data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\\left(\\frac58, \\frac38 \\right)$ is greater than or equal to $\\f... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 2300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 12 students in a classroom; 6 of them are Democrats and 6 of them are Republicans. Every hour the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change his... | math | {
"style": "rule",
"ground_truth": "\\frac{341}{55}"
} | {
"split": "train",
"index": 2301,
"solution": "When the party distribution is $6-6$, the situation can change (to $3-9$ ) only when a group of three contains three people from the same party, and the remaining three are distributed evenly across the other three groups (to be converted). To compute the probabilit... |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\frac{\\sin (\\beta+\\gamma) \\sin (\\gamma+\\alpha)}{\\cos \\alpha \\cos \\gamma}=\\frac{4}{9}\\), find the value of \\(\\frac{\\sin (\\beta+\\gamma) \\sin (\\gamma+\\alpha)}{\\cos (\\alpha+\\beta+\\gamma) \\cos \\gamma}\\). Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 2302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that two trainees were born on the same day (not necessarily the same year)? Note: There are 62 trainees. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99.59095749"
} | {
"split": "train",
"index": 2303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the volume of the region in space defined by\n\\[ |x + 2y + z| + |x + 2y - z| \\le 12 \\]\nand $x, y \\geq 0$, $z \\geq -2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 2304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all three-digit integers \\( abc = n \\) such that \\( \\frac{2n}{3} = a! \\cdot b! \\cdot c! \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 2305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ with an angle of 120° between them, find the projection of $\\overrightarrow{a} + \\overrightarrow{b}$ onto the direction of $\\overrightarrow{b}$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the addition problem: \\begin{tabular}{ccccc} & C & A & S & H \\\\ + & & & M & E \\\\ \\hline O & S & I & D & E \\end{tabular} where each letter represents a base-ten digit, and $C, M, O \\neq 0$. (Distinct letters are allowed to represent the same digit) How many ways... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 2307,
"solution": "Clearly, $C A S H$ and $M E$ cannot add up to 11000 or more, so $O=1$ and $S=0$. By examining the units digit, we find that $H=0$. Then $C A S H+M E<9900+99<10000$, so there are no solutions."
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, let the line $l: \\begin{cases} x=2-t \\\\ y=2t \\end{cases} (t \\text{ is a parameter})$, and the curve $C_{1}: \\begin{cases} x=2+2\\cos \\theta \\\\ y=2\\sin \\theta \\end{cases} (\\theta \\text{ is a parameter})$. In the polar coo... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 2308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of triples of natural numbers \\( m, n, k \\) that are solutions to the equation \\( m + \\sqrt{n+\\sqrt{k}} = 2023 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27575680773"
} | {
"split": "train",
"index": 2309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose point P is on the curve $y=x^2+1$ (where $x \\geq 0$), and point Q is on the curve $y=\\sqrt{x-1}$ (where $x \\geq 1$). Then the minimum value of the distance $|PQ|$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 2310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x, y, z \\in \\mathbf{R} \\) such that \\( x^2 + y^2 + xy = 1 \\), \\( y^2 + z^2 + yz = 2 \\), \\( x^2 + z^2 + xz = 3 \\), find \\( x + y + z \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3 + \\sqrt{6}}"
} | {
"split": "train",
"index": 2311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$ are both arithmetic sequences, where the sum of the first $n$ terms of $\\{a_{n}\\}$ is $S_{n}$ and the sum of the first $n$ terms of $\\{b_{n}\\}$ is $T_{n}$. If $\\frac{S_{n}}{T_{n}}=\\frac{2n+1}{3n+2}$, then find the v... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{29}"
} | {
"split": "train",
"index": 2312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $4+10\\div2-2\\cdot3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $x^{x^{x^{.^{.^.}}}}=4$ is satisfied when $x$ is equal to:\nA) 2\nB) $\\sqrt[3]{4}$\nC) $\\sqrt{4}$\nD) $\\sqrt{2}$\nE) None of these Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 2314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "049"
} | {
"split": "train",
"index": 2315,
"solution": "Because the LCM of all of the numbers we are dividing by is $60$, we know that all of the remainders are $0$ again at $60$, meaning that we have a cycle that repeats itself every $60$ numbers.\n After listing all of the remainders up to $60$, we find that $35$, $58$... |
deepscaler | [
{
"role": "user",
"content": "Maya owns 16 pairs of shoes, consisting of 8 identical black pairs, 4 identical brown pairs, 3 identical grey pairs, and 1 pair of white shoes. If Maya randomly picks two shoes, what is the probability that they are the same color and that one is a left shoe and the other is a ... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{248}"
} | {
"split": "train",
"index": 2316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a district of Shanghai, the government convened the heads of 5 companies for an annual experience exchange meeting. Among them, Company A had 2 representatives attending, while the other 4 companies each had 1 representative attending. If 3 representatives are to be selected... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability that a car driving on this road does not stop at point A and the probability that it does not stop at point B and the probability that it does not stop at point C are $\\left(1-\\frac{25}{60}\\right)$, $\\left(1-\\frac{35}{60}\\right)$, and $\\left(1-\\frac{45}{... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{192}"
} | {
"split": "train",
"index": 2318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For nonnegative integers $a$ and $b$ with $a + b \\leq 6$, let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \\leq 6$. Find the remainder when $S$ is divided by $1000$.\... | math | {
"style": "rule",
"ground_truth": "564"
} | {
"split": "train",
"index": 2319,
"solution": "Let $c=6-(a+b)$, and note that $\\binom{6}{a + b}=\\binom{6}{c}$. The problem thus asks for the sum $\\binom{6}{a} \\binom{6}{b} \\binom{6}{c}$ over all $a,b,c$ such that $a+b+c=6$. Consider an array of 18 dots, with 3 columns of 6 dots each. The desired expression ... |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point on the ellipse $\\frac{x^2}{16} + \\frac{y^2}{9} =1$, and $F_1$, $F_2$ be the left and right foci of the ellipse, respectively. If $\\angle F_1 PF_2 = \\frac{\\pi}{3}$, find the area of $\\triangle F_1 PF_2$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 2320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-13.5, 13.5]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "756"
} | {
"split": "train",
"index": 2321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard six-sided die is rolled, and $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a^2 + b^2 = 2017c^2$, calculate the value of $\\frac{\\tan C}{\\tan A} + \\frac{\\tan C}{\\tan B}$. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1008}"
} | {
"split": "train",
"index": 2323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A boy named Vasya wrote down the nonzero coefficients of a tenth-degree polynomial \\( P(x) \\) in his notebook. He then calculated the derivative of the resulting polynomial and wrote down its nonzero coefficients, and continued this process until he arrived at a constant, whi... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of non-positive integers for which the values of the quadratic polynomial \\(2x^2 + 2021x + 2019\\) are non-positive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 2325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $y=\\sin (2x+\\varphi)$ is translated to the left by $\\dfrac {\\pi}{8}$ units along the $x$-axis and results in a graph of an even function, then determine one possible value of $\\varphi$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\pi}{4}"
} | {
"split": "train",
"index": 2326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how m... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2327,
"solution": "We are tasked with finding the number of ways six people can rearrange themselves around a circular table such that no one sits in the same or adjacent seat as they originally occupied. We will analyze this problem by considering the possible permutations of the six... |
deepscaler | [
{
"role": "user",
"content": "Suppose the 9-digit number $\\overline{32 x 35717 y}$ is a multiple of 72, and $P = xy$. Find the value of $P$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 2328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive real numbers $s$, let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2... | math | {
"style": "rule",
"ground_truth": "736"
} | {
"split": "train",
"index": 2329,
"solution": "Note: Archimedes15 Solution which I added an answer here are two cases. Either the $4$ and $10$ are around an obtuse angle or the $4$ and $10$ are around an acute triangle. If they are around the obtuse angle, the area of that triangle is $<20$ as we have $\\frac{1}... |
deepscaler | [
{
"role": "user",
"content": "Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?\nA) $\\frac{273}{288}$ \nB) $\\frac{275}{288}$ \nC) $\\frac{277}{288}$ \nD) $\\frac{279}{288}$ Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{277}{288}"
} | {
"split": "train",
"index": 2330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ ... | math | {
"style": "rule",
"ground_truth": "298"
} | {
"split": "train",
"index": 2331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the number of multiples of 4 that are between 100 and 500. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 2332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct sets are there such that each set contains only non-negative powers of $2$ or $3$ and sum of its elements is $2014$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 2333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 2,$ and $DE = 9$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?\n Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 2334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $$C: \\frac {x^{2}}{4}+ \\frac {y^{2}}{b^{2}}=1(0＜b＜2)$$, a straight line with a slope angle of $$\\frac {3π}{4}$$ intersects the ellipse C at points A and B. The midpoint of the line segment AB is M, and O is the coordinate origin. The angle between $$\\overr... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 2335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n\\,$ fish for various values of $n\\,$.\n$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline n & 0 & 1 & 2 & 3 & \\dots & 13 & 14 & 15 \\\\ \\hline \\t... | math | {
"style": "rule",
"ground_truth": "943"
} | {
"split": "train",
"index": 2336,
"solution": "Let $f$ be the total number of fish caught by the contestants who didn't catch $0, 1, 2, 3, 13, 14$, or $15$ fish and let $a$ be the number of contestants who didn't catch $0, 1, 2, 3, 13, 14$, or $15$ fish. From $\\text{(b)}$, we know that $\\frac{69+f+65+28+15}{a+... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence where each term is either 1 or 2, begins with the term 1, and between the $k$-th term 1 and the $(k+1)$-th term 1 there are $2^{k-1}$ terms of 2 (i.e., $1,2,1,2,2,1,2,2,2,2,1,2,2,2,2,2,2,2,2,1, \\cdots$), what is the sum of the first 1998 terms in this sequence... | math | {
"style": "rule",
"ground_truth": "3985"
} | {
"split": "train",
"index": 2337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of functions $f$ from $\\{0, 1, 2, 3, 4, 5, 6\\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and \\[|x - y| \\leq |f(x) - f(y)| \\leq 3|x - y|\\] for all $x$ and $y$ in $\\{0, 1, 2, 3, 4, 5, 6\\}$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"split": "train",
"index": 2338,
"solution": "Because $f(n)$ and $f(n+1)$ can differ by at most 3 for $n=0,1,2,3,4,5$, $f$ can decrease at most once. This is because if $f$ decreases $2$ times (or more) then we get a contradiction: $12 = f(6) \\le 3\\cdot 4 + (-1)\\cdot 2 = 10$.\n If $f$ never decreases, then $... |
deepscaler | [
{
"role": "user",
"content": "Let \\( F_{1} \\) and \\( F_{2} \\) be the left and right foci of the hyperbola \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 \\) (where \\( a > 0 \\) and \\( b > 0 \\)). There exists a point \\( P \\) on the right branch of the hyperbola such that \\( \\left( \\overrightarrow... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} + 1"
} | {
"split": "train",
"index": 2339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all two-digit primes greater than 30 but less than 99, which are still prime when their two digits are interchanged? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "388"
} | {
"split": "train",
"index": 2340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = |\\lg(x+1)|$, where $a$ and $b$ are real numbers, and $a < b$ satisfies $f(a) = f(- \\frac{b+1}{b+2})$ and $f(10a + 6b + 21) = 4\\lg2$. Find the value of $a + b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{11}{15}"
} | {
"split": "train",
"index": 2341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that there exists a complex number $z$, with positive real and imaginary part, satisfying $z^{n}=(\\bar{z})^{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2342,
"solution": "Since $|z|=|\\bar{z}|$ we may divide by $|z|$ and assume that $|z|=1$. Then $\\bar{z}=\\frac{1}{z}$, so we are looking for the smallest positive integer $n$ such that there is a $2 n^{\\text {th }}$ root of unity in the first quadrant. Clearly there is a sixth root ... |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d \\neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\\{a_n^2\\}$ is $S_n$. A geometric sequence $\\{b_n\\}$ has a common ratio $q$ less than 1 and consists of rational sine v... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain sequence, the first term is $a_1 = 101$ and the second term is $a_2 = 102$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = n + 2$ for all $n \\geq 1$. Determine $a_{50}$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 2344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 2010 boxes labeled $B_1, B_2, \\dots, B_{2010}$, and $2010n$ balls have been distributed\namong them, for some positive integer $n$. You may redistribute the balls by a sequence of moves,\neach of which consists of choosing an $i$ and moving \\emph{exactly} $i$ balls ... | math | {
"style": "rule",
"ground_truth": "n \\geq 1005"
} | {
"split": "train",
"index": 2345,
"solution": "It is possible if and only if $n \\geq 1005$.\nSince\n\\[\n1 + \\cdots + 2009 = \\frac{2009 \\times 2010}{2} = 2010 \\times 1004.5,\n\\]\nfor $n \\leq 1004$, we can start with an initial distribution in which each box\n$B_i$ starts with at most $i-1$ balls (so in pa... |
deepscaler | [
{
"role": "user",
"content": "A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? Let's think step... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 2346,
"solution": "Let's denote the total number of flowers as $F$. We are given several pieces of information about the distribution of these flowers:\n\n1. Six tenths of the flowers are pink, so the number of pink flowers is $\\frac{6}{10}F$.\n2. One third of the pink flowers are ro... |
deepscaler | [
{
"role": "user",
"content": "Seven students shared the cost of a $\\$26.00$ pizza. Each student paid either $\\$3.71$ or $\\$3.72$. How many students paid $\\$3.72$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2347,
"solution": "We can rephrase the given information by saying that each of the seven students paid $\\$3.71$ and some of the students paid an extra $\\$0.01$. Since $7 \\times \\$3.71=\\$25.97$ and the pizza cost $\\$26.00$ in total, then the students who paid the extra $\\$0.01$... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all two-digit positive integers whose squares end with the digits 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "495"
} | {
"split": "train",
"index": 2348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store is having a sale for a change of season, offering discounts on a certain type of clothing. If each item is sold at 40% of the marked price, there is a loss of 30 yuan per item, while selling it at 70% of the marked price yields a profit of 60 yuan per item.\nFind:\n(1) ... | math | {
"style": "rule",
"ground_truth": "50\\%"
} | {
"split": "train",
"index": 2349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $sin({α+\\frac{π}{4}})=\\frac{{12}}{{13}}$, and $\\frac{π}{4}＜α＜\\frac{{3π}}{4}$, find the value of $\\cos \\alpha$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{2}}{26}"
} | {
"split": "train",
"index": 2350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5$, no collection of $k$ pairs made by... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 2351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of all integers between 50 and 350 which end in 1 is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5880"
} | {
"split": "train",
"index": 2352,
"solution": "1. **Identify the sequence**: The problem asks for the sum of all integers between 50 and 350 that end in 1. These integers are $51, 61, 71, \\ldots, 341$.\n\n2. **Determine the sequence type**: The sequence is an arithmetic sequence where each term increases by 10.... |
deepscaler | [
{
"role": "user",
"content": "If I have a $5\\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 2353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 5. Point $K$ is such that $\\overrightarrow{L K}=\\overrightarrow{F B}-3 \\overrightarrow{A B}$. Determine whether point $K$ lies inside, on the boundary, or o... | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 2354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two standard dice are rolled. What is the expected number of 1's obtained? 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": "\\frac{1}{3}"
} | {
"split": "train",
"index": 2355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $n$ is a positive integer, let $s(n)$ denote the sum of the digits of $n$. We say that $n$ is zesty if there exist positive integers $x$ and $y$ greater than 1 such that $x y=n$ and $s(x) s(y)=s(n)$. How many zesty two-digit numbers are there? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 2356,
"solution": "Let $n$ be a zesty two-digit number, and let $x$ and $y$ be as in the problem statement. Clearly if both $x$ and $y$ are one-digit numbers, then $s(x) s(y)=n \\neq s(n)$. Thus either $x$ is a two-digit number or $y$ is. Assume without loss of generality that it is $... |
deepscaler | [
{
"role": "user",
"content": "Given a set of data arranged in ascending order, which are -1, 0, 4, x, 7, 14, and the median is 5, find the variance of this set of data. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{74}{3}"
} | {
"split": "train",
"index": 2357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 2358,
"solution": "1. **Assign Side Lengths**: Let the side length of each square $ABCD$, $EFGH$, and $GHIJ$ be $s = 1$ for simplicity.\n\n2. **Identify Key Points and Relationships**:\n - Points $C$ and $D$ are midpoints of sides $IH$ and $HE$ respectively, so $HC = CI = \\frac{1}{... |
deepscaler | [
{
"role": "user",
"content": "Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of $4$ to $3$. His wife got twice as much as the son. If the cook received a bequest of $\\textdollar{500}$, then the entire estate was:... | math | {
"style": "rule",
"ground_truth": "7000"
} | {
"split": "train",
"index": 2359,
"solution": "Let's denote the total estate by $E$. According to the problem, the daughter and son together received half of the estate, and they share this half in the ratio of $4:3$. Let's denote the shares of the daughter and son as $4x$ and $3x$ respectively. Therefore, we ha... |
deepscaler | [
{
"role": "user",
"content": "Xiaoming shot a total of 80 times in 3 minutes and scored 50 goals. Calculate the frequency of Xiaoming's scoring. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.625"
} | {
"split": "train",
"index": 2360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Choose one of the three conditions: 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/>Question: Doe... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 2361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, point $D$ is on side $BC$ such that $BD:DC = 1:2$. A line through $A$ and $D$ intersects $BC$ at $E$. If the area of triangle $ABE$ is $30$, find the total area of triangle $ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 2362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three friends are in the park. Bob and Clarise are standing at the same spot and Abe is standing 10 m away. Bob chooses a random direction and walks in this direction until he is 10 m from Clarise. What is the probability that Bob is closer to Abe than Clarise is to Abe? Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 2363,
"solution": "We call Clarise's spot $C$ and Abe's spot $A$. Consider a circle centred at $C$ with radius 10 m. Since $A$ is 10 m from $C$, then $A$ is on this circle. Bob starts at $C$ and picks a direction to walk, with every direction being equally likely to be chosen. We mode... |
deepscaler | [
{
"role": "user",
"content": "Calculate the product: $100 \\times 29.98 \\times 2.998 \\times 1000 = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2998^2"
} | {
"split": "train",
"index": 2364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( P \\) and \\( Q \\) are located on side \\( BC \\) of triangle \\( ABC \\), with \\( BP: PQ: QC = 1: 2: 3 \\). Point \\( R \\) divides side \\( AC \\) of this triangle such that \\( AR: RC = 1: 2 \\). What is the ratio of the area of quadrilateral \\( PQST \\) to the... | math | {
"style": "rule",
"ground_truth": "5/24"
} | {
"split": "train",
"index": 2365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)=2\\sin(wx+\\varphi+ \\frac {\\pi}{3})+1$$ where $|\\varphi|< \\frac {\\pi}{2}$ and $w>0$, is an even function, and the distance between two adjacent axes of symmetry of the function $f(x)$ is $$\\frac {\\pi}{2}$$. \n(1) Find the value of $$f( \\frac {... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 2366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cashier, upon checking the account before leaving work, finds that the cash is 153 yuan less than the account book. She knows the actual amount collected cannot be wrong, so it must be due to a decimal point error during bookkeeping. What is the actual amount of the cash that... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 2367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x^3 - 9x^2 + 27x - 25$ and let $g(f(x)) = 3x + 4$. What is the sum of all possible values of $g(7)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 2368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{m} = (a\\cos x, \\cos x)$ and $\\overrightarrow{n} = (2\\cos x, b\\sin x)$, with a function $f(x) = \\overrightarrow{m} \\cdot \\overrightarrow{n}$ that satisfies $f(0) = 2$ and $f\\left(\\frac{\\pi}{3}\\right) = \\frac{1}{2} + \\frac{\\sqrt{3}}{... | math | {
"style": "rule",
"ground_truth": "-\\frac{4 + \\sqrt{7}}{3}"
} | {
"split": "train",
"index": 2369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mahmoud tosses three coins. What is the probability that he will get at least one head? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 2370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Kang exercises every day by doing several sets of push-ups, 8 per set, and several sets of pull-ups, 5 per set. On the first day, he did a total of 41 reps (combining both exercises), and he increases the number by 1 each day until the 100th day. If the number of sets of p... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 2371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer $n$ such that $n^2 - 11n +24 \\leq 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 2372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the angles of a triangle at points \\( A, B \\), and \\( C \\) are such that \\( \\angle ABC = 50^\\circ \\) and \\( \\angle ACB = 30^\\circ \\), calculate the value of \\( x \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 2373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers 407 and 370 equal the sum of the cubes of their digits. For example, \\( 4^3 = 64 \\), \\( 0^3 = 0 \\), and \\( 7^3 = 343 \\). Adding 64, 0, and 343 gives you 407. Similarly, the cube of 3 (27), added to the cube of 7 (343), gives 370.\n\nCould you find a number, no... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 2374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are $(3,17)$ and $(48,281)$? (Include both endpoints of the segment in your count.) Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 2375,
"solution": "1. **Identify the coordinates of the endpoints and calculate the differences in coordinates:**\n - The endpoints of the line segment are $(3,17)$ and $(48,281)$.\n - The difference in the $x$-coordinates is $48 - 3 = 45$.\n - The difference in the $y$-coordina... |
deepscaler | [
{
"role": "user",
"content": "To complete the grid below, each of the digits 1 through 4 must occur once\nin each row and once in each column. What number will occupy the lower\nright-hand square?\n\\[\\begin{tabular}{|c|c|c|c|}\\hline 1 & & 2 &\\ \\hline 2 & 3 & &\\ \\hline & &&4\\ \\hline & &&\\ \\hline\\... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 2376,
"solution": "We are given a 4x4 grid where each digit from 1 through 4 must appear exactly once in each row and once in each column. The initial grid is:\n\n\\[\n\\begin{array}{|c|c|c|c|}\n\\hline\n1 & & 2 & \\\\\n\\hline\n2 & 3 & & \\\\\n\\hline\n & & & 4 \\\\\n\\h... |
deepscaler | [
{
"role": "user",
"content": "Find $\\cos \\frac{5 \\pi}{4}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{\\sqrt{2}}"
} | {
"split": "train",
"index": 2377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers 1, 2, 3, 4, 5, a five-digit number is formed with digits not repeating. What is the probability of randomly selecting a five-digit number $\\overline{abcde}$ that satisfies the condition \"$a < b > c < d > e$\"? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "2/15"
} | {
"split": "train",
"index": 2378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be the midpoint of side $AC$ of the triangle $ABC$ . Let $P$ be a point on the side $BC$ . If $O$ is the point of intersection of $AP$ and $BM$ and $BO = BP$ , determine the ratio $\\frac{OM}{PC}$ . Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 2379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $21 \\cdot \\frac{8}{15} \\cdot \\frac{1}{14}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 2380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cubical cake with edge length $2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $M$ is the midpoint of a top edge. The piece whose top is triangle $B$ contains $c$ cubic inches of cake and $s$ square inches of... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{5}"
} | {
"split": "train",
"index": 2381,
"solution": "\n1. **Understanding the Problem:**\n - We have a cube with edge length $2$ inches, which is iced on the sides and the top.\n - The cube is cut into three pieces, and we focus on the piece with a triangular top view labeled as triangle $B$.\n - We need to find... |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\sin ^{4}\\left(\\frac{x}{2}\\right) \\cos ^{4}\\left(\\frac{x}{2}\\right) d x\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{8}"
} | {
"split": "train",
"index": 2382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer whose digits are the same when read forwards or backwards is called a palindrome. An example of a palindrome is 13931. What is the sum of the digits of the next palindrome greater than 13931? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the probability that a baby born in a certain hospital will speak in the next day is 1/4, what is the probability that at least 2 babies out of a cluster of 5 babies will speak tomorrow? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{47}{128}"
} | {
"split": "train",
"index": 2384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Write the product of the digits of each natural number from 1 to 2018 (for example, the product of the digits of the number 5 is 5; the product of the digits of the number 72 is \\(7 \\times 2=14\\); the product of the digits of the number 607 is \\(6 \\times 0 \\times 7=0\\), ... | math | {
"style": "rule",
"ground_truth": "184320"
} | {
"split": "train",
"index": 2385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particular convex polygon with seven sides has exactly one right angle. How many diagonals does this seven-sided polygon have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 2386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The values of $a$, $b$, $c$ and $d$ are 1, 2, 3 and 4, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $bc$, $cd$ and $da$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 2387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function is given by\n$$\nf(x)=\\ln (a x+b)+x^{2} \\quad (a \\neq 0).\n$$\n(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$.\n\n(2) If $f(x) \\leqslant x^{2}+x$ always holds, find the maximum value of $ab$. Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{e}{2}"
} | {
"split": "train",
"index": 2388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $E$ with eccentricity $\\frac{\\sqrt{2}}{2}$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ that passes through point $A(1, \\frac{\\sqrt{2}}{2})$.\n\n$(1)$ Find the equation of ellipse $E$; \n$(2)$ If a line $l$: $y= \\frac{\\sqrt{2}}{2}x+m$ that ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 2389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 2390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three pastures full of grass. The first pasture is 33 acres and can feed 22 cows for 27 days. The second pasture is 28 acres and can feed 17 cows for 42 days. How many cows can the third pasture, which is 10 acres, feed for 3 days (assuming the grass grows at a unifor... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A convex polyhedron S has vertices U1, U2, …, Um, and 120 edges. This polyhedron is intersected by planes Q1, Q2, …, Qm, where each plane Qk intersects only those edges that are connected to vertex Uk. No two planes intersect within the volume or on the surface of S. As a resul... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 2392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $A,B$ and $C$ are non-zero distinct digits in base $6$ such that $\\overline{ABC}_6 + \\overline{BCA}_6+ \\overline{CAB}_6 = \\overline{AAA0}_6$, find $B+C$ in base $6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 2393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\log_{10}\\sin x + \\log_{10}\\cos x= -1$ and that $\\log_{10}(\\sin x+\\cos\nx)=\\frac{1}{2}(\\log_{10}n-1)$, find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a circle of radius $4$ with center $O_1$ , a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$ , and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$ . The centers of the circle are... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 2395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Segments $BD$ and $AE$ intersect at $C$, with $AB = BC$ and $CD = DE = EC$. Additionally, $\\angle A = 4 \\angle B$. Determine the degree measure of $\\angle D$.\nA) 45\nB) 50\nC) 52.5\nD) 55\nE) 60 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52.5"
} | {
"split": "train",
"index": 2396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $\\dfrac{\\sqrt[3]{81}}{\\sqrt[4]{81}}$ in terms of 81 raised to what power? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 2397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(PQR\\), \\(PQ = 8\\), \\(QR = 15\\), and \\(PR = 17\\). If \\(S\\) and \\(T\\) are points on \\(\\overline{PQ}\\) and \\(\\overline{PR}\\) respectively, such that \\(PS = 3\\) and \\(PT = 10\\), find the area of triangle \\(PST\\). Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{225}{17}"
} | {
"split": "train",
"index": 2398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin x_{1}=\\sin x_{2}=\\frac{1}{3}$ and $0 \\lt x_{1} \\lt x_{2} \\lt 2\\pi$, find $\\cos |\\overrightarrow{a}|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{9}"
} | {
"split": "train",
"index": 2399,
"solution": ""
} |
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