data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The graph of the function \\( y = x^2 + ax + b \\) is drawn on a board. Julia drew two lines parallel to the \\( O x \\) axis on the same diagram. The first line intersects the graph at points \\( A \\) and \\( B \\), and the second line intersects the graph at points \\( C \\)... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 23549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: (2m+1)x + (m+1)y - 7m-4 = 0$, where $m \\in \\mathbb{R}$. Find the minimum value of the chord length $|AB|$ cut by line $l$ on circle $C$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 23550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $b>1$, $x>0$, and $(2x)^{\\log_b 2}-(3x)^{\\log_b 3}=0$, then $x$ is 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": 23551,
"solution": "Given the equation:\n\\[\n(2x)^{\\log_b 2} - (3x)^{\\log_b 3} = 0\n\\]\n\nWe can rewrite the terms using the properties of exponents:\n\\[\n(2x)^{\\log_b 2} = 2^{\\log_b 2} \\cdot x^{\\log_b 2} = b^{\\log_b 2} \\cdot x^{\\log_b 2} = 2 \\cdot x^{\\log_b 2}\n\\]\n\\[... |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "193"
} | {
"split": "train",
"index": 23552,
"solution": "Since $XY = WZ$, $PQ = PQ$ and the areas of the trapezoids $PQZW$ and $PQYX$ are the same, then the heights of the trapezoids are the same. Thus both trapezoids have area $\\frac{1}{2} \\cdot \\frac{19}{2}(XY + PQ) = \\frac{19}{4}(XY + 87)$. This number is also equ... |
deepscaler | [
{
"role": "user",
"content": "A and B plays a game on a pyramid whose base is a $2016$ -gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the ... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 23553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many lattice points lie on the hyperbola \\(x^2 - y^2 = 999^2\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ways to partition a set of $10$ elements, $S = \\{1, 2, 3, . . . , 10\\}$ into two parts; that is, the number of unordered pairs $\\{P, Q\\}$ such that $P \\cup Q = S$ and $P \\cap Q = \\emptyset$ . Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "511"
} | {
"split": "train",
"index": 23555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $θ \\in (0,π)$, and $\\sin ( \\frac {π}{4}-θ)= \\frac { \\sqrt {2}}{10}$, find $\\tan 2θ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {24}{7}"
} | {
"split": "train",
"index": 23556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_1=1$, $na_{n+1}=(n+1)a_n+n(n+1)$, and $b_n=a_n\\cos \\frac {2n\\pi}{3}$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\\{b_n\\}$. Calculate $S_{24}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "304"
} | {
"split": "train",
"index": 23557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the mystical mountain, there are only two types of legendary creatures: Nine-Headed Birds and Nine-Tailed Foxes. A Nine-Headed Bird has nine heads and one tail, while a Nine-Tailed Fox has nine tails and one head.\n\nA Nine-Headed Bird discovers that, excluding itself, the t... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 23558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"example\" are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5040"
} | {
"split": "train",
"index": 23559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is inside right triangle $\\triangle ABC$ with $\\angle B = 90^\\circ$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$, respectively. Given that $PQ = 2$, $PR = 3$, and $PS = 4$, what is... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{5}"
} | {
"split": "train",
"index": 23560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two circles C<sub>1</sub>: $x^{2}+y^{2}-x+y-2=0$ and C<sub>2</sub>: $x^{2}+y^{2}=5$, determine the positional relationship between the two circles; if they intersect, find the equation of the common chord and the length of the common chord. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 23561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real number $x$, the symbol $[x]$ represents the integer part of $x$, i.e., $[x]$ is the largest integer not exceeding $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. This function $[x]$ is called the \"floor function\", which has wide applications in mathematics and... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 23562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $0=x_0<x_1<\\cdots<x_n=1$ .Find the largest real number $ C$ such that for any positive integer $ n $ , we have $$ \\sum_{k=1}^n x^2_k (x_k - x_{k-1})>C $$ 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": 23563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Keisha's basketball team must decide on a new uniform. The seventh-graders will pick the color of the shorts (black, gold, or red) and the eighth-graders will pick the color of the jersey (black, white, gold, or blue), and each group will not confer with the other. Additionally... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 23564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\\ldots,x_t$ with \\[x^3_1+x^3_2+\\,\\ldots\\,+x^3_t=2002^{2002}\\,?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 23565,
"solution": "\nTo determine the smallest positive integer \\( t \\) such that there exist integers \\( x_1, x_2, \\ldots, x_t \\) satisfying\n\n\\[\nx_1^3 + x_2^3 + \\cdots + x_t^3 = 2002^{2002},\n\\]\n\nwe will apply Fermat's Last Theorem and results regarding sums of cubes.\n... |
deepscaler | [
{
"role": "user",
"content": "If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $1000x^3+27= (ax^2 + bx +c )(d x^2 +ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11,\\!090"
} | {
"split": "train",
"index": 23566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( g_{1}(x) = \\sqrt{2-x} \\), and for integers \\( n \\geq 2 \\), define \n\\[ g_{n}(x) = g_{n-1}\\left(\\sqrt{(n+1)^2 - x}\\right) \\].\nLet \\( M \\) be the largest value of \\( n \\) for which the domain of \\( g_n \\) is non-empty. For this value of \\( M \\), the dom... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 23567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle, side AB is 8 units, and side BC is 15 units. Calculate $\\cos C$ and $\\sin C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"split": "train",
"index": 23568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first bl... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 23569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 100 people in a room with ages $1,2, \\ldots, 100$. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room? Let's think s... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 23570,
"solution": "For a cute pair $(a, b)$ we would have $$a \\geq \\frac{b}{2}+7, b \\geq \\frac{a}{2}+7$$ Solving the system, we get that $a$ and $b$ must both be at least 14. However 14 could only be paired with itself or a smaller number; therefore, only people with age 15 or ab... |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x) = (x-3)^3 + x - 1$, and $\\{a_n\\}$ be an arithmetic sequence with a non-zero common difference. If $f(a_1) + f(a_2) + \\ldots + f(a_7) = 14$, then calculate the value of $a_1 + a_2 + \\ldots + a_7$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27a^2"
} | {
"split": "train",
"index": 23572,
"solution": "1. **Identify the expression for $r$:** \n Given that $r$ is the result when both the base and the exponent of $a^b$ are tripled, we have:\n \\[\n r = (3a)^{3b}\n \\]\n\n2. **Relate $r$ to the given product $a^b x^b$:** \n According to the problem, $r$ ... |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of a hyperbola $E$, and point $P$ is on $E$, with $\\angle F_1 P F_2 = \\frac{\\pi}{6}$ and $(\\overrightarrow{F_2 F_1} + \\overrightarrow{F_2 P}) \\cdot \\overrightarrow{F_1 P} = 0$, determine the eccentricity $e$ of hyper... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} + 1}{2}"
} | {
"split": "train",
"index": 23573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a, b, c, d$ are real numbers such that $$|a-b|+|c-d|=99 ; \\quad|a-c|+|b-d|=1$$ Determine all possible values of $|a-d|+|b-c|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 23574,
"solution": "99 If $w \\geq x \\geq y \\geq z$ are four arbitrary real numbers, then $|w-z|+|x-y|=$ $|w-y|+|x-z|=w+x-y-z \\geq w-x+y-z=|w-x|+|y-z|$. Thus, in our case, two of the three numbers $|a-b|+|c-d|,|a-c|+|b-d|,|a-d|+|b-c|$ are equal, and the third one is less than or eq... |
deepscaler | [
{
"role": "user",
"content": "How many sequences of ten binary digits are there in which neither two zeroes nor three ones ever appear in a row? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 23575,
"solution": "Let $a_{n}$ be the number of binary sequences of length $n$ satisfying the conditions and ending in 0 , let $b_{n}$ be the number ending in 01 , and let $c_{n}$ be the number ending in 11 . From the legal sequences of length $201,11,10$, we find that $a_{2}=b_{2}=c... |
deepscaler | [
{
"role": "user",
"content": "There are 5 students on a team for a math competition. The math competition has 5 subject tests. Each student on the team must choose 2 distinct tests, and each test must be taken by exactly two people. In how many ways can this be done? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "2040"
} | {
"split": "train",
"index": 23576,
"solution": "We can model the situation as a bipartite graph on 10 vertices, with 5 nodes representing the students and the other 5 representing the tests. We now simply want to count the number of bipartite graphs on these two sets such that there are two edges incident on eac... |
deepscaler | [
{
"role": "user",
"content": "Determine the ratio $\\frac{s}{r}$, where $r$ is the total number of rectangles and $s$ is the number of squares formed by the grid of a $7\\times7$ checkerboard. Express $\\frac{s}{r}$ in its simplest form and find the sum of the numerator and denominator. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 23577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jerry cuts a wedge from a 6-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "603"
} | {
"split": "train",
"index": 23578,
"solution": "1. **Identify the dimensions of the cylinder**: The problem states that the cylinder has a radius of $6$ cm. However, the solution incorrectly uses $4$ cm as the radius. We need to correct this and use the correct radius of $6$ cm.\n\n2. **Calculate the volume of t... |
deepscaler | [
{
"role": "user",
"content": "Given vectors satisfying $\\overrightarrow{a}\\cdot (\\overrightarrow{a}-2\\overrightarrow{b})=3$ and $|\\overrightarrow{a}|=1$, with $\\overrightarrow{b}=(1,1)$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {3\\pi}{4}"
} | {
"split": "train",
"index": 23579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passes through the distinct vectors $\\mathbf{a}$ and $\\mathbf{b}.$ Then for a certain value of $k,$ the vector\n\\[k \\mathbf{a} + \\frac{3}{4} \\mathbf{b}\\]must also lie on the line. Find $k.$ 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": 23580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let b be a real number randomly selected from the interval $[-17,17]$. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation $x^4+25b^2=(4b^2-10b)x^2$ has $\\textit{at least}$ two distinct real solutions. Find the value of $... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 23581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \\( IJ \\) and exits through \\( LK \\), such that \\( AI = DL = 4 \\) cm, \\( JF = KG = 3 \\) cm, and the section \\( IJKL \\) is a rectangle. The total surface area of the two resul... | math | {
"style": "rule",
"ground_truth": "1176"
} | {
"split": "train",
"index": 23582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a \\neq b$ be positive real numbers and $m, n$ be positive integers. An $m+n$-gon $P$ has the property that $m$ sides have length $a$ and $n$ sides have length $b$. Further suppose that $P$ can be inscribed in a circle of radius $a+b$. Compute the number of ordered pairs $... | math | {
"style": "rule",
"ground_truth": "940"
} | {
"split": "train",
"index": 23583,
"solution": "Letting $x=\\frac{a}{a+b}$, we have to solve $$m \\arcsin \\frac{x}{2}+n \\arcsin \\frac{1-x}{2}=\\pi$$ This is convex in $x$, so if it is to have a solution, we must find that the LHS exceeds $\\pi$ at one of the endpoints. Thus $\\max (m, n) \\geq 7$. If $\\min (... |
deepscaler | [
{
"role": "user",
"content": "In a 6 by 6 grid, each of the 36 small squares measures 3 cm by 3 cm and is initially shaded. Five unshaded squares of side 1.5 cm are placed on the grid in such a way that some may overlap each other and/or the edges of the grid. If such overlap occurs, the overlapping parts a... | math | {
"style": "rule",
"ground_truth": "335.25"
} | {
"split": "train",
"index": 23584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When should the tires be swapped so that they wear out at the same time? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9375"
} | {
"split": "train",
"index": 23585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Between 1000 and 9999, how many four-digit integers with all different digits have an absolute difference of 2 between the first and last digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 23586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of sets of composite numbers less than 23 that sum to 23. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 23587,
"solution": "Because 23 is odd, we must have an odd number of odd numbers in our set. Since the smallest odd composite number is 9, we cannot have more than 2 odd numbers, as otherwise the sum would be at least 27. Therefore, the set has exactly one odd number. The only odd com... |
deepscaler | [
{
"role": "user",
"content": "Given a convex quadrilateral \\(ABCD\\), \\(X\\) is the midpoint of the diagonal \\(AC\\). It is known that \\(CD \\parallel BX\\). Find \\(AD\\), given that \\(BX = 3\\), \\(BC = 7\\), and \\(CD = 6\\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 23588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $b_1, b_2, \\ldots$ be a sequence determined by the rule $b_n= \\frac{b_{n-1}}{2}$ if $b_{n-1}$ is even and $b_n=3b_{n-1}+1$ if $b_{n-1}$ is odd. For how many positive integers $b_1 \\le 1000$ is it true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$? Let's think s... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 23589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\tan Y$ in the right triangle shown below.\n\n[asy]\npair X,Y,Z;\nX = (0,0);\nY = (24,0);\nZ = (0,7);\ndraw(X--Y--Z--X);\ndraw(rightanglemark(Y,X,Z,23));\nlabel(\"$X$\",X,SW);\nlabel(\"$Y$\",Y,SE);\nlabel(\"$Z$\",Z,N);\nlabel(\"$25$\",(Y+Z)/2,NE);\nlabel(\"$24$\",Y/2,S);... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{24}"
} | {
"split": "train",
"index": 23590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Employees from department X are 30, while the employees from department Y are 20. Since employees from the same department do not interact, the number of employees from department X that will shake hands with the employees from department Y equals 30, and the number of employee... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 23591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different wa... | math | {
"style": "rule",
"ground_truth": "170"
} | {
"split": "train",
"index": 23592,
"solution": "1. **Understanding the Problem:**\n - We have 6 individuals: Adam, Benin, Chiang, Deshawn, Esther, and Fiona.\n - Each individual has the same number of internet friends within this group.\n - The number of friends each person has ranges from 1 to 4, as no on... |
deepscaler | [
{
"role": "user",
"content": "If the sum of the second and fourth terms of an arithmetic sequence is $6$, what is the third term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 23593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain class, with any distribution of 200 candies, there will be at least two students who receive the same number of candies (possibly none). What is the minimum number of students in such a class? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A normal line (a line that passes through a point of tangency and is perpendicular to the tangent line) is drawn to the parabola \\( y = x^2 \\) at point \\( A \\). The normal line intersects the parabola at another point \\( B \\). Let \\( O \\) be the origin of coordinates. W... | math | {
"style": "rule",
"ground_truth": "\\frac{-3+\\sqrt{33}}{24}"
} | {
"split": "train",
"index": 23595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From milk with a fat content of $5\\%$, cottage cheese with a fat content of $15.5\\%$ is produced, while there remains whey with a fat content of $0.5\\%$. How much cottage cheese is obtained from 1 ton of milk? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 23596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a school of 2100 students, a sample of 30 students is randomly selected. The time (in minutes) each student spends on homework outside of class is as follows: \n75, 80, 85, 65, 95, 100, 70, 55, 65, 75, 85, 110, 120, 80, 85, 80, 75, 90, 90, 95, 70, 60, 60, 75, 90, 95, 65, ... | math | {
"style": "rule",
"ground_truth": "630"
} | {
"split": "train",
"index": 23597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\[\\frac 3{\\log_5{3000^5}} + \\frac 4{\\log_7{3000^5}},\\] giving your answer as a fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 23598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 23599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}$ where $a,b \\in \\mathbb{R}$. If the function $f(x)$ has an extremum of $10$ at $x=1$, then the value of $b$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 23600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points were marked on the sides of triangle \\(ABC\\): 12 points on side \\(AB\\), 9 points on side \\(BC\\), and 10 points on side \\(AC\\). None of the vertices of the triangle are marked. How many triangles can be formed with vertices at the marked points? Let's think step b... | math | {
"style": "rule",
"ground_truth": "4071"
} | {
"split": "train",
"index": 23601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a set of points \\( M \\) on a plane and seven different circles \\( C_{1}, C_{2}, \\cdots, C_{7} \\). Circle \\( C_{7} \\) passes through exactly 7 points in \\( M \\), circle \\( C_{6} \\) passes through exactly 6 points in \\( M \\), and so on, with circle \\( C_{1}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the square has a perimeter of $48$ and the triangle has a height of $48.$ If the square and the triangle have the same area, what is the value of $x?$ [asy]\ndraw((0,0)--(2,0)--(2,2)--(0,2)--cycle);\ndraw((3,0)--(6,0)--(6,5)--cycle);\ndraw((5.8,0)--(5.8,.2)--(6,... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 23603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What value of $k$ will make $x^2 - 16x + k$ the square of a binomial? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 23604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the plane passing through the origin with normal vector $\\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix}.$ Find the matrix $\\mathbf{P}$ such that for any vector $\\mathbf{v},$ $\\mathbf{P} \\mathbf{v}$ is the projection of $\\mathbf{v}$ onto plane $P.$ Let's think... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} \\frac{5}{6} & \\frac{1}{3} & -\\frac{1}{6} \\\\ \\frac{1}{3} & \\frac{1}{3} & \\frac{1}{3} \\\\ -\\frac{1}{6} & \\frac{1}{3} & \\frac{5}{6} \\end{pmatrix}"
} | {
"split": "train",
"index": 23605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pirate is searching for buried treasure on 6 islands. On each island, there is a $\\frac{1}{4}$ chance that the island has buried treasure and no traps, a $\\frac{1}{12}$ chance that the island has traps but no treasure, and a $\\frac{2}{3}$ chance that the island has neither... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{54}"
} | {
"split": "train",
"index": 23606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $m$: $(m-4)^3 = \\left(\\frac 18\\right)^{-1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 23607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle \\( \\triangle ABC \\) with circumcenter \\( O \\) and orthocenter \\( H \\), and \\( O \\neq H \\). Let \\( D \\) and \\( E \\) be the midpoints of sides \\( BC \\) and \\( CA \\) respectively. Let \\( D' \\) and \\( E' \\) be the reflections of \\( D \\) and \\... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"split": "train",
"index": 23608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the point $X$ is chosen on the edge $A_{1} D_{1}$ and the point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. F... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the sides \\( AB \\) and \\( AD \\) of a square \\( ABCD \\) with side length 108, semicircles are constructed inward. Find the radius of a circle that touches one side of the square and the semicircles: one externally and the other internally. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 23610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the leftmost digit is odd and less than 5, the second digit is an even number less than 6, all four digits are different, and the number is divisible by 5, find the number of four-digit numbers that satisfy these conditions. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 23611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $b-a=-6$ and $ab=7$, find the value of $a^2b-ab^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-42"
} | {
"split": "train",
"index": 23612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three different numbers from the set $\\{ -3, -2, -1, 4, 5 \\}$ are multiplied, the largest possible product is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 23613,
"solution": "To find the largest possible product when three different numbers from the set $\\{ -3, -2, -1, 4, 5 \\}$ are multiplied, we need to consider the signs and magnitudes of the products.\n\n1. **Identify the possible combinations**:\n - Three positive numbers: Not p... |
deepscaler | [
{
"role": "user",
"content": "An angle can be represented by two uppercase letters on its sides and the vertex letter, such as $\\angle A O B$ (where “ $\\angle$ \" represents an angle), or by $\\angle O$ if the vertex has only one angle. In the triangle $\\mathrm{ABC}$ shown below, $\\angle B A O=\\angle C... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $k$ does the equation $\\frac{x-1}{x-2} = \\frac{x-k}{x-6}$ have no solution for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 23615,
"solution": "1. **Identify the domain**: The equation given is $\\frac{x-1}{x-2} = \\frac{x-k}{x-6}$. We must exclude values of $x$ that make the denominators zero, hence the domain is $\\mathbb{R} \\setminus \\{2,6\\}$.\n\n2. **Cross-multiply to eliminate fractions**: \n \\[... |
deepscaler | [
{
"role": "user",
"content": "Choose three digits from the odd numbers 1, 3, 5, 7, 9 and two digits from the even numbers 2, 4, 6, 8 to form a five-digit number with no repeating digits, such that the odd and even digits alternate. How many such five-digit numbers can be formed? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 23616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function $f(x) = \\cos(2x + \\frac{\\pi}{3}) + \\sqrt{3}\\sin 2x + 2m$, where $x \\in \\mathbb{R}$ and $m \\in \\mathbb{R}$.\n(I) Determine the smallest positive period of $f(x)$ and its intervals of monotonic increase.\n(II) If $f(x)$ has a minimum value of 0 when... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 23617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 23618,
"solution": "1. **Given Information**: The mean, median, and mode of the data set $60, 100, x, 40, 50, 200, 90$ are all equal to $x$.\n\n2. **Calculating the Mean**:\n The mean of the data set is given by:\n \\[\n \\text{Mean} = \\frac{60 + 100 + x + 40 + 50 + 200 + 90}{7... |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be the roots of $x^3 + px + q = 0.$ Compute the determinant\n\\[\\begin{vmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function \\\\(f(x)\\\\) defined on \\\\(\\mathbb{R}\\\\) that satisfies: the graph of \\\\(y=f(x-1)\\\\) is symmetric about the point \\\\((1,0)\\\\), and when \\\\(x \\geqslant 0\\\\), it always holds that \\\\(f(x+2)=f(x)\\\\). When \\\\(x \\in [0,2)\\\\), \\\\(f(x)=e... | math | {
"style": "rule",
"ground_truth": "1-e"
} | {
"split": "train",
"index": 23620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all four-digit numbers that are equal to the cube of the sum of their digits (leading zeros are not allowed)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10745"
} | {
"split": "train",
"index": 23621,
"solution": "We want to find all integers $x$ between 1000 and 9999 that are the cube of the sum of their digits. Of course, our search is only restricted to perfect cubes. The smallest such cube is $10^{3}=1000$ and the largest such cube is $21^{3}=9261$. This means we only ha... |
deepscaler | [
{
"role": "user",
"content": "Let $x = \\cos \\frac{2 \\pi}{7} + i \\sin \\frac{2 \\pi}{7}.$ Compute the value of\n\\[(2x + x^2)(2x^2 + x^4)(2x^3 + x^6)(2x^4 + x^8)(2x^5 + x^{10})(2x^6 + x^{12}).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 23622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit whole numbers have no 7's and no 9's as digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "448"
} | {
"split": "train",
"index": 23623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z} \\) be a function with the following properties:\n\n(i) \\( f(1)=0 \\),\n\n(ii) \\( f(p)=1 \\) for all prime numbers \\( p \\),\n\n(iii) \\( f(xy)=y f(x)+x f(y) \\) for all \\( x, y \\in \\mathbb{Z}_{>0} \\).\n\nDetermine the... | math | {
"style": "rule",
"ground_truth": "3125"
} | {
"split": "train",
"index": 23624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 4 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 23625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expansion of $(\\sqrt{x} + \\frac{2}{x^2})^n$, the ratio of the coefficient of the fifth term to the coefficient of the third term is 56:3. \n(Ⅰ) Find the constant term in the expansion; \n(Ⅱ) When $x=4$, find the term with the maximum binomial coefficient in the ex... | math | {
"style": "rule",
"ground_truth": "\\frac{63}{256}"
} | {
"split": "train",
"index": 23626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$, $\\beta$, $\\gamma$ represent three different planes, and $a$, $b$, $c$ represent three different lines. Consider the following five propositions: \n(1) If $a \\parallel \\alpha$, $b \\parallel \\beta$, and $a \\parallel b$, then $\\alpha \\parallel \\beta$; \n... | math | {
"style": "rule",
"ground_truth": "(2)"
} | {
"split": "train",
"index": 23627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?\n\n[asy]\ndraw((-75,0)--(-45,0)--(-60,26)--cycle);\ndraw((0... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 23628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $A=4-3i$, $M=-4+i$, $S=i$, and $P=2$, find $A-M+S-P$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6-3i"
} | {
"split": "train",
"index": 23629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \\ge 1$, once the bee reaches point $P_j$, she turns $30^{\\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015},$ how far from $P... | math | {
"style": "rule",
"ground_truth": "1008 \\sqrt{6} + 1008 \\sqrt{2}"
} | {
"split": "train",
"index": 23630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As we enter the autumn and winter seasons, the air becomes dry. A certain appliance store is preparing to purchase a batch of humidifiers. The cost price of each unit is $80$ yuan. After market research, the selling price is set at $100$ yuan per unit. The store can sell $500$ ... | math | {
"style": "rule",
"ground_truth": "12250"
} | {
"split": "train",
"index": 23631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given quadrilateral ABCD, ∠A = 120∘, and ∠B and ∠D are right angles. Given AB = 13 and AD = 46, find the length of AC. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 23632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regi... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 23633,
"solution": "By checking all the possibilities, one can show that $T$ has height 4 and base lengths 4 and 5. Orient $T$ so that the shorter base is on the top. Then, the length of the cut parallel to the bases is $\\frac{4+5}{2}=\\frac{9}{2}$. Thus, the top two pieces are trape... |
deepscaler | [
{
"role": "user",
"content": "Kevin starts with the vectors \\((1,0)\\) and \\((0,1)\\) and at each time step, he replaces one of the vectors with their sum. Find the cotangent of the minimum possible angle between the vectors after 8 time steps. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "987"
} | {
"split": "train",
"index": 23634,
"solution": "Say that the vectors Kevin has at some step are \\((a, b)\\) and \\((c, d)\\). Notice that regardless of which vector he replaces with \\((a+c, b+d)\\), the area of the triangle with vertices \\((0,0),(a, b)\\), and \\((c, d)\\) is preserved with the new coordinate... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, there is a curve $C_{1}: x+y=4$, and another curve $C_{2}$ defined by the parametric equations $\\begin{cases} x=1+\\cos \\theta, \\\\ y=\\sin \\theta \\end{cases}$ (with $\\theta$ as the parameter). A polar coordinate system is establi... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{4}(\\sqrt{2}+1)"
} | {
"split": "train",
"index": 23635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight coins are arranged in a circle heads up. A move consists of flipping over two adjacent coins. How many different sequences of six moves leave the coins alternating heads up and tails up? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7680"
} | {
"split": "train",
"index": 23636,
"solution": "Imagine we flip over two adjacent coins by pushing a button halfway between them. Then the outcome depends only on the parities of the number of times that each button is pushed. To flip any coin, we must push the two buttons adjacent to that coin a total of an odd... |
deepscaler | [
{
"role": "user",
"content": "At a conference, there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\\frac{3}{7}$. What fraction of the people in the conference are married men? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"split": "train",
"index": 23637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangle $ABC$, the side lengths are given as $AB=\\sqrt{2}$, $BC=\\sqrt{5}$, and $AC=3$. Compare the measure of the angle $\\angle BOC$ to $112.5^{\\circ}$, where $O$ is the center of the circle inscribed in triangle $ABC$. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"split": "train",
"index": 23638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Li Yun is sitting by the window in a train moving at a speed of 60 km/h. He sees a freight train with 30 cars approaching from the opposite direction. When the head of the freight train passes the window, he starts timing, and he stops timing when the last car passes the window... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 23639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Thompson teaches science to 20 students. She noticed that when she had recorded the grades for 19 of these students, their average was 76. After including Oliver's grade, the new average grade for all 20 students became 78. Calculate Oliver's score on the test. Let's think... | math | {
"style": "rule",
"ground_truth": "116"
} | {
"split": "train",
"index": 23641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function $y=ax^{2}-4ax+3+b\\left(a\\neq 0\\right)$. \n$(1)$ Find the axis of symmetry of the graph of the quadratic function; \n$(2)$ If the graph of the quadratic function passes through the point $\\left(1,3\\right)$, and the integers $a$ and $b$ satisfy $... | math | {
"style": "rule",
"ground_truth": "t = \\frac{5}{2}"
} | {
"split": "train",
"index": 23642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all pairs of integer solutions $(n, m)$ to $2^{3^{n}}=3^{2^{m}}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,0) \\text{ and } (1,1)"
} | {
"split": "train",
"index": 23643,
"solution": "We find all solutions of $2^{x}=3^{y}-1$ for positive integers $x$ and $y$. If $x=1$, we obtain the solution $x=1, y=1$, which corresponds to $(n, m)=(0,0)$ in the original problem. If $x>1$, consider the equation modulo 4. The left hand side is 0, and the right ha... |
deepscaler | [
{
"role": "user",
"content": "In a chess tournament, each pair of players plays exactly one game. The winner of each game receives 2 points, the loser receives 0 points, and in case of a draw, both players receive 1 point each. Four scorers have recorded the total score of the tournament, but due to neglige... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 23644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \\in \\{1, 2, 3, \\ldots, 1000\\}$ and which are divisible by 121. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"split": "train",
"index": 23645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to color the dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.) ... | math | {
"style": "rule",
"ground_truth": "7983360"
} | {
"split": "train",
"index": 23646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the regular quadrangular pyramid \\(P-ABCD\\), \\(M\\) and \\(N\\) are the midpoints of \\(PA\\) and \\(PB\\) respectively. If the tangent of the dihedral angle between a side face and the base is \\(\\sqrt{2}\\), find the cosine of the angle between skew lines \\(DM\\) and ... | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"split": "train",
"index": 23647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S(n)$ denote the sum of digits of a natural number $n$ . Find all $n$ for which $n+S(n)=2004$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"split": "train",
"index": 23648,
"solution": ""
} |
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