data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If the sum $1 + 2 + 3 + \\cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1, 8, and 49"
} | {
"split": "train",
"index": 9100,
"solution": "To solve this problem, we need to determine the values of $K$ for which the sum $1 + 2 + 3 + \\cdots + K$ is a perfect square $N^2$, and $N$ is less than $100$.\n\n1. **Sum Formula**: The sum of the first $K$ natural numbers is given by the formula:\n \\[\n S = ... |
deepscaler | [
{
"role": "user",
"content": "How many continuous paths from $A$ to $B$, along segments of the figure, do not revisit any of the six labeled points?\n\n[asy]\ndraw((0,0)--(3,0)--(3,2)--(0,2)--(0,0)--cycle,linewidth(2));\ndraw((0,2)--(1,0)--(3,2)--(0,2)--cycle,linewidth(2));\ndraw((0,2)--(1.5,3.5)--(3,2),lin... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 9101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$, let $O$ be the intersection of diagonals $\\overline{AC}$ and $\\overline{BD}$. Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$, and angle $ACB$ is $r$ times as large as angle $AOB$. Find $r.$ Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 9102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the two regular tetrahedra \\(A-OBC\\) and \\(D-OBC\\) with coinciding bases, \\(M\\) and \\(N\\) are the centroids of \\(\\triangle ADC\\) and \\(\\triangle BDC\\) respectively. Let \\(\\overrightarrow{OA}=\\boldsymbol{a}, \\overrightarrow{OB}=\\boldsymbol{b}, \\overrightar... | math | {
"style": "rule",
"ground_truth": "439"
} | {
"split": "train",
"index": 9103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\{a_1,a_2,a_3,\\ldots,a_n\\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \\cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \\cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty s... | math | {
"style": "rule",
"ground_truth": "368"
} | {
"split": "train",
"index": 9104,
"solution": "We note that the number of subsets (for now, including the empty subset, which we will just define to have a power sum of zero) with $9$ in it is equal to the number of subsets without a $9$. To easily see this, take all possible subsets of $\\{1,2,\\ldots,8\\}$. Si... |
deepscaler | [
{
"role": "user",
"content": "Find the number of pairs $(n,C)$ of positive integers such that $C\\leq 100$ and $n^2+n+C$ is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 9105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the units digit of the sum of the nine terms of the sequence $1! + 1, \\, 2! + 2, \\, 3! + 3, \\, ..., \\, 8! + 8, \\, 9! + 9$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 9106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $||\\overrightarrow{OA}||=||\\overrightarrow{OB}||=2$, point $C$ is on line segment $AB$, and the minimum value of $||\\overrightarrow{OC}||$ is $1$, find the minimum value of $||\\overrightarrow{OA}-t\\overrightarrow{OB}||$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 9107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum sum of the distances from a point in space to the vertices of a regular tetrahedron with side length 1 is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 9108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \\ldots,\na_k$ . It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there are exactly $a_1$ numbers of ... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 9109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man chooses two positive integers \\( m \\) and \\( n \\). He defines a positive integer \\( k \\) to be good if a triangle with side lengths \\( \\log m \\), \\( \\log n \\), and \\( \\log k \\) exists. He finds that there are exactly 100 good numbers. Find the maximum possi... | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 9110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest solution to the equation \\[\\lfloor x^2 \\rfloor - \\lfloor x \\rfloor^2 = 19.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{119}"
} | {
"split": "train",
"index": 9111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square can be divided into four congruent figures as shown: If each of the congruent figures has area 1, what is the area of the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 9112,
"solution": "There are four congruent figures with area 1, so the area of the square is 4."
} |
deepscaler | [
{
"role": "user",
"content": "Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\\circ$ angle with $AB$. At the ... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 9113,
"solution": "Label the point of intersection as $C$. Since $d = rt$, $AC = 8t$ and $BC = 7t$. According to the law of cosines,\n[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=16*expi(pi/3); D(B--A); D(A--C); D(B--C,dashed); MP(\"A\",A,SW);MP(\"B\",B,... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\\frac {(a+b)^{2}-c^{2}}{3ab}=1$. \n$(1)$ Find $\\angle C$; \n$(2)$ If $c= \\sqrt {3}$ and $b= \\sqrt {2}$, find $\\angle B$ and the area o... | math | {
"style": "rule",
"ground_truth": "\\frac {3+ \\sqrt {3}}{4}"
} | {
"split": "train",
"index": 9114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four years ago you invested some money at $10\\%$ interest. You now have $\\$439.23$ in the account. If the interest was compounded yearly, how much did you invest 4 years ago? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 9115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. The altitude from $A$ intersects $B C$ at $D$. Let $\\omega_{1}$ and $\\omega_{2}$ be the incircles of $A B D$ and $A C D$, and let the common external tangent of $\\omega_{1}$ and $\\omega_{2}$ (other than $B C$) interse... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 9116,
"solution": "Solution 1: Let $I_{1}, I_{2}$ be the centers of $\\omega_{1}, \\omega_{2}$, respectively, $X_{1}, X_{2}$ be the tangency points of $\\omega_{1}, \\omega_{2}$ with $B C$, respectively, and $Y_{1}, Y_{2}$ be the tangency points of $\\omega_{1}, \\omega_{2}$ with $A D... |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\left(\\sqrt[6]{27} - \\sqrt{6 \\frac{3}{4} }\\right)^2$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 9117,
"solution": "1. **Simplify $\\sqrt[6]{27}$:**\n \\[\n \\sqrt[6]{27} = (3^3)^{\\frac{1}{6}} = 3^{\\frac{3}{6}} = 3^{\\frac{1}{2}} = \\sqrt{3}\n \\]\n\n2. **Simplify $\\sqrt{6 \\frac{3}{4}}$:**\n \\[\n 6 \\frac{3}{4} = 6 + \\frac{3}{4} = \\frac{24}{4} + \\frac{3}{4} = \\... |
deepscaler | [
{
"role": "user",
"content": "Find all polynomials $P$ with integer coefficients such that $P (0)\\ne 0$ and $$P^n(m)\\cdot P^m(n)$$ is a square of an integer for all nonnegative integers $n, m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "P(x) = x + 1"
} | {
"split": "train",
"index": 9118,
"solution": "\nWe are tasked with finding all polynomials \\( P(x) \\) with integer coefficients such that \\( P(0) \\neq 0 \\) and for all nonnegative integers \\( n, m \\), the expression \\( P^n(m) \\cdot P^m(n) \\) is a square of an integer. The polynomial \\( P^n(m) \\) den... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $2 \\cdot \\sqrt[4]{2^7 + 2^7 + 2^8}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8 \\cdot \\sqrt[4]{2}"
} | {
"split": "train",
"index": 9119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have modified my walking game. On move 1, I still do nothing, but for each move $n$ where $2 \\le n \\le 30$, I walk one step backward if $n$ is prime and three steps forward if $n$ is composite. After all 30 moves, I assess how far I am from my starting point. How many total... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 9120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the altitude and the median from vertex $C$ each divide the angle $ACB$ into three equal parts. Determine the ratio of the sides of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 : \\sqrt{3} : 1"
} | {
"split": "train",
"index": 9121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $1000^{100}$ is expanded out, the result is $1$ followed by how many zeros? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 9122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the square roots of a positive number are $a+2$ and $2a-11$, find the positive number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 9123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $5(3x + 2) - 2 = -2(1 - 7x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 9124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two gre... | math | {
"style": "rule",
"ground_truth": "310"
} | {
"split": "train",
"index": 9125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Parallelogram $ABCD$ has vertices $A(3,4)$, $B(-2,1)$, $C(-5,-2)$, and $D(0,1)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is left of the $y$-axis? Express your answer as a common fraction. Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 9126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 9127,
"solution": "Obviously $n\\le 90$. We see that the problem's condition is equivalent to: 96 is the smallest number that can be formed which is 1 mod 5, and 92, 93, 94 can be formed (95 can always be formed). Now divide this up into cases. If $n\\equiv 0\\pmod{5}$, then 91 can be... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt{2^2\\cdot 3^4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 9128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide the sequence successively into groups with the first parenthesis containing one number, the second parenthesis two numbers, the third parenthesis three numbers, the fourth parenthesis four numbers, the fifth parenthesis one number, and so on in a cycle: $(3)$, $(5,7)$, $... | math | {
"style": "rule",
"ground_truth": "2072"
} | {
"split": "train",
"index": 9129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an equilateral triangle with one vertex at the origin and the other two vertices on the parabola $y^2 = 2\\sqrt{3}x$, find the length of the side of this equilateral triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 9130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lauren plays basketball with her friends. She makes 10 baskets. Each of these baskets is worth either 2 or 3 points. Lauren scores a total of 26 points. How many 3 point baskets did she make? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 9131,
"solution": "Suppose that Lauren makes $x$ baskets worth 3 points each. Since she makes 10 baskets, then $10-x$ baskets that she made are worth 2 points each. Since Lauren scores 26 points, then $3 x+2(10-x)=26$ and so $3 x+20-x=26$ which gives $x=6$. Therefore, Lauren makes 6 b... |
deepscaler | [
{
"role": "user",
"content": "The circle $2x^2 = -2y^2 + 12x - 4y + 20$ is inscribed inside a square which has a pair of sides parallel to the x-axis. What is the area of the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 9132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 9133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the inequality system about $y$ $\\left\\{\\begin{array}{l}{2y-6≤3(y-1)}\\\\{\\frac{1}{2}a-3y＞0}\\end{array}\\right.$, if it has exactly $4$ integer solutions, then the product of all integer values of $a$ that satisfy the conditions is ______. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 9134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of length 15 consisting of zeros and ones, find the number of sequences that have either all zeros consecutive or all ones consecutive, but not both. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "238"
} | {
"split": "train",
"index": 9135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many pairs of consecutive integers in $\\{1000,1001,1002,\\ldots,2000\\}$ is no carrying required when the two integers are added? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 9136,
"solution": "Consider what carrying means: If carrying is needed to add two numbers with digits $abcd$ and $efgh$, then $h+d\\ge 10$ or $c+g\\ge 10$ or $b+f\\ge 10$. 6. Consider $c \\in \\{0, 1, 2, 3, 4\\}$. $1abc + 1ab(c+1)$ has no carry if $a, b \\in \\{0, 1, 2, 3, 4\\}$. This... |
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 lies below the $x$-axis? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 9137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bag A has 3 white marbles and 4 black marbles. Bag B has 6 yellow marbles and 4 blue marbles. Bag C has 2 yellow marbles and 5 blue marbles. A marble is drawn at random from Bag A. If it is white, a marble is drawn at random from Bag B, otherwise, if it is black, a marble i... | math | {
"style": "rule",
"ground_truth": "\\frac{103}{245}"
} | {
"split": "train",
"index": 9138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define \\[P(x) =(x-1^2)(x-2^2)\\cdots(x-50^2).\\] How many integers $n$ are there such that $P(n)\\leq 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1300"
} | {
"split": "train",
"index": 9139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a collection of red, blue, and green marbles, there are $25\\%$ more red marbles than blue marbles, and there are $60\\%$ more green marbles than red marbles. Suppose that there are $r$ red marbles. What is the total number of marbles in the collection? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "3.4r"
} | {
"split": "train",
"index": 9140,
"solution": "Let's denote the number of blue marbles as $b$ and the number of green marbles as $g$. According to the problem, we have the following relationships:\n1. There are $25\\%$ more red marbles than blue marbles, which translates to:\n \\[\n r = b + 0.25b = 1.25b\n ... |
deepscaler | [
{
"role": "user",
"content": "If \\( x = \\frac{2}{3} \\) and \\( y = \\frac{3}{2} \\), find the value of \\( \\frac{1}{3}x^8y^9 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 9141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest three-digit number in a format $abc$ (where $a, b, c$ are digits, $a \\neq 0$) such that when multiplied by 111, the result is not a palindrome. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 9142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$, and $\\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\\overline{BK}$ and $\\overlin... | math | {
"style": "rule",
"ground_truth": "072"
} | {
"split": "train",
"index": 9143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation $a^3 + b^3 + c^3 = 2001$ in positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(10,10,1), (10,1,10), (1,10,10)"
} | {
"split": "train",
"index": 9144,
"solution": "Note that for all positive integers $n,$ the value $n^3$ is congruent to $-1,0,1$ modulo $9.$ Since $2001 \\equiv 3 \\pmod{9},$ we find that $a^3,b^3,c^3 \\equiv 1 \\pmod{9}.$ Thus, $a,b,c \\equiv 1 \\pmod{3},$ and the only numbers congruent to $1$ modulo $3$ are ... |
deepscaler | [
{
"role": "user",
"content": "Let $x_1<x_2< \\ldots <x_{2024}$ be positive integers and let $p_i=\\prod_{k=1}^{i}(x_k-\\frac{1}{x_k})$ for $i=1,2, \\ldots, 2024$ . What is the maximal number of positive integers among the $p_i$ ? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"split": "train",
"index": 9145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three people, A, B, and C, stand on a staircase with 7 steps. If each step can accommodate at most 2 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 9146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand the following expression: $16(2x+5)$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32x+80"
} | {
"split": "train",
"index": 9147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set \\( S = \\left\\{ z \\mid |z - 7 - 8i| = |z_1^4 + 1 - 2z_1^2| ; z, z_1 \\in \\mathbb{C}, |z_1| = 1 \\right\\} \\), find the area of the region corresponding to \\( S \\) in the complex plane. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"split": "train",
"index": 9148,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the cells of a $7 \\times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 9149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\det \\mathbf{A} = 2$ and $\\det \\mathbf{B} = 12,$ then find $\\det (\\mathbf{A} \\mathbf{B}).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 9150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 9151,
"solution": "1. **Identify the Geometry of the Hexagon and Midpoints:**\n Given a regular hexagon $P_1P_2P_3P_4P_5P_6$, each side of the hexagon is equal, and each internal angle is $120^\\circ$. The apothem, which is the distance from the center to the midpoint of any side, i... |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the following expression, simplified as a fraction: $$1+\\cfrac{3}{2+\\cfrac{5}{6}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{17}"
} | {
"split": "train",
"index": 9152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: \n\\[x=x^2+y^2\\] \\[y=2xy\\] \nis Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 9153,
"solution": "We are given the system of equations:\n1. \\(x = x^2 + y^2\\)\n2. \\(y = 2xy\\)\n\nWe will analyze this system by considering different cases.\n\n#### Case 1: \\(y = 0\\)\nSubstituting \\(y = 0\\) into the first equation:\n\\[ x = x^2 + 0^2 \\]\n\\[ x = x^2 \\]\n\\[... |
deepscaler | [
{
"role": "user",
"content": "The number of triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations\n$ab+bc=44$\n$ac+bc=23$\nis Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 9154,
"solution": "1. **Factor the second equation**: Given $ac + bc = 23$, we can factor out $c$:\n \\[\n c(a + b) = 23\n \\]\n Since $23$ is a prime number, the possible values for $c$ are $1$ and $23$.\n\n2. **Analyze the possible values of $c$**:\n - If $c = 23$, then $a... |
deepscaler | [
{
"role": "user",
"content": "The polynomial $\\frac{1}{5}{x^2}{y^{|m|}}-(m+1)y+\\frac{1}{7}$ is a cubic binomial in terms of $x$ and $y$. Find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 9155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expression \\((x+y+z)^{2034}+(x-y-z)^{2034}\\), the brackets were expanded, and like terms were combined. How many monomials of the form \\(x^{a} y^{b} z^{c}\\) have a non-zero coefficient? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1036324"
} | {
"split": "train",
"index": 9156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit positive integers are there such that the product of their digits equals 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 9157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.67\\%"
} | {
"split": "train",
"index": 9158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 9159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ is chosen at random from the interior of a square $ABCD$. What is the probability that the triangle $ABP$ has a greater area than each of the triangles $BCP$, $CDP$, and $DAP$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 9160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ Determine the area of $\\triangle ABC.$\n\n[asy]\ndraw((0,0)--(2,5)--(11,5)--(18,0)--cycle);\ndraw((11,5)--(0,0));\nlabel(\"$D$\",(0,0),W);\nlabel(\"$A$\",(2,5),NW);\nlabel(\"$B$... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 9161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For an integer $x \\geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \\ldots$ define... | math | {
"style": "rule",
"ground_truth": "142"
} | {
"split": "train",
"index": 9162,
"solution": "\nWe are given a sequence \\( x_0, x_1, x_2, \\ldots \\) defined by \\( x_0 = 1 \\) and the recursive formula\n\n\\[\nx_{n+1} = \\frac{x_n p(x_n)}{q(x_n)}\n\\]\n\nwhere \\( p(x) \\) is the least prime that does not divide \\( x \\) and \\( q(x) \\) is the product of... |
deepscaler | [
{
"role": "user",
"content": "How many even numbers are greater than 300 and less than 600? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "149"
} | {
"split": "train",
"index": 9163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two fair dice and their sides are positive integers \\( a_{1}, \\ldots, a_{6} \\) and \\( b_{1}, \\ldots, b_{6} \\), respectively. After throwing them, the probability of getting a sum of \\( 2, 3, 4, \\ldots, 12 \\) respectively is the same as that of throwing two no... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 9164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The archipelago consists of $N \\geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island and that among any 7 islands, there are always two that are connected by a bridge. What is the maximum p... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 9165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse E: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}$$\\=1 (a>b>0) passing through point P ($$\\sqrt {3}$$, $$\\frac {1}{2}$$) with its left focus at F ($$-\\sqrt {3}$$, 0).\n\n1. Find the equation of ellipse E.\n2. If A is the right vertex of ellipse E, and the l... | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {5}+ \\sqrt {15}}{4}"
} | {
"split": "train",
"index": 9166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α \\in \\left( \\frac{π}{2}, π \\right)$ and $3\\cos 2α = \\sin \\left( \\frac{π}{4} - α \\right)$, find the value of $\\sin 2α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{17}{18}"
} | {
"split": "train",
"index": 9167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the volume of a cube with edge length six inches to the volume of a cube with edge length one foot? 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}{8}"
} | {
"split": "train",
"index": 9168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A research study group is investigating the traffic volume at a certain intersection near the school during the rush hour from 8:00 to 10:00. After long-term observation and statistics, they have established a simple function model between traffic volume and average vehicle spe... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{3}"
} | {
"split": "train",
"index": 9169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of children in the families $A$, $B$, $C$, $D$, and $E$ are shown in the table below:\n\n| | $A$ | $B$ | $C$ | $D$ | $E$ |\n|---------|-----|-----|-----|-----|-----|\n| Boys | $0$ | $1$ | $0$ | $1$ | $1$ |\n| Girls | $0$ | $0$ | $1$ | $1$ | $2$ |\n\n$(1)... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"split": "train",
"index": 9170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 24x^2 + 50x - 42 = 0.$ Find the value of $\\frac{a}{\\frac{1}{a}+bc} + \\frac{b}{\\frac{1}{b}+ca} + \\frac{c}{\\frac{1}{c}+ab}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{476}{43}"
} | {
"split": "train",
"index": 9171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Emma had just been given some coins by her parents. On the way to school she lost exactly half of them, and then by retracing her steps she found exactly four-fifths of the coins she had lost. What fraction of the coins that she received from her parents were still missing af... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 9172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. W... | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 9173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The random variable $X$ follows a normal distribution $N(1,4)$. Given that $P(X \\geqslant 2) = 0.2$, calculate the probability that $0 \\leqslant X \\leqslant 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 9174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\\cos B = \\frac{4}{5}$ and $b = 2$.\n1. Find the value of $a$ when $A = \\frac{\\pi}{6}$.\n2. Find the value of $a + c$ when the area of $\\triangle ABC$ is $... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"split": "train",
"index": 9175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to select 4 representatives from a group of 5 male students and 4 female students to participate in an activity, ensuring that there are at least two males and at least one female among the representatives. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 9176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with center \\( Q \\) and radius 2 rolls around the inside of a right triangle \\( DEF \\) with side lengths 9, 12, and 15, always remaining tangent to at least one side of the triangle. When \\( Q \\) first returns to its original position, through what distance has \... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 9177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Beaumont High School, there are 12 players on the baseball team. All 12 players are taking at least one of biology or chemistry. If 7 players are taking biology and 2 players are taking both sciences, how many players are taking chemistry? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 9178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 9179,
"solution": "1. **Understanding the problem**: We start with a square piece of paper with each side measuring 4 inches. The paper is folded in half vertically, which means the fold divides the square into two equal rectangles each with dimensions $4 \\times 2$ inches.\n\n2. **Cu... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{P}$ be a regular $2022$ -gon with area $1$ . Find a real number $c$ such that, if points $A$ and $B$ are chosen independently and uniformly at random on the perimeter of $\\mathcal{P}$ , then the probability that $AB \\geq c$ is $\\tfrac{1}{2}$ . \n... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2/\\pi}"
} | {
"split": "train",
"index": 9180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jeremy wakes up at 6:00 a.m., catches the school bus at 7:00 a.m., has 7 classes that last 45 minutes each, enjoys 45 minutes for lunch, and spends an additional 2.25 hours (which includes 15 minutes for miscellaneous activities) at school. He takes the bus home and arrives at ... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 9181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 9182,
"solution": "1. **Identify the total number of marbles**: Initially, Tyrone has 97 marbles and Eric has 11 marbles. The total number of marbles is:\n \\[\n 97 + 11 = 108\n \\]\n\n2. **Set up the equation after redistribution**: Let's denote the number of marbles Tyrone giv... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $\\sin A = \\frac{3}{5}$ and $\\cos B = \\frac{5}{13}.$ Find $\\cos C.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{65}"
} | {
"split": "train",
"index": 9183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer sequence \\(a_1, a_2, a_3, \\dots\\) is defined as follows: \\(a_1 = 1\\). For \\(n \\geq 1\\), \\(a_{n+1}\\) is the smallest integer greater than \\(a_n\\) such that for all \\(i, j, k \\in \\{1, 2, \\dots, n+1\\}\\), the condition \\(a_i + a_j \\neq 3a_k\\) is sat... | math | {
"style": "rule",
"ground_truth": "66016"
} | {
"split": "train",
"index": 9184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the polynomial \\( P(x)=x^{3}+x^{2}-x+2 \\). Determine all real numbers \\( r \\) for which there exists a complex number \\( z \\) not in the reals such that \\( P(z)=r \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "r>3, r<49/27"
} | {
"split": "train",
"index": 9185,
"solution": "Because such roots to polynomial equations come in conjugate pairs, we seek the values \\( r \\) such that \\( P(x)=r \\) has just one real root \\( x \\). Considering the shape of a cubic, we are interested in the boundary values \\( r \\) such that \\( P(x)-r \\) ... |
deepscaler | [
{
"role": "user",
"content": "What is the expected value of the roll of a standard 6-sided die? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 9186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 35 groups of students, each group containing 3 students. Among these groups:\n- 10 groups have only 1 boy.\n- 19 groups have at least 2 boys.\n- The number of groups with 3 boys is twice the number of groups with 3 girls.\n\nHow many boys are there? Let's think step b... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 9187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the areas of all distinct rectangles that can be formed from 9 squares (not necessarily all), if the side of each square is $1 \\text{ cm}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 9188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given six people, including A, B, and C, are arranged in a row for a photo, and A and B are not on the same side as C, calculate the total number of different arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 9189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the operation $Z$ is defined as $a Z b = b + 10a - a^2$, what is the value of $2Z6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 9190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dani wrote the integers from 1 to \\( N \\). She used the digit 1 fifteen times. She used the digit 2 fourteen times.\nWhat is \\( N \\) ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 9191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "272"
} | {
"split": "train",
"index": 9192,
"solution": "Using the trapezoid $ABCD$ mentioned above, draw an altitude of the trapezoid passing through point $B$ onto $AD$ at point $J$. Now, we can use the pythagorean theorem: $(22^2-(r-10)^2)+10^2=r^2$. Expanding and combining like terms gives us the quadratic \\[r^2-10r-... |
deepscaler | [
{
"role": "user",
"content": "Both roots of the quadratic equation $x^2 - 63 x + k = 0$ are prime numbers. How many possible values of $k$ are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 9193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[\\begin{aligned} a &= \\sqrt{2}+\\sqrt{3}+\\sqrt{6}, \\\\ b &= -\\sqrt{2}+\\sqrt{3}+\\sqrt{6}, \\\\ c&= \\sqrt{2}-\\sqrt{3}+\\sqrt{6}, \\\\ d&=-\\sqrt{2}-\\sqrt{3}+\\sqrt{6}. \\end{aligned}\\]Evaluate $\\left(\\frac1a + \\frac1b + \\frac1c + \\frac1d\\right)^2.$ Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{96}{529}"
} | {
"split": "train",
"index": 9194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $θ∈[0,π]$, then the probability of $\\sin (θ+ \\frac {π}{3}) > \\frac {1}{2}$ being true is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 9195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $t,$ the point of intersection of the lines $tx - 2y - 3t = 0$ and $x - 2ty + 3 = 0$ is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Let'... | math | {
"style": "rule",
"ground_truth": "\\text{(E)}"
} | {
"split": "train",
"index": 9196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\(x_1+1=x_2+2=x_3+3=\\cdots=x_{2010}+2010=x_1+x_2+x_3+\\cdots+x_{2010}+2011\\). Find the value of \\(\\left\\lfloor|T|\\right\\rfloor\\), where \\(T=\\sum_{n=1}^{2010}x_n\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1005"
} | {
"split": "train",
"index": 9197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a moving point $A$ on the curve $y=x^{2}$, let $m$ be the tangent line to the curve at point $A$. Let $n$ be a line passing through point $A$, perpendicular to line $m$, and intersecting the curve at another point $B$. Determine the minimum length of the line segment $AB$... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 9198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^2 + c^2 - b^2 = ac$, $c=2$, and point $G$ satisfies $|\\overrightarrow{BG}| = \\frac{\\sqrt{19}}{3}$ and $\\overrightarrow{BG} = \\frac{1}{3}(\\overrightarrow{BA}... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{21}}{14}"
} | {
"split": "train",
"index": 9199,
"solution": ""
} |
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