data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "For a finite sequence $A=(a_1,a_2,\\dots,a_n)$ of numbers, the Cesaro sum of $A$ is defined to be\n\\[\\frac{S_1+\\cdots+S_n}{n},\\]where $S_k=a_1+\\cdots+a_k$ and $1\\leq k\\leq n$.\n\nIf the Cesaro sum of the 99-term sequence $(a_1,\\dots,a_{99})$ is 1000, what is the Cesaro... | math | {
"style": "rule",
"ground_truth": "991"
} | {
"split": "train",
"index": 9000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the 9 factors $\\left(1 - \\frac12\\right)\\left(1 - \\frac13\\right)\\left(1 - \\frac14\\right) \\cdots \\left(1 - \\frac {1}{10}\\right) =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 9001,
"solution": "1. **Identify the Factors**: The given expression is a product of factors of the form $\\left(1 - \\frac{1}{n}\\right)$ for $n$ ranging from $2$ to $10$. This can be rewritten as:\n \\[\n \\left(1 - \\frac{1}{2}\\right)\\left(1 - \\frac{1}{3}\\right)\\left(1 - \... |
deepscaler | [
{
"role": "user",
"content": "Calculate the area of the parallelogram formed by the vectors $\\begin{pmatrix} 4 \\\\ 2 \\\\ -3 \\end{pmatrix}$ and $\\begin{pmatrix} 2 \\\\ -4 \\\\ 5 \\end{pmatrix}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{30}"
} | {
"split": "train",
"index": 9002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "191"
} | {
"split": "train",
"index": 9003,
"solution": "We get around the condition that each man can't be opposite to another man by simply considering all $7$ diagonals, and choosing $5$ where there will be a single man. For each diagonal, the man can go on either side, and there are $\\binom{14}{5}$ ways to arrange th... |
deepscaler | [
{
"role": "user",
"content": "John went to the bookstore and purchased 20 notebooks totaling $62. Some notebooks were priced at $2 each, some at $5 each, and some at $6 each. John bought at least one notebook of each type. Let x be the number of $2 notebooks, y be the number of $5 notebooks, and z be the nu... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 9004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[z = \\frac{-\\sqrt{3} + i}{2}.\\]Compute $z^6.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 9005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, and $r$ be the roots of the polynomial $x^3 - x - 1 = 0$. Find the value of $\\frac{1}{p-2} + \\frac{1}{q-2} + \\frac{1}{r-2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{7}"
} | {
"split": "train",
"index": 9006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x) = 3\\sqrt[3]{x}$, and $Q(x) = x^3$. Determine $P(Q(P(Q(P(Q(4))))))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 9007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular dodecagon $A B C D E F G H I J K L$ inscribed in a circle with a radius of $6 \\mathrm{~cm}$, determine the perimeter of the pentagon $A C F H K$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18 + 12\\sqrt{2}"
} | {
"split": "train",
"index": 9008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA linear function \\( f(x) \\) is given. It is known that the distance between the points of intersection of the graphs \\( y = x^{2} \\) and \\( y = f(x) \\) is \\( 2 \\sqrt{3} \\), and the distance between the points of intersection of the graphs \\( y = x^{2}-2 \\) and \\(... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{11}"
} | {
"split": "train",
"index": 9009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $50$-gon \\(Q_1\\) is drawn in the Cartesian plane where the sum of the \\(x\\)-coordinates of the \\(50\\) vertices equals \\(150\\). A constant scaling factor \\(k = 1.5\\) applies only to the \\(x\\)-coordinates of \\(Q_1\\). The midpoints of the sides of \\(Q_1\\) form a ... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 9010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an equilateral triangle $ABC$ with a circle of radius $3$ tangent to line $AB$ at $B$ and to line $AC$ at $C$, find the area of the circle that passes through vertices $A$, $B$, and $C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"split": "train",
"index": 9011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a table, there are 2 candles, each 20 cm long, but of different diameters. The candles burn evenly, with the thin candle burning completely in 4 hours and the thick candle in 5 hours. After how much time will the thin candle become twice as short as the thick candle if they ... | math | {
"style": "rule",
"ground_truth": "20/3"
} | {
"split": "train",
"index": 9012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a_{1}, a_{2}, \\cdots, a_{k}\\) be a finite arithmetic sequence, such that \\(a_{4} + a_{7} + a_{10} = 17\\), and \\(a_{4} + a_{5} + a_{6} + \\cdots + a_{14} = 77\\), and \\(a_{k} = 13\\). Calculate the value of \\(k\\). Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 9013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 9014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ for which the matrix\n\\[\\begin{pmatrix} 1 + x & 7 \\\\ 3 - x & 8 \\end{pmatrix}\\]is not invertible. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{15}"
} | {
"split": "train",
"index": 9015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The owner of an individual clothing store purchased 30 dresses for $32 each. The selling price of the 30 dresses varies for different customers. Using $47 as the standard price, any excess amount is recorded as positive and any shortfall is recorded as negative. The results are... | math | {
"style": "rule",
"ground_truth": "472"
} | {
"split": "train",
"index": 9016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\\pi$. Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\pi}"
} | {
"split": "train",
"index": 9017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the real numbers \\( x_1, x_2, \\ldots, x_{2001} \\) satisfy \\( \\sum_{k=1}^{2000} \\left|x_k - x_{k+1}\\right| = 2001 \\). Let \\( y_k = \\frac{1}{k} \\left( x_1 + x_2 + \\cdots + x_k \\right) \\) for \\( k = 1, 2, \\ldots, 2001 \\). Find the maximum possible value of \... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 9018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. The area of the triangle is given by $S= \\frac{a^{2}+b^{2}-c^{2}}{4}$ and $\\sin A= \\frac{3}{5}$.\n\n1. Find $\\sin B$.\n2. If side $c=5$, find the area of $\\tri... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{2}"
} | {
"split": "train",
"index": 9019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$, $y$, $z$ are positive numbers satisfying \n\\[x+\\frac{y}{z}=y+\\frac{z}{x}=z+\\frac{x}{y}=2.\\]\nFind all the possible values of $x+y+z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 9020,
"solution": "\nWe are given that \\(x\\), \\(y\\), and \\(z\\) are positive numbers satisfying the system of equations:\n\\[\nx + \\frac{y}{z} = 2,\n\\]\n\\[\ny + \\frac{z}{x} = 2,\n\\]\n\\[\nz + \\frac{x}{y} = 2.\n\\]\n\nOur goal is to find all possible values of \\(x + y + z\\... |
deepscaler | [
{
"role": "user",
"content": "Given that $a = \\sin (2015\\pi - \\frac {\\pi}{6})$ and the function $f(x) = \\begin{cases} a^{x}, & x > 0 \\\\ f(-x), & x < 0 \\end{cases}$, calculate the value of $f(\\log_{2} \\frac {1}{6})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{6}"
} | {
"split": "train",
"index": 9021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of $x+6$, $6x+2$, and $2x+7$ is $4x-7$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 9022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a photograph, Aristotle, David, Flora, Munirah, and Pedro are seated in a random order in a row of 5 chairs. If David is seated in the middle of the row, what is the probability that Pedro is seated beside him? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 9023,
"solution": "After David is seated, there are 4 seats in which Pedro can be seated, of which 2 are next to David. Thus, the probability that Pedro is next to David is $\frac{2}{4}$ or $\frac{1}{2}$."
} |
deepscaler | [
{
"role": "user",
"content": "James and his sister each spin a spinner once. The modified spinner has six congruent sectors numbered from 1 to 6. If the absolute difference of their numbers is 2 or less, James wins. Otherwise, his sister wins. What is the probability that James wins? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 9024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $\\dfrac {x^{2}}{a^{2}}+ \\dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, there is a point $P$ on the ellipse such that the distance to the two foci of the ellipse, $F_{1}$ and $F_{2}$, satisfies $|PF_{1}|+|PF_{2}|=10$, and the eccentricity $e= \\dfrac {4}{5}$. \n$(1)$ F... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt {3}"
} | {
"split": "train",
"index": 9025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the square of 1033 without a calculator. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1067089"
} | {
"split": "train",
"index": 9026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1, x_2, ... , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \\ge {\\frac{1}{540}}$. Let $p$ and $q$ be positive relatively prime integers such that $\\frac{p}{q}$ is the maximum possible value of $x_1 x_2 x_3... | math | {
"style": "rule",
"ground_truth": "559"
} | {
"split": "train",
"index": 9027,
"solution": "Note that neither the constraint nor the expression we need to maximize involves products $x_i x_j$ with $i \\equiv j \\pmod 3$. Factoring out say $x_1$ and $x_4$ we see that the constraint is $x_1(x_3x_5) + x_4(x_2x_6) \\ge {\\frac1{540}}$, while the expression we ... |
deepscaler | [
{
"role": "user",
"content": "Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face\ncomes up with probability $1/2$ and the other five faces have equal probability of coming up. He\nunfortunately does not know which side is weighted, but he knows each f... | math | {
"style": "rule",
"ground_truth": "3/26"
} | {
"split": "train",
"index": 9028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\\mathcal{P}$ with 2022 edges, which satisfies the following properties:\n[list]\n[*]The degrees of the vertices of $\\mathcal{P}$ don’t differ by more than one, and\n[*]It is possible to colour... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 9029,
"solution": "\nWe are tasked with finding the largest positive integer \\( k \\) such that there exists a convex polyhedron \\(\\mathcal{P}\\) with 2022 edges, which satisfies the following conditions:\n1. The degrees of the vertices of \\(\\mathcal{P}\\) do not differ by more t... |
deepscaler | [
{
"role": "user",
"content": "The sum of the distances from one vertex of a square with sides of length $2$ to the midpoints of each of the sides of the square is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2+2\\sqrt{5}"
} | {
"split": "train",
"index": 9030,
"solution": "Let's consider a square with vertices labeled as $A$, $B$, $C$, and $D$, and let the side length of the square be $2$. Assume without loss of generality that $A$ is the vertex from which we are measuring the distances to the midpoints of the sides.\n\n1. **Identify ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=kx^{2}+2kx+1$ defined on the interval $[-3,2]$, the maximum value of the function is $4$. Determine the value of the real number $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 9031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a \\) is an integer, if \\( 50! \\) is divisible by \\( 2^a \\), find the largest possible value of \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 9032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[\\mathbf{A} = \\begin{pmatrix} 4 & 1 \\\\ -9 & -2 \\end{pmatrix}.\\]Compute $\\mathbf{A}^{100}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 301 & 100 \\\\ -900 & -299 \\end{pmatrix}"
} | {
"split": "train",
"index": 9033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=2\\sqrt{x} + \\frac{12}{\\sqrt{x}}$ and let $g(x)=2x^2-2x-3$. What is $f(g(3))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 9034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $M(\\frac{p}{2}$,$p)$ is a point on the parabola $y^{2}=2px\\left(p \\gt 0\\right)$, and the distance from $M$ to the point $\\left(1,0\\right)$ is 1 greater than the distance to the $y$-axis, find:<br/>\n$(1)$ The equation of the parabola.<br/>\n$(2)$ Let line $l$ intersect... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 9035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Maximum difference in weights of two bags is achieved by taking the largest and smallest possible values of the two different brands. Given the weights of three brands of flour are $\\left(25\\pm 0.1\\right)kg$, $\\left(25\\pm 0.2\\right)kg$, and $\\left(25\\pm 0.3\\right)kg$, ... | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 9036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A spherical decoration was suspended in a cylindrical container when the water inside it froze. The decoration was removed (without breaking the ice), leaving a hole 30 cm across at the top and 10 cm deep. If the water froze up to a height of 5 cm from the top of the sphere, wh... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{13}"
} | {
"split": "train",
"index": 9037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\(\\frac{1}{2} \\leq x \\leq 2\\) and \\(\\frac{4}{3} \\leq y \\leq \\frac{3}{2}\\). Determine the minimum value of\n$$\n\\frac{x^{3} y^{3}}{x^{6}+3 x^{4} y^{2}+3 x^{3} y^{3}+3 x^{2} y^{4}+y^{6}}.\n$$ Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "27/1081"
} | {
"split": "train",
"index": 9038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a, b, c \\) are positive integers, and the parabola \\( y = ax^2 + bx + c \\) intersects the x-axis at two distinct points \\( A \\) and \\( B \\). If the distances from \\( A \\) and \\( B \\) to the origin are both less than 1, find the minimum value of \\( a +... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 9039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f$ defined by $f(x)= \\frac{ax+b}{cx+d}$, where $a$,$b$,$c$ and $d$ are nonzero real numbers, has the properties $f(19)=19$, $f(97)=97$ and $f(f(x))=x$ for all values except $\\frac{-d}{c}$. Find the unique number that is not in the range of $f$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 9040,
"solution": "Because there are no other special numbers other than $19$ and $97$, take the average to get $\\boxed{58}$. (Note I solved this problem the solution one way but noticed this and this probably generalizes to all $f(x)=x, f(y)=y$ questions like these)"
} |
deepscaler | [
{
"role": "user",
"content": "In three-dimensional space, let $S$ be the region of points $(x, y, z)$ satisfying $-1 \\leq z \\leq 1$. Let $S_{1}, S_{2}, \\ldots, S_{2022}$ be 2022 independent random rotations of $S$ about the origin ( $0,0,0$). The expected volume of the region $S_{1} \\cap S_{2} \\cap \\c... | math | {
"style": "rule",
"ground_truth": "271619"
} | {
"split": "train",
"index": 9041,
"solution": "Consider a point $P$ of distance $r$ from the origin. The distance from the origin of a random projection of $P$ onto a line is uniform from 0 to $r$. Therefore, if $r<1$ then the probability of $P$ being in all the sets is 1, while for $r \\geq 1$ it is $r^{-2022}$... |
deepscaler | [
{
"role": "user",
"content": " How many (possibly empty) sets of lattice points $\\{P_1, P_2, ... , P_M\\}$ , where each point $P_i =(x_i, y_i)$ for $x_i\n, y_i \\in \\{0, 1, 2, 3, 4, 5, 6\\}$ , satisfy that the slope of the line $P_iP_j$ is positive for each $1 \\le i < j \\le M$ ? An infinite slop... | math | {
"style": "rule",
"ground_truth": "3432"
} | {
"split": "train",
"index": 9042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rhombus $PQRS^{}_{}$ is inscribed in rectangle $ABCD^{}_{}$ so that vertices $P^{}_{}$, $Q^{}_{}$, $R^{}_{}$, and $S^{}_{}$ are interior points on sides $\\overline{AB}$, $\\overline{BC}$, $\\overline{CD}$, and $\\overline{DA}$, respectively. It is given that $PB^{}_{}=15$, $BQ... | math | {
"style": "rule",
"ground_truth": "677"
} | {
"split": "train",
"index": 9043,
"solution": "We can just use areas. Let $AP = b$ and $AS = a$. $a^2 + b^2 = 625$. Also, we can add up the areas of all 8 right triangles and let that equal the total area of the rectangle, $(a+20)(b+15)$. This gives $3a + 4b = 120$. Solving this system of equation gives $\\frac{... |
deepscaler | [
{
"role": "user",
"content": "Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.\n\n$M_{1}(2, 3, 1)$\n\n$M_{2}(4, 1, -2)$\n\n$M_{3}(6, 3, 7)$\n\n$M_{0}(-5, -4, 8)$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 9044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y, z, u, v \\in \\mathbf{R}_{+} \\). Determine the maximum value of \\( f = \\frac{xy + yz + zu + uv}{2x^2 + y^2 + 2z^2 + u^2 + 2v^2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 9045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a > 1$, $b > 1$, and $$\\frac {1}{a-1} + \\frac {1}{b-1} = 1$$, find the minimum value of $a + 4b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 9046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 60? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4087"
} | {
"split": "train",
"index": 9047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $N$ such that any \"hydra\" with 100 necks, where each neck connects two heads, can be defeated by cutting at most $N$ strikes. Here, one strike can sever all the necks connected to a particular head $A$, and immediately after, $A$ grows new n... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 9048,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a road, there are three locations $A$, $O$, and $B$. $O$ is between $A$ and $B$, and $A$ is 1360 meters away from $O$. Two individuals, Jia and Yi, start simultaneously from points $A$ and $O$ towards point $B$. At the 10th minute after departure, both Jia and Yi are equidis... | math | {
"style": "rule",
"ground_truth": "2040"
} | {
"split": "train",
"index": 9049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\\heartsuit$ and $\\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\\spadesuit$ and $\\clubsuit$, called 'spades' and 'clubs') are black. The car... | math | {
"style": "rule",
"ground_truth": "2652"
} | {
"split": "train",
"index": 9050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Toad has a class of 2017 students, with unhappiness levels $1,2, \\ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, a... | math | {
"style": "rule",
"ground_truth": "1121"
} | {
"split": "train",
"index": 9051,
"solution": "One can show that the optimal configuration is $\\{1\\},\\{2\\}, \\ldots,\\{14\\},\\{15, \\ldots, 2017\\}$. This would give us an answer of $1+2+\\cdots+14+\\frac{15+2017}{2}=105+1016=1121$."
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of \\( a \\) for which the equation \\( x^{2} + 2ax = 8a \\) has two distinct integer roots. Record the product of all such \\( a \\), rounding to the nearest hundredth if necessary. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "506.25"
} | {
"split": "train",
"index": 9052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from at least half of the other 7 balls. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{128}"
} | {
"split": "train",
"index": 9053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word \"SUCCESS\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 9054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many paths are there from point $C$ to point $D$ on a grid, if every step must be either to the right or upwards, and the grid dimensions are now 7 steps to the right and 9 steps upward? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11440"
} | {
"split": "train",
"index": 9055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A new city is constructed in the form of 12 rectangular blocks of houses, divided by streets; at each corner of the block and at every street intersection, there is a mailbox. The blocks either touch each other along entire sides or share only a single corner.\n\nAssuming that ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 9056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag has 3 red marbles and 5 white marbles. Two marbles are drawn from the bag and not replaced. What is the probability that the first marble is red and the second marble is white? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{15}{56}"
} | {
"split": "train",
"index": 9057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The maximum value of the function \\( y = \\tan x - \\frac{2}{|\\cos x|} \\) is to be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 9058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $105 \\cdot 77 \\cdot 132 \\equiv m \\pmod{25}$, where $0 \\le m < 25$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 9059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the expression\n\n$$\n\\left(x_{1}-x_{2}\\right)^{2}+\\left(x_{2}-x_{3}\\right)^{2}+\\ldots+\\left(x_{2010}-x_{2011}\\right)^{2}+\\left(x_{2011}-x_{1}\\right)^{2}\n$$\n\nwhere \\(x_{1}, \\ldots, x_{2011} \\in [0, 1]\\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"split": "train",
"index": 9060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 9061,
"solution": "1. **Define the problem**: We need to find the smallest possible value for the sum of the digits of $S$, where $S$ is the sum of two three-digit numbers $a$ and $b$. Each digit among $a$ and $b$ is unique.\n\n2. **Set constraints on $a$ and $b$**: Since $a$ and $b$ ... |
deepscaler | [
{
"role": "user",
"content": "A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 9062,
"solution": "\nLet's denote the square as $ABCD$ with side length $1$. Assume the center of the square is $O$ and the points on the sides where $O$ connects are $P$, $Q$, and $R$ on sides $AB$, $BC$, and $CD$ respectively. The trapezoids are formed by $APRO$ and $CQRO$, and the ... |
deepscaler | [
{
"role": "user",
"content": "Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{108}"
} | {
"split": "train",
"index": 9063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z = \\cos \\frac{4 \\pi}{7} + i \\sin \\frac{4 \\pi}{7}.$ Compute\n\\[\\frac{z}{1 + z^2} + \\frac{z^2}{1 + z^4} + \\frac{z^3}{1 + z^6}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 9064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include th... | math | {
"style": "rule",
"ground_truth": "166"
} | {
"split": "train",
"index": 9065,
"solution": "[asy] import three; size(250);defaultpen(0.7+fontsize(9)); real unit = 0.5; real r = 2.8; triple O=(0,0,0), P=(0,0,unit+unit/(r-1)); dot(P); draw(O--P); draw(O--(unit,0,0)--(unit,0,unit)--(0,0,unit)); draw(O--(0,unit,0)--(0,unit,unit)--(0,0,unit)); draw((unit,0,0)--... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 9066,
"solution": "Okay, an exercise in counting (lots of binomials to calculate!). In base 2, the first number is $11111111$, which is the only way to choose 8 1's out of 8 spaces, or $\\binom{8}{8}$. What about 9 spaces? Well, all told, there are $\\binom{9}{8}=9$, which includes th... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ for which $(8-x)^2=x^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 9067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2014$, and $a^2 - b^2 + c^2 - d^2 = 2014$. Find the number of possible values of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "502"
} | {
"split": "train",
"index": 9068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\sqrt {3}\\sin 2x+2\\cos ^{2}x-1$.\n(I) Find the smallest positive period of $f(x)$:\n(II) Find the maximum and minimum values of $f(x)$ in the interval $\\[- \\dfrac {π}{6}, \\dfrac {π}{4}\\]$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 9069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = 1,$ $\\|\\mathbf{b}\\| = 5,$ $\\|\\mathbf{c}\\| = 3,$ and\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = \\mathbf{c}.\\]If $\\theta$ is the angle between $\\mathbf{a}$ and $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 9070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bouncy ball is dropped from a height of 100 meters. After each bounce, it reaches a height that is half of the previous one. What is the total distance the ball has traveled when it hits the ground for the 10th time? (Round the answer to the nearest whole number) Let's think ... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 9071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Crestview's school has expanded its official colors to include blue along with the original purple and gold. The students need to design a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different flags are possible... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 9072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which value of $x$ does the function $f(x) = \\frac{2x^2 - 5x - 7}{x^2 - 4x + 1}$ cross its horizontal asymptote? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 9073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_i$ be the set of all integers $n$ such that $100i\\leq n < 100(i + 1)$. For example, $S_4$ is the set ${400,401,402,\\ldots,499}$. How many of the sets $S_0, S_1, S_2, \\ldots, S_{999}$ do not contain a perfect square?\n Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "708"
} | {
"split": "train",
"index": 9074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $l$: $x=my+1$ passes through the right focus $F$ of the ellipse $C$: $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the focus of the parabola $x^{2}=4\\sqrt{3}y$ is the upper vertex of the ellipse $C$, and the line $l$ intersects the ellipse $C$ at po... | math | {
"style": "rule",
"ground_truth": "-\\frac{8}{3}"
} | {
"split": "train",
"index": 9075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the product $\\frac{1}{2}\\cdot\\frac{4}{1}\\cdot\\frac{1}{8}\\cdot\\frac{16}{1} \\dotsm \\frac{1}{16384}\\cdot\\frac{32768}{1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 9076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where \\[g(n) = \\left\\{\n\\begin{array}{cl}\n8 & \\text{ if } n \\text{ is a multiple of 3 and 4}, \\\\\n3 & \\text{ if } n \\text{ is only a multipl... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 9077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the equation $x+y=\\left\\lfloor x^{2}+y^{2}\\right\\rfloor$ consists of several line segments. Compute the sum of their lengths. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4+\\sqrt{6}-\\sqrt{2}"
} | {
"split": "train",
"index": 9078,
"solution": "We split into cases on the integer $k=\\left\\lfloor x^{2}+y^{2}\\right\\rfloor$. Note that $x+y=k$ but $x^{2}+y^{2} \\geq$ $\\frac{1}{2}(x+y)^{2}=\\frac{1}{2} k^{2}$ and $x^{2}+y^{2}<k+1$, which forces $k \\leq 2$. If $k=0$, the region defined by $0 \\leq x^{2}+y^{... |
deepscaler | [
{
"role": "user",
"content": "Let $S = \\{2^0,2^1,2^2,\\ldots,2^{10}\\}$. Consider all possible positive differences of pairs of elements of $S$. Let $N$ be the sum of all of these differences. Find $N.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16398"
} | {
"split": "train",
"index": 9079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $50$. Determine how many values of $n$ make $q+r$ divisible by $13$.\nA) 7000\nB) 7200\nC) 7400\nD) 7600 Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "7200"
} | {
"split": "train",
"index": 9080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose Harvard Yard is a $17 \\times 17$ square. There are 14 dorms located on the perimeter of the Yard. If $s$ is the minimum distance between two dorms, the maximum possible value of $s$ can be expressed as $a-\\sqrt{b}$ where $a, b$ are positive integers. Compute $100a+b$.... | math | {
"style": "rule",
"ground_truth": "602"
} | {
"split": "train",
"index": 9081,
"solution": "If two neighboring dorms are separated by a distance of more than $s$, we can move them slightly closer together and adjust the other dorms, increasing $s$. Therefore, in an optimal arrangement, the dorms form an equilateral 14-gon with side length $s$. By scaling, ... |
deepscaler | [
{
"role": "user",
"content": "Monsieur and Madame Dubois are traveling from Paris to Deauville, where their children live. Each is driving their own car. They depart together and arrive in Deauville at the same time. However, Monsieur Dubois spent on stops one-third of the time during which his wife continu... | math | {
"style": "rule",
"ground_truth": "8/9"
} | {
"split": "train",
"index": 9082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers... | math | {
"style": "rule",
"ground_truth": "554"
} | {
"split": "train",
"index": 9083,
"solution": "Since $\\angle{BAD}=90$ and $\\angle{EAF}=60$, it follows that $\\angle{DAF}+\\angle{BAE}=90-60=30$. Rotate triangle $ADF$ $60$ degrees clockwise. Note that the image of $AF$ is $AE$. Let the image of $D$ be $D'$. Since angles are preserved under rotation, $\\angle{... |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of line $l$ as $\\begin{cases}x=-\\frac{1}{2}+\\frac{\\sqrt{2}}{2}t \\\\ y=\\frac{1}{2}+\\frac{\\sqrt{2}}{2}t\\end{cases}$ and the parametric equation of ellipse $C$ as $\\begin{cases}x=2\\cos\\theta \\\\ y=\\sqrt{3}\\sin\\theta\\end{cases}$, in a ... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{14}"
} | {
"split": "train",
"index": 9084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company has a total of 60 employees. In order to carry out club activities, a questionnaire survey was conducted among all employees. There are 28 people who like sports, 26 people who like literary and artistic activities, and 12 people who do not like either sports or liter... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 9085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A small ball is released from a height \\( h = 45 \\) m without an initial velocity. The collision with the horizontal surface of the Earth is perfectly elastic. Determine the moment in time after the ball starts falling when its average speed equals its instantaneous speed. Th... | math | {
"style": "rule",
"ground_truth": "4.24"
} | {
"split": "train",
"index": 9086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon is inscribed in another regular hexagon such that each vertex of the inscribed hexagon divides a side of the original hexagon into two parts in the ratio 2:1. Find the ratio of the area of the inscribed hexagon to the area of the larger hexagon. Let's think st... | math | {
"style": "rule",
"ground_truth": "7/9"
} | {
"split": "train",
"index": 9087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of triangle $ABC$ is 1. Let $A_1, B_1, C_1$ be the midpoints of the sides $BC, CA, AB$ respectively. Points $K, L, M$ are taken on the segments $AB_1, CA_1, BC_1$ respectively. What is the minimum area of the common part of triangles $KLM$ and $A_1B_1C_1$? Let's think ... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 9088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Comparing two rectangular parallelepiped bars, it was found that the length, width, and height of the second bar are each 1 cm greater than those of the first bar, and the volume and total surface area of the second bar are 18 cm³ and 30 cm² greater, respectively, than those of... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 9089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three numbers are stored in a computer's memory. Every second, the following operation is performed: each number in this triplet is replaced by the sum of the other two numbers. For example, the triplet \\((1; 3; 7)\\) becomes \\((10; 8; 4)\\). What will be the difference betwe... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 9090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x \\sim N(-1,36)$ and $P(-3 \\leqslant \\xi \\leqslant -1) = 0.4$, calculate $P(\\xi \\geqslant 1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.1"
} | {
"split": "train",
"index": 9091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nGiven a triangle \\(ABC\\) where \\(AB = AC\\) and \\(\\angle A = 80^\\circ\\). Inside triangle \\(ABC\\) is a point \\(M\\) such that \\(\\angle MBC = 30^\\circ\\) and \\(\\angle MCB = 10^\\circ\\). Find \\(\\angle AMC\\). Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 9092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the set of values of the parameter \\(a\\) for which the sum of the cubes of the roots of the equation \\(x^{2}-a x+a+2=0\\) is equal to -8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 9093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given right triangle $ABC$ with a right angle at vertex $C$ and $AB = 2BC$, calculate the value of $\\cos A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 9094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(a\\) and \\(b\\) are real numbers, satisfying:\n\\[\n\\sqrt[3]{a} - \\sqrt[3]{b} = 12, \\quad ab = \\left( \\frac{a + b + 8}{6} \\right)^3.\n\\]\n\nFind \\(a - b\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "468"
} | {
"split": "train",
"index": 9095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of distances from a point on the ellipse $7x^{2}+3y^{2}=21$ to its two foci. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 9096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common divisor of 1407 and 903? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 9097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = \\sin^6\\left(\\frac{x}{4}\\right) + \\cos^6\\left(\\frac{x}{4}\\right) \\) for all real numbers \\( x \\). Determine \\( f^{(2008)}(0) \\) (i.e., \\( f \\) differentiated 2008 times and then evaluated at \\( x = 0 \\)). Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 9098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of an isosceles right triangle is $2p$. Its area is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3-2\\sqrt{2})p^2"
} | {
"split": "train",
"index": 9099,
"solution": "1. **Identify the properties of the triangle**: We are given an isosceles right triangle. In such a triangle, the two legs are equal, and the hypotenuse is $\\sqrt{2}$ times the length of a leg. Let the length of each leg be $x$.\n\n2. **Write the equation for the p... |
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