linewidth(1pt)+fontsize(10pt));\npa... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 23485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two water droplets fall freely one after another from a $300 \\mathrm{~m}$ high cliff. The first droplet has already fallen $\\frac{1}{1000} \\mathrm{~mm}$ when the second one starts falling.\n\nHow many millimeters apart will the two droplets be at the moment the first one rea... | math | {
"style": "rule",
"ground_truth": "34.6"
} | {
"split": "train",
"index": 23486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the 256 smallest positive integers written in base 8 use the digit 6 or 7 (or both)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-axis? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 23488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jim borrows $1500$ dollars from Sarah, who charges an interest rate of $6\\%$ per month (which compounds monthly). What is the least integer number of months after which Jim will owe more than twice as much as he borrowed? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya is thinking of a four-digit number of the form \\( \\overline{20 * *} \\).\n\nVasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For exampl... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 23490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \\le k \\le 150$, how many $b_k$ are divisible by 9? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 23491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first operation divides the bottom-left square of diagram $\\mathrm{a}$ into four smaller squares, as shown in diagram b. The second operation further divides the bottom-left smaller square of diagram b into four even smaller squares, as shown in diagram c; continuing this ... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 23492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence ${\\_{a\\_n}}$ with a non-zero common difference $d$, and $a\\_7$, $a\\_3$, $a\\_1$ are three consecutive terms of a geometric sequence ${\\_{b\\_n}}$.\n\n(1) If $a\\_1=4$, find the sum of the first 10 terms of the sequence ${\\_{a\\_n}}$, denoted a... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 23493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $2^{16} \\div 16^2$? Write your answer as an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 23494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four tiles marked X, and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that the two outermost positions in the arrangement are occupied by X and the middle one by O? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"split": "train",
"index": 23495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 15 red marbles, 10 blue marbles, and 5 green marbles. Four marbles are selected at random and without replacement. What is the probability that two marbles are red, one is blue, and one is green? Express your answer as a common fraction. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{350}{1827}"
} | {
"split": "train",
"index": 23496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that A and B are any two points on the line l, and O is a point outside of l. If there is a point C on l that satisfies the equation $\\overrightarrow {OC}= \\overrightarrow {OA}cosθ+ \\overrightarrow {OB}cos^{2}θ$, find the value of $sin^{2}θ+sin^{4}θ+sin^{6}θ$. Let's th... | math | {
"style": "rule",
"ground_truth": "\\sqrt {5}-1"
} | {
"split": "train",
"index": 23497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given $\\cos(15°+\\alpha) = \\frac{15}{17}$, with $\\alpha \\in (0°, 90°)$, find the value of $\\sin(15°-\\alpha)$. \n(2) Given $\\cos\\alpha = \\frac{1}{7}$, $\\cos(\\alpha-\\beta) = \\frac{13}{14}$, and $0 < \\beta < \\alpha < \\frac{\\pi}{2}$, find the value of $\\beta$... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 23498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have 5 books, two of which are identical copies of the same math book (and all of the rest of the books are different). In how many ways can I arrange them on a shelf? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 23500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "797"
} | {
"split": "train",
"index": 23501,
"solution": "\\[5^n \\equiv n \\pmod{8}\\] Note that $5^n \\equiv 5,1,5,1,...$ and $n \\equiv 0,..,7$ so $n$ is periodic every $[2,8]=8$. Easy to check that only $n\\equiv 5 \\pmod{8}$ satisfy. Let $n=8a+5$. Note that by binomial theorem, \\[2^{8a+5}=32\\cdot 2^{8a} \\equiv 7(1... |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with a common difference of 5. If $\\frac{S_{2n}}{S_n}$ is a constant that does not depend on $n$ for all positive integers $n$, find the first term. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 23502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is \n$$\n\\begin{cases} \nx=2- \\frac { \\sqrt {2}}{2}t \\\\\ny=1+ \\frac { \\sqrt {2}}{2}t\n\\end{cases}\n(t \\text{ is the parameter}).\n$$\nIn the polar coordinate system (using the same unit lengt... | math | {
"style": "rule",
"ground_truth": "\\sqrt {14}"
} | {
"split": "train",
"index": 23503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nSquirrels $A$, $B$, and $C$ have several pine cones in total. Initially, squirrel $A$ has 26 pine cones, and it takes 10 pine cones to evenly divide between $B$ and $C$. Then, squirrel $B$ takes 18 pine cones and evenly divides them between $A$ and $C$. Finally, squirrel $C$ ... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 23504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1, a_2, \\ldots$ and $b_1, b_2, \\ldots$ be arithmetic progressions such that $a_1 = 50, b_1 = 100$, and $a_{50} + b_{50} = 850$. Find the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \\ldots$ Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "25000"
} | {
"split": "train",
"index": 23505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the length of the parametric curve described by\n\\[(x,y) = (2 \\sin t, 2 \\cos t)\\]from $t = 0$ to $t = \\pi.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\pi"
} | {
"split": "train",
"index": 23506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 3 (\\angle D$) and $\\angle BAC = t \\pi$ in radians, then find $t$.\n\n[asy]\nimport graph;\n\nunitsize(2 ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"split": "train",
"index": 23507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set \\( S = \\{1, 2, \\cdots, 100\\} \\), determine the smallest possible value of \\( m \\) such that in any subset of \\( S \\) with \\( m \\) elements, there exists at least one number that is a divisor of the product of the remaining \\( m-1 \\) numbers. Let's thi... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 23508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is $3$ , then the minimum possible volume of the cone can... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many perfect cubes exist between \\(3^6 + 1\\) and \\(3^{12} + 1\\), inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 23510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to ... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 23511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The height \\(CH\\), dropped from the vertex of the right angle of the triangle \\(ABC\\), bisects the angle bisector \\(BL\\) of this triangle. Find the angle \\(BAC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 23512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on Geo... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{40}"
} | {
"split": "train",
"index": 23513,
"solution": "Let Brian's roll be $d$ and let George's rolls be $x, y, z$. By pairing the situation $d, x, y, z$ with $21-d, 7-x, 7-y, 7-z$, we see that the probability that Brian rolls higher is the same as the probability that George rolls higher. Given any of George's rolls $... |
deepscaler | [
{
"role": "user",
"content": "A house and store were sold for $12,000 each. The house was sold at a loss of 20% of the cost, and the store at a gain of 20% of the cost. The entire transaction resulted in: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-$1000"
} | {
"split": "train",
"index": 23514,
"solution": "1. **Identify the cost price of the house and store:**\n - Let $h$ be the cost price of the house and $s$ be the cost price of the store.\n - The house was sold at a loss of $20\\%$. Therefore, the selling price of the house is $80\\%$ of $h$, which is given by... |
deepscaler | [
{
"role": "user",
"content": "Two teachers are taking a photo with 3 male students and 3 female students lined up in a row. The teachers can only stand at the ends, and the male students cannot be adjacent. How many different ways are there to arrange the photo? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 23515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum value of the function \\( f(x)=(x+1)(x+2)(x+3)(x+4)+35 \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 23516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A science student is asked to find the coefficient of the $x^2$ term in the expansion of $(x^2-3x+2)^4$. The coefficient is \\_\\_\\_\\_\\_\\_. (Answer with a number) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "248"
} | {
"split": "train",
"index": 23517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the result of $120 \\div (6 \\div 2 \\times 3)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{120}{9}"
} | {
"split": "train",
"index": 23518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Company XYZ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A, B$, and $C$. There are 1,5 , and 4 workers at $A, B$, and $C$, respectiv... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 23519,
"solution": "We want to minimize $1 \\cdot P A+5 \\cdot P B+4 \\cdot P C$. By the triangle inequality, $(P A+P B)+4(P B+P C) \\geq A B+4 B C=13+56=69$, with equality precisely when $P=[A B] \\cap[B C]=B$."
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = 5x^2 - 2x - 1$, then $f(x + h) - f(x)$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "h(10x+5h-2)"
} | {
"split": "train",
"index": 23520,
"solution": "To find $f(x+h) - f(x)$ for the function $f(x) = 5x^2 - 2x - 1$, we start by substituting $x+h$ into the function and then subtracting $f(x)$.\n\n1. **Substitute $x+h$ into $f$:**\n \\[\n f(x+h) = 5(x+h)^2 - 2(x+h) - 1\n \\]\n Expand $(x+h)^2$:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( A B C \\), angle \\( B \\) equals \\( 45^\\circ \\) and angle \\( C \\) equals \\( 30^\\circ \\). Circles are constructed on the medians \\( B M \\) and \\( C N \\) as diameters, intersecting at points \\( P \\) and \\( Q \\). The chord \\( P Q \\) intersects si... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"split": "train",
"index": 23521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of angles between 0 and $2 \\pi,$ other than integer multiples of $\\frac{\\pi}{2},$ such that $\\sin \\theta,$ $\\cos \\theta$, and $\\tan \\theta$ form a geometric sequence in some order. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 23522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What's the coefficient of the $m^4n^4$ term in the expansion of $(m+n)^8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 23523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces labelled $1,1,6,6,8,8$. - Dice of type $C$ have faces labelled $3,3,5,5,7,7$. All three players simultane... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"split": "train",
"index": 23524,
"solution": "Short version: third player doesn't matter; against 1 opponent, by symmetry, you'd both play the same strategy. Type A beats B, B beats C, and C beats A all with probability $5 / 9$. It can be determined that choosing each die with probability $1 / 3$ is the best s... |
deepscaler | [
{
"role": "user",
"content": "Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23525,
"solution": "To solve this problem, we need to ensure that each face of the cube has exactly two edges labeled with $1$ and two edges labeled with $0$. We will analyze the possible configurations that satisfy this condition.\n\n#### Step 1: Understanding the Cube's Faces and Ed... |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{a}$ and $\\mathbf{b}$ are two unit vectors, with an angle of $\\frac{\\pi}{3}$ between them, then compute the volume of the parallelepiped generated by $\\mathbf{a},$ $\\mathbf{b} + \\mathbf{b} \\times \\mathbf{a},$ and $\\mathbf{b}.$ Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 23526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number $m$ is randomly selected from the set $\\{11,13,15,17,19\\}$, and a number $n$ is randomly selected from $\\{1999,2000,2001,\\ldots,2018\\}$. What is the probability that $m^n$ has a units digit of $1$? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{20}"
} | {
"split": "train",
"index": 23527,
"solution": "1. **Reduction to Units Digits**:\n Since we are interested in the units digit of $m^n$, we can reduce the set $\\{11, 13, 15, 17, 19\\}$ to their respective units digits, resulting in the set $A = \\{1, 3, 5, 7, 9\\}$.\n\n2. **Case Analysis**:\n - **Case $m = ... |
deepscaler | [
{
"role": "user",
"content": "Suppose $d\\not=0$. We can write $\\left(12d+13+14d^2\\right)+\\left(2d+1\\right)$, in the form $ad+b+cd^2$, where $a$, $b$, and $c$ are integers. Find $a+b+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 23528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya wrote consecutive natural numbers \\(N\\), \\(N+1\\), \\(N+2\\), and \\(N+3\\) in rectangular boxes. Below each rectangle, he wrote the sum of the digits of the corresponding number in a circle.\n\nThe sum of the numbers in the first and second circles equals 200, and the... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 23529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, square \\(PQRS\\) has side length 40. Points \\(J, K, L,\\) and \\(M\\) are on the sides of \\(PQRS\\), so that \\(JQ = KR = LS = MP = 10\\). Line segments \\(JZ, KW, LX,\\) and \\(MY\\) are drawn parallel to the diagonals of the square so that \\(W\\) is on \\(... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 23530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If Alex does not sing on Saturday, then she has a $70 \\%$ chance of singing on Sunday; however, to rest her voice, she never sings on both days. If Alex has a $50 \\%$ chance of singing on Sunday, find the probability that she sings on Saturday. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"split": "train",
"index": 23531,
"solution": "Let $p$ be the probability that Alex sings on Saturday. Then, the probability that she sings on Sunday is $.7(1-p)$; setting this equal to .5 gives $p=\\frac{2}{7}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $A,B$ be the points on the coordinate plane with coordinates $(t-4,-1)$ and $(-2,t+3)$, respectively. The square of the distance between the midpoint of $\\overline{AB}$ and an endpoint of $\\overline{AB}$ is equal to $t^2/2$. What is the value of $t$? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 23532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( S \\) contain 2012 elements, where the ratio of any two elements is not an integer. An element \\( x \\) in \\( S \\) is called a \"good element\" if there exist distinct elements \\( y \\) and \\( z \\) in \\( S \\) such that \\( x^2 \\) divides \\( y \\cdot z ... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"split": "train",
"index": 23533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The terms of the sequence $(a_i)$ defined by $a_{n + 2} = \\frac {a_n + 2009} {1 + a_{n + 1}}$ for $n \\ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 23534,
"solution": "If $a_{n} \\ne \\frac {2009}{a_{n+1}}$, then either \\[a_{n} = \\frac {a_{n}}{1} < \\frac {a_{n} + 2009}{1 + a_{n+1}} < \\frac {2009}{a_{n+1}}\\]\n or\n\\[\\frac {2009}{a_{n+1}} < \\frac {2009 + a_{n}}{a_{n+1} + 1} < \\frac {a_{n}}{1} = a_{n}\\]\n All the integers ... |
deepscaler | [
{
"role": "user",
"content": "An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. \nIf the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance \nbetween the centers of the pulleys in inches is Let's... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 23535,
"solution": "1. **Identify the setup**: We have two pulleys with radii $14$ inches and $4$ inches respectively. The distance between the points of contact of the belt with the pulleys is $24$ inches. We denote the centers of the smaller and larger pulleys as $A$ and $B$ respect... |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Semicircles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Express your answer in terms of \\(\\pi\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{4}\\pi"
} | {
"split": "train",
"index": 23537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two decimals are multiplied, and the resulting product is rounded to 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "27.55"
} | {
"split": "train",
"index": 23538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a-b=1$ and $ab=-2$, then $\\left(a+1\\right)\\left(b-1\\right)=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 23539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square, points \\(P\\) and \\(Q\\) are placed such that \\(P\\) is the midpoint of the bottom side and \\(Q\\) is the midpoint of the right side of the square. The line segment \\(PQ\\) divides the square into two regions. Calculate the fraction of the square's area that i... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 23540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\alpha, \\beta \\in \\left(0, \\frac{\\pi}{2}\\right)\\) and \\(\\sin \\beta = 2 \\cos (\\alpha + \\beta) \\cdot \\sin \\alpha \\left(\\alpha + \\beta \\neq \\frac{\\pi}{2}\\right)\\), find the maximum value of \\(\\tan \\beta\\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 23541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Label one disk \"$1$\", two disks \"$2$\", three disks \"$3$\", ..., fifty disks \"$50$\". Put these $1+2+3+ \\cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee dra... | math | {
"style": "rule",
"ground_truth": "415"
} | {
"split": "train",
"index": 23542,
"solution": "1. **Understanding the Problem:**\n We are given disks labeled from 1 to 50, where the label corresponds to the number of disks with that label. We need to determine the minimum number of disks that must be drawn to ensure that at least ten disks with the same la... |
deepscaler | [
{
"role": "user",
"content": "On an island, there are knights who always tell the truth and liars who always lie. At the main celebration, 100 islanders sat around a large round table. Half of the attendees said the phrase: \"both my neighbors are liars,\" while the remaining said: \"among my neighbors, the... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 23543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the only value of \\( x \\) in the open interval \\((- \\pi / 2, 0)\\) that satisfies the equation\n$$\n\\frac{\\sqrt{3}}{\\sin x} + \\frac{1}{\\cos x} = 4.\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4\\pi}{9}"
} | {
"split": "train",
"index": 23544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive integers $a$, $b$, and $c$ are chosen so that $a<b<c$, and the system of equations \\[\n2x + y = 2003\\quad\\text{and}\\quad y = |x-a| + |x-b| + |x-c|\n\\]has exactly one solution. What is the minimum value of $c$? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"split": "train",
"index": 23545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(A B C D\\) is a rectangle, \\(P\\) is on \\(B C\\), \\(Q\\) is on \\(C D\\), and \\(R\\) is inside \\(A B C D\\). Also, \\(\\angle P R Q = 30^\\circ\\), \\(\\angle R Q D = w^\\circ\\), \\(\\angle P Q C = x^\\circ\\), \\(\\angle C P Q = y^\\circ\\), and \\(\\a... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 23546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 23547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 23548,
"solution": ""
} |
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