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 $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\\circ$ rotation about $G.$ What is the area of the union of the two regions enclos... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To implement the \"double reduction\" policy and promote intangible cultural heritage, a certain school plans to organize extracurricular interest classes for students to visit the following projects: $A$ (Xiu Shui pottery art), $B$ (Xiu Shui tea-picking opera), $C$ (Jiujiang m... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chord PQ of the left branch of the hyperbola $x^2 - y^2 = 4$ passes through its left focus $F_1$, and the length of $|PQ|$ is 7. If $F_2$ is the right focus of the hyperbola, then the perimeter of $\\triangle PF_2Q$ is. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy]\nimport graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;\n\nfor(int i = -4; i... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\frac{35}{12 \\pi^2}"
} | {
"split": "train",
"index": 104,
"solution": "\\textbf{First solution:} (by Daniel Kane) The probability is $1 - \\frac{35}{12\\pi^2}$. We start with some notation and simplifications. For simplicity, we assume without loss of generality that the circle has radius 1. Let $E$ denote the expected value of a random... |
deepscaler | [
{
"role": "user",
"content": "Given the inequality $(|x|-1)^2+(|y|-1)^2<2$, determine the number of lattice points $(x, y)$ that satisfy it. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of a circle whose circumference is $24\\pi$ is $k\\pi$. What is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 106,
"solution": "1. **Identify the relationship between circumference and diameter**: \n The circumference $C$ of a circle is given by the formula $C = 2\\pi r$, where $r$ is the radius of the circle. Given that the circumference is $24\\pi$, we can set up the equation:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "The surface area of a sphere with radius $r$ is $4\\pi r^2$. Including the area of its circular base, what is the total surface area of a hemisphere with radius 6 cm? Express your answer in terms of $\\pi$.\n\n[asy]\n\nimport markers;\nsize(150);\nimport geometry;\ndraw((0,-7)-... | math | {
"style": "rule",
"ground_truth": "108\\pi"
} | {
"split": "train",
"index": 107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three circles with radii 2, 3, and 3 touch each other. What is the area of the triangle formed by joining the centers of these circles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Around the outside of a $4$ by $4$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tang... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 109,
"solution": "1. **Identify the radius of the semicircles**: The original square has a side length of $4$. Each semicircle is constructed with the side of the square as its diameter. Therefore, the radius of each semicircle is half the side length of the square, which is $\\frac{4... |
deepscaler | [
{
"role": "user",
"content": "Determine the residue of $-998\\pmod{28}$. Your answer should be an integer in the range $0,1,2,\\ldots,25,26,27$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A residential building has a construction investment of 250 yuan per square meter. Considering a service life of 50 years and an annual interest rate of 5%, calculate the monthly rent per square meter in yuan required to recoup the total investment. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "1.14"
} | {
"split": "train",
"index": 111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square root of $x$ is greater than 3 and less than 4. How many integer values of $x$ satisfy this condition? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $m$ and $n∈\\{lg2+lg5,lo{g}_{4}3,{(\\frac{1}{3})}^{-\\frac{3}{5}},tan1\\}$, the probability that the function $f\\left(x\\right)=x^{2}+2mx+n^{2}$ has two distinct zeros is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many values of $x$, $-17<x<100$, satisfy $\\cos^2 x + 3\\sin^2 x = \\cot^2 x$? (Note: $x$ is measured in radians.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a subset of $\\{1,2,3, \\ldots, 12\\}$ such that it is impossible to partition $S$ into $k$ disjoint subsets, each of whose elements sum to the same value, for any integer $k \\geq 2$. Find the maximum possible sum of the elements of $S$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 115,
"solution": "We note that the maximum possible sum is 78 (the entire set). However, this could be partitioned into 2 subsets with sum 39: $\\{1,2,3,10,11,12\\}$ and $\\{4,5,6,7,8,9\\}$. The next largest possible sum is 77 (the entire set except 1). If $k \\geq 2$ subsets each had... |
deepscaler | [
{
"role": "user",
"content": "While walking on a plane surface, a traveler first headed 18 miles north, then 11 miles west, then 6 miles south and finally 6 miles east. How many miles from the starting point was the traveler after these four legs of the journey? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$ in the equation $(-1)(2)(x)(4)=24$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 117,
"solution": "Since $(-1)(2)(x)(4)=24$, then $-8x=24$ or $x=\\frac{24}{-8}=-3$."
} |
deepscaler | [
{
"role": "user",
"content": "In the figure, $\\angle ABC$ and $\\angle ADB$ are each right angles. Additionally, $AC = 17.8$ units and $AD = 5$ units. What is the length of segment $DB$? [asy]\nimport olympiad; import geometry; size(150); defaultpen(linewidth(0.8));\ntriangle t = triangle((0,0),(sqrt(89),0... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many different negative values of $x$ is $\\sqrt{x +150}$ a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a \"Modulo $m$ graph paper,\" with $m = 17$, forming a grid of $17^2$ points. Each point $(x, y)$ represents integer pairs where $0 \\leq x, y < 17$. We graph the congruence $$5x \\equiv 3y + 2 \\pmod{17}.$$ The graph has single $x$-intercept $(x_0, 0)$ and single $y$-... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x \\diamondsuit y = 3x + 5y$ for all $x$ and $y$, then what is the value of $2 \\diamondsuit 7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system \\( xOy \\), find the area of the graph formed by all points \\( (x, y) \\) that satisfy \\( \\lfloor x \\rfloor \\cdot \\lfloor y \\rfloor = 2013 \\), where \\( \\lfloor x \\rfloor \\) represents the greatest integer less than or equal to t... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the quadratic equation $x^{2}-(r+7) x+r+87=0$ where $r$ is a real number. This equation has two distinct real solutions $x$ which are both negative exactly when $p<r<q$, for some real numbers $p$ and $q$. What is the value of $p^{2}+q^{2}$? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "8098"
} | {
"split": "train",
"index": 123,
"solution": "A quadratic equation has two distinct real solutions exactly when its discriminant is positive. For the quadratic equation $x^{2}-(r+7) x+r+87=0$, the discriminant is $\\Delta=(r+7)^{2}-4(1)(r+87)=r^{2}+14 r+49-4 r-348=r^{2}+10 r-299$. Since $\\Delta=r^{2}+10 r-299=(... |
deepscaler | [
{
"role": "user",
"content": "Suppose that there are real numbers $a, b, c \\geq 1$ and that there are positive reals $x, y, z$ such that $$\\begin{aligned} a^{x}+b^{y}+c^{z} & =4 \\\\ x a^{x}+y b^{y}+z c^{z} & =6 \\\\ x^{2} a^{x}+y^{2} b^{y}+z^{2} c^{z} & =9 \\end{aligned}$$ What is the maximum possible va... | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{4}"
} | {
"split": "train",
"index": 124,
"solution": "The Cauchy-Schwarz inequality states that given 2 sequences of $n$ real numbers $x_{1}, x_{2}, \\ldots, x_{n}$ and $y_{1}, y_{2}, \\ldots, y_{n}$, then $\\left(x_{1}^{2}+x_{2}^{2}+\\ldots+x_{n}^{2}\\right)\\left(y_{1}^{2}+y_{2}^{2}+\\ldots+y_{n}^{2}\\right) \\geq\\le... |
deepscaler | [
{
"role": "user",
"content": "Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \\[\n\\{-8, -6, -4, -1, 3, 5, 7, 10\\}.\n\\] What is the minimum possible value of \\[\n(p+q+r+s)^{2} + (t+u+v+w)^{2}?\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the solution set of the system of linear inequalities in one variable $x$, $\\left\\{{\\begin{array}{l}{x-1≥2x+1}\\\\{2x-1＜a}\\end{array}}\\right.$, is $x\\leqslant -2$, and the solution of the fractional equation in variable $y$, $\\frac{{y-1}}{{y+1}}=\\frac{a}{{y+1}}-2$, i... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be nonzero real numbers. Find the minimum value of\n\\[ x^2 + y^2 + \\frac{4}{x^2} + \\frac{2y}{x}. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\( x_3 > \\frac{1}{2}(x_1 + x_2) \\) are sati... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the range of the function\n\\[f(x) = \\left( \\arccos \\frac{x}{2} \\right)^2 + \\pi \\arcsin \\frac{x}{2} - \\left( \\arcsin \\frac{x}{2} \\right)^2 + \\frac{\\pi^2}{12} (x^2 + 6x + 8).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left[ \\frac{\\pi^2}{4}, \\frac{9 \\pi^2}{4} \\right]"
} | {
"split": "train",
"index": 129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $e = \\dfrac{\\sqrt{6}}{3}$, and the distance between the left focus and one endpoint of the minor axis is $\\sqrt{3}$.\n\n$(I)$ Find the standard equation of the ellipse;\n\n$(II)$ Gi... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{6}"
} | {
"split": "train",
"index": 130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), a circle \\(\\omega\\) with center \\(O\\) passes through \\(B\\) and \\(C\\) and intersects segments \\(\\overline{AB}\\) and \\(\\overline{AC}\\) again at \\(B'\\) and \\(C'\\), respectively. Suppose that the circles with diameters \\(BB'\\) and \\(CC'\... | math | {
"style": "rule",
"ground_truth": "65/3"
} | {
"split": "train",
"index": 131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Albert's choice of burgers, sides, and drinks are independent events. How many different meals can Albert get if there are 5 choices of burgers, 3 choices of sides, and 12 choices of drinks? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 132,
"solution": "The total number of different meals that Albert can get is: $$5 \\times 3 \\times 12=180$$"
} |
deepscaler | [
{
"role": "user",
"content": "Regular octagon $C H I L D R E N$ has area 1. Find the area of pentagon $C H I L D$. 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": 133,
"solution": "The pentagon $C H I L D$ is congruent to the pentagon $N E R D C$, as their corresponding angles and sides are congruent. Moreover, the two pentagons together compose the entire octagon, so each pentagon must have area one-half of the area of the octagon, or $\\frac{... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x) = \\ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}\\left(e - \\dfrac{1}{e}\\right)"
} | {
"split": "train",
"index": 134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a bijective function $g : R \\to R$ , we say that a function $f : R \\to R$ is its superinverse if it satisfies the following identity $(f \\circ g)(x) = g^{-1}(x)$ , where $g^{-1}$ is the inverse of $g$ . Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its super... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( f(x) \\) is a polynomial of degree \\( n \\) with non-negative integer coefficients, and that \\( f(1)=6 \\) and \\( f(7)=3438 \\), find \\( f(2) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measure of angle $ABC$ is $50^\\circ$, $\\overline{AD}$ bisects angle $BAC$, and $\\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "115^\\circ"
} | {
"split": "train",
"index": 137,
"solution": "1. **Identify Given Information and Define Variables:**\n - Given that $\\angle ABC = 50^\\circ$.\n - $\\overline{AD}$ bisects $\\angle BAC$, so let $\\angle CAD = \\angle BAD = x$.\n - $\\overline{DC}$ bisects $\\angle BCA$, so let $\\angle ACD = \\angle BCD =... |
deepscaler | [
{
"role": "user",
"content": "A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the cent... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 138,
"solution": "We use coordinates. Let the circle have center $(0,0)$ and radius $\\sqrt{50}$; this circle has equation $x^2 + y^2 = 50$. Let the coordinates of $B$ be $(a,b)$. We want to find $a^2 + b^2$. $A$ and $C$ with coordinates $(a,b+6)$ and $(a+2,b)$, respectively, both lie... |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\\cos B - b\\cos A = c$, and $C = \\frac{π}{5}$, then find the measure of angle $B$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{10}"
} | {
"split": "train",
"index": 139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a trapezoid with $AB \\parallel CD$. The bisectors of $\\angle CDA$ and $\\angle DAB$ meet at $E$, the bisectors of $\\angle ABC$ and $\\angle BCD$ meet at $F$, the bisectors of $\\angle BCD$ and $\\angle CDA$ meet at $G$, and the bisectors of $\\angle DAB$ and $\... | math | {
"style": "rule",
"ground_truth": "\\frac{256}{7}"
} | {
"split": "train",
"index": 140,
"solution": "Let $M, N$ be the midpoints of $AD, BC$ respectively. Since $AE$ and $DE$ are bisectors of supplementary angles, the triangle $AED$ is right with right angle $E$. Then $EM$ is the median of a right triangle from the right angle, so triangles $EMA$ and $EMD$ are isosc... |
deepscaler | [
{
"role": "user",
"content": "In different historical periods, the conversion between \"jin\" and \"liang\" was different. The idiom \"ban jin ba liang\" comes from the 16-based system. For convenience, we assume that in ancient times, 16 liang equaled 1 jin, with each jin being equivalent to 600 grams in t... | math | {
"style": "rule",
"ground_truth": "2800"
} | {
"split": "train",
"index": 141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all positive integers $n$ for which the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has an integer as a solution. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n=1"
} | {
"split": "train",
"index": 142,
"solution": "If $n$ is even, $x^{n}+(2+x)^{n}+(2-x)^{n}>0$, so $n$ is odd. For $n=1$, the equation reduces to $x+(2+x)+(2-x)=0$, which has the unique solution $x=-4$. For $n>1$, notice that $x$ is even, because $x, 2-x$, and $2+x$ have all the same parity. Let $x=2 y$, so the equ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(\\cos x) = -f'(\\frac{1}{2})\\cos x + \\sqrt{3}\\sin^2 x$, find the value of $f(\\frac{1}{2})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many prime numbers are between 20 and 30? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{x}{x-1}"
} | {
"split": "train",
"index": 145,
"solution": "1. **Given expressions for $x$ and $y$:**\n \\[\n x = 1 + 2^p \\quad \\text{and} \\quad y = 1 + 2^{-p}\n \\]\n\n2. **Express $2^p$ in terms of $x$:**\n \\[\n x = 1 + 2^p \\implies 2^p = x - 1\n \\]\n\n3. **Substitute $2^p$ in the expression for $y$:**\n ... |
deepscaler | [
{
"role": "user",
"content": "Consider the function $z(x, y)$ describing the paraboloid $z=(2 x-y)^{2}-2 y^{2}-3 y$. Archimedes and Brahmagupta are playing a game. Archimedes first chooses $x$. Afterwards, Brahmagupta chooses $y$. Archimedes wishes to minimize $z$ while Brahmagupta wishes to maximize $z$. A... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"split": "train",
"index": 146,
"solution": "Viewing $x$ as a constant and completing the square, we find that $z =-\\left(y+\\frac{4 x+3}{2}\\right)^{2}+\\left(\\frac{4 x+3}{2}\\right)^{2}+4 x^{2}$. Brahmagupta wishes to maximize $z$, so regardless of the value of $x$, he will pick $y=-\\frac{4 x+3}{2}$. The e... |
deepscaler | [
{
"role": "user",
"content": "There is an angle $\\theta$ in the range $0^\\circ < \\theta < 45^\\circ$ which satisfies\n\\[\\tan \\theta + \\tan 2 \\theta + \\tan 3 \\theta = 0.\\]Calculate $\\tan \\theta$ for this angle. 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": 147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the random variable $\\xi$ follows a normal distribution $N(1,4)$, if $p(\\xi > 4)=0.1$, then $p(-2 \\leqslant \\xi \\leqslant 4)=$ _____ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 148,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log_{25}(x-4)=\\frac{1}{2}$, find $\\frac{1}{\\log_{x}3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The whole numbers from 1 to \\( 2k \\) are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of \\( k \\)? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a\\) and \\(b\\) be real numbers such that\n\\[\n\\frac{a}{2b} + \\frac{a}{(2b)^2} + \\frac{a}{(2b)^3} + \\dots = 6.\n\\]\nFind\n\\[\n\\frac{a}{a + 2b} + \\frac{a}{(a + 2b)^2} + \\frac{a}{(a + 2b)^3} + \\dots.\n\\] Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For an integer $n>2$, the tuple $(1, 2, \\ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\\{1, 2... | math | {
"style": "rule",
"ground_truth": "n \\geq 14"
} | {
"split": "train",
"index": 152,
"solution": "\nGiven the problem, we start with the sequence \\( (1, 2, \\ldots, n) \\) on a blackboard. The challenge is to determine for which integers \\( n > 2 \\), it is possible to obtain every permutation of \\( \\{1, 2, \\ldots, n\\} \\) by repeatedly swapping two numbers... |
deepscaler | [
{
"role": "user",
"content": "Find the product of all positive integral values of $n$ such that $n^2 - 21n + 110 = p$ for some prime number $p$. Assume there is at least one such $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To test the quality of a certain product, it was decided to use the random number table method to draw 5 samples from 300 products for inspection. The products are numbered from 000, 001, 002, ..., to 299. The following are the 7th and 8th rows of the random number table. If we... | math | {
"style": "rule",
"ground_truth": "057"
} | {
"split": "train",
"index": 154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sequence $A$ is a geometric sequence. Sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300.$ What is the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B?$\n\n$\\... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the real sequence $-1$, $a$, $b$, $c$, $-2$ forms a geometric sequence, find the value of $abc$. 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": 156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the following expression: $\\frac{1}{3}-\\frac{1}{9}+\\frac{1}{27}-\\frac{1}{81}+\\frac{1}{243}$? 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{61}{243}"
} | {
"split": "train",
"index": 157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Morse code, each symbol is represented by a sequence of dashes and dots. How many distinct symbols can be represented using sequences of 1, 2, 3, or 4 total dots and/or dashes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x$ if $\\log_x32 = \\dfrac{5}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(15) Given the following propositions:\n\n(1) \"If $x > 2$, then $x > 0$\" - the negation of the proposition\n\n(2) \"For all $a \\in (0, +\\infty)$, the function $y = a^x$ is strictly increasing on its domain\" - the negation\n\n(3) \"$π$ is a period of the function $y = \\sin... | math | {
"style": "rule",
"ground_truth": "(2)(3)"
} | {
"split": "train",
"index": 160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $n, n+1, n+2, n+3, n+4$ are five consecutive integers.\n\nDetermine a simplified expression for the sum of these five consecutive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5n+10"
} | {
"split": "train",
"index": 161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) that satisfies the following conditions:\n1. The units digit of \\( n \\) is 6.\n2. If the units digit 6 is moved to the front of the number, the new number is 4 times \\( n \\). Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"split": "train",
"index": 162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $M = \\{1,2,3,4\\}$, let $A$ be a subset of $M$. The product of all elements in set $A$ is called the \"cumulative value\" of set $A$. It is stipulated that if set $A$ has only one element, its cumulative value is the value of that element, and the cumulative valu... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Choose one digit from the set {0, 2} and two different digits from the set {1, 3, 5} to form a three-digit number without any repeating digits. The total number of such odd three-digit numbers is _________. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $n$, let $\\tau(n)$ denote the number of positive divisors of $n$. If $n$ is a positive integer such that $\\frac{\\tau\\left(n^{2}\\right)}{\\tau(n)}=3$, compute $\\frac{\\tau\\left(n^{7}\\right)}{\\tau(n)}$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 165,
"solution": "Let the prime factorization of $n$ be $n=p_{1}^{e_{1}} p_{2}^{e_{2}} \\cdots p_{k}^{e_{k}}$. Then, the problem condition is equivalent to $$\\prod_{i=1}^{k} \\frac{2 e_{i}+1}{e_{i}+1}=3$$ Note that since $\\frac{2 x+1}{x+1} \\geq 1.5$ for $x \\geq 1$, and $1.5^{3}>3$... |
deepscaler | [
{
"role": "user",
"content": "From the following infinite list of numbers, how many are integers? $$\\sqrt{4096},\\sqrt[3]{4096},\\sqrt[4]{4096},\\sqrt[5]{4096},\\sqrt[6]{4096},\\ldots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Players A and B participate in a two-project competition, with each project adopting a best-of-five format (the first player to win 3 games wins the match, and the competition ends), and there are no ties in each game. Based on the statistics of their previous matches, player A... | math | {
"style": "rule",
"ground_truth": "\\frac{209}{162}"
} | {
"split": "train",
"index": 167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 4 distinct books that are to be distributed evenly between two students, find the probability that the Chinese language book and the Mathematics book are given to the same student. 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": 168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[f(x)=\\cos(x^3-4x^2+5x-2).\\]\nIf we let $f^{(k)}$ denote the $k$ th derivative of $f$ , compute $f^{(10)}(1)$ . For the sake of this problem, note that $10!=3628800$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "907200"
} | {
"split": "train",
"index": 169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability that Asha will lose a game is $\\frac{4}{9}$. If it is impossible to tie, what is the probability that Asha will win the game? 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{5}{9}"
} | {
"split": "train",
"index": 170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a \\) and \\( b \\) be positive integers such that \\( 79 \\mid (a + 77b) \\) and \\( 77 \\mid (a + 79b) \\). Find the smallest possible value of \\( a + b \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "193"
} | {
"split": "train",
"index": 171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What positive integer can $n$ represent if it is known that by erasing the last three digits of the number $n^{3}$, we obtain the number $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters), and all digits except for the combination 000. Tanya considers a license plate happy if the first letter is a co... | math | {
"style": "rule",
"ground_truth": "384000"
} | {
"split": "train",
"index": 173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circular sheet of paper with radius of $6$ cm is cut into three congruent sectors. What is the height of the cone in centimeters that can be created by rolling one of the sections until the edges meet? Express your answer in simplest radical form. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2}"
} | {
"split": "train",
"index": 174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. Calculate the diameter of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Altitudes $\\overline{AP}$ and $\\overline{BQ}$ of an acute triangle $\\triangle ABC$ intersect at point $H$. If $HP=8$ and $HQ=3$, then calculate $(BP)(PC)-(AQ)(QC)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 4 tan, 3 pink, 5 violet, and 2 green chips. If all 14 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the 4 tan chips, the 3 pink chips, and the 5 violet chips are each drawn consecutively, and there is a... | math | {
"style": "rule",
"ground_truth": "\\frac{1440}{14!}"
} | {
"split": "train",
"index": 177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle $PQR$ where $\\angle R = 90^\\circ$, the lengths of sides $PQ = 15$ and $PR = 9$. Find $\\sin Q$ and $\\cos Q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 60-degree angle contains five circles, where each subsequent circle (starting from the second) touches the previous one. By how many times is the sum of the areas of all five circles greater than the area of the smallest circle? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "7381"
} | {
"split": "train",
"index": 179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $({x}^{2}+1){(2x+1)}^{9}={a}_{0}+{a}_{1}(x+2)+{a}_{2}{(x+2)}^{2}+\\cdots +{a}_{11}{(x+2)}^{11}$, find the value of $({a}_{0}+{a}_{1}+{a}_{2}+\\cdots +{a}_{11})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $x^{2}- \\frac{y^{2}}{24}=1$, let the focal points be F<sub>1</sub> and F<sub>2</sub>, respectively. If P is a point on the left branch of the hyperbola such that $|PF_{1}|=\\frac{3}{5}|F_{1}F_{2}|$, find the area of triangle $\\triangle PF_{1}F_{2}$. Let's ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right cone has a base with a circumference of $16\\pi$ inches and a height of 30 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $192\\pi$ cubic inches. What is the ratio of the shorter height to the origina... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $n \\in \\mathbb{N}^*$, the coefficient of the second term in the expansion of $(x+2)^n$ is $\\frac{1}{5}$ of the coefficient of the third term.\n(1) Find the value of $n$;\n(2) Find the term with the maximum binomial coefficient in the expansion;\n(3) If $(x+2)^n = ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a polynomial \\( P(x) \\) with integer coefficients. It is known that \\( P(1) = 2013 \\), \\( P(2013) = 1 \\), and \\( P(k) = k \\), where \\( k \\) is some integer. Find \\( k \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"split": "train",
"index": 184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sin(\\frac{\\pi}{6} - \\alpha) = \\frac{2}{3}$, find the value of $\\cos(\\frac{2\\pi}{3} - \\alpha)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n$$\n\\int_{1}^{\\sqrt{3}} x^{2 x^{2}+1}+\\ln \\left(x^{2 x^{2 x^{2}+1}}\\right) d x. \\text{ }$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\\text{th}}$ pirate to take a share takes $\\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for whi... | math | {
"style": "rule",
"ground_truth": "1925"
} | {
"split": "train",
"index": 187,
"solution": "To solve this problem, we need to determine the smallest number of coins in the chest initially, such that each pirate receives a whole number of coins when taking their share according to the given rule. The rule states that the $k^{\\text{th}}$ pirate takes $\\frac... |
deepscaler | [
{
"role": "user",
"content": "Compute the least positive value of $t$ such that\n\\[\\arcsin (\\sin \\alpha), \\ \\arcsin (\\sin 2 \\alpha), \\ \\arcsin (\\sin 7 \\alpha), \\ \\arcsin (\\sin t \\alpha)\\]is a geometric progression for some $\\alpha$ with $0 < \\alpha < \\frac{\\pi}{2}.$ Let's think step by ... | math | {
"style": "rule",
"ground_truth": "9 - 4 \\sqrt{5}"
} | {
"split": "train",
"index": 188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $a$ and $b$, define $a \\diamond b = \\sqrt{a^2 + b^2}$. What is the value of $(5 \\diamond 12) \\diamond ((-12) \\diamond (-5))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13\\sqrt{2}"
} | {
"split": "train",
"index": 189,
"solution": "1. **Calculate $5 \\diamond 12$:**\n \\[\n 5 \\diamond 12 = \\sqrt{5^2 + 12^2} = \\sqrt{25 + 144} = \\sqrt{169} = 13\n \\]\n\n2. **Calculate $(-12) \\diamond (-5)$:**\n \\[\n (-12) \\diamond (-5) = \\sqrt{(-12)^2 + (-5)^2} = \\sqrt{144 + 25} = \\sqrt{169} =... |
deepscaler | [
{
"role": "user",
"content": "Given that Jessie moves from 0 to 24 in six steps, and travels four steps to reach point x, then one more step to reach point z, and finally one last step to point y, calculate the value of y. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaopang, Xiaodingding, Xiaoya, and Xiaoqiao have a total of 8 parents and 4 children in their four families. They are going to an amusement park together. The ticket pricing is as follows: Adult tickets are 100 yuan per person, children's tickets are 50 yuan per person. If the... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular parallelepiped with edge lengths AB = 5, BC = 3, and CG = 4, and point M as the midpoint of diagonal AG, determine the volume of the rectangular pyramid with base BCHE and apex M. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{34}"
} | {
"split": "train",
"index": 192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If $13$ married couples attended, how many handshakes were there among these $26$ people? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "234"
} | {
"split": "train",
"index": 193,
"solution": "To solve this problem, we need to calculate the total number of handshakes at the party under the given conditions: each man shakes hands with every other man and with every woman except his own spouse. No handshakes occur between women.\n\n1. **Calculate handshakes ... |
deepscaler | [
{
"role": "user",
"content": "Eleven positive integers from a list of fifteen positive integers are $3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23$. What is the largest possible value of the median of this list of fifteen positive integers? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Under the call for the development of the western region by the country, a certain western enterprise received a $4$ million yuan interest-free loan for equipment renewal. It is predicted that after the equipment renewal, the income of the enterprise in the first month is $200,... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer that must divide the product of any $5$ consecutive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of the $x^2$ term in the expansion of the product $(ax^3 + 3x^2 - 2x)(bx^2 - 7x - 4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function\n$$\nf(x) = |a \\sin x + b \\cos x - 1| + |b \\sin x - a \\cos x| \\quad (a, b \\in \\mathbf{R})\n$$\nattains a maximum value of 11, then $a^{2} + b^{2} = \\, \\, \\, $ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 199,
"solution": ""
} |
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