data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The Tigers beat the Sharks 2 out of the 3 times they played. They then played $N$ more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for $N$? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 6500,
"solution": "1. **Initial Setup**: The Tigers and Sharks have played 3 games, with the Tigers winning 2 games and the Sharks winning 1 game. Let $N$ be the number of additional games played, all of which are won by the Sharks (to maximize the Sharks' winning percentage).\n\n2. *... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\frac{\\ln x}{x+1}$.\n(1) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;\n(2) For $t < 0$, and $x > 0$ with $x\\neq 1$, the inequality $f(x)-\\frac{t}{x} > \\frac{\\ln x}{x-1}$ holds. Find the maximum value of the ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{n+1} = \\frac{4}{7}a_n + \\frac{3}{7}a_{n-1}$ and $a_0 = 1$ , $a_1 = 2$ . Find $\\lim_{n \\to \\infty} a_n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.7"
} | {
"split": "train",
"index": 6502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A total of 6 letters are used to spell the English word \"theer\". Calculate the probability that the person spells this English word incorrectly. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{59}{60}"
} | {
"split": "train",
"index": 6503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 6504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(In the 15th Jiangsu Grade 7 First Trial) On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The building numbered $k$ has exactly $k$ (where $k=1, 2, 3, 4, 5$) workers from Factory A. The distance between two adjacent buildings is 50 mete... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 6505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $| \\vec{e} | = 1$ and it satisfies $| \\vec{a} + \\vec{e} | = | \\vec{a} - 2\\vec{e} |$, find the projection of vector $\\vec{a}$ in the direction of $\\vec{e}$. 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": 6506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 6507,
"solution": "To solve this problem, we need to count the number of paths from Jack's house to Jill's house, avoiding the dangerous intersection. We can represent Jack's house as the origin (0,0) on a coordinate grid and Jill's house as the point (3,2). The dangerous intersection... |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, \\(E, F, G, H\\) are the midpoints of the sides \\(AB, BC, CD, DA\\) of the quadrilateral \\(ABCD\\). The intersection of \\(BH\\) and \\(DE\\) is \\(M\\), and the intersection of \\(BG\\) and \\(DF\\) is \\(N\\). What is \\(\\frac{S_{\\mathrm{BMND}}}{S... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 6508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the expression \\(\\left\\lfloor \\frac{8(a+b)}{c} \\right\\rfloor + \\left\\lfloor \\frac{8(a+c)}{b} \\right\\rfloor + \\left\\lfloor \\frac{8(b+c)}{a} \\right\\rfloor\\), where \\(a\\), \\(b\\), and \\(c\\) are arbitrary natural numbers. Let's think ... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 6509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the new operation $n\\heartsuit m=n^{3+m}m^{2+n}$, evaluate $\\frac{2\\heartsuit 4}{4\\heartsuit 2}$. 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": 6510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of seven integers is $-1$. What is the maximum number of the seven integers that can be larger than $13$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6511,
"solution": "To solve this problem, we need to determine the maximum number of integers among the seven that can exceed 13 while still achieving a total sum of $-1$.\n\n1. **Assume the maximum number of integers greater than 13**: Let's denote these integers that are greater tha... |
deepscaler | [
{
"role": "user",
"content": "What number can be added to both the numerator and denominator of $\\frac{3}{5}$ so that the resulting fraction will be equivalent to $\\frac{5}{6}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 6512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive integers such that $xy - 8x + 7y = 775$, what is the minimal possible value of $|x - y|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "703"
} | {
"split": "train",
"index": 6513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be positive real numbers. Find the smallest possible value of\n\\[6a^3 + 9b^3 + 32c^3 + \\frac{1}{4abc}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\frac{1}{{{x^2}+2x+1}}\\cdot (1+\\frac{3}{x-1})\\div \\frac{x+2}{{{x^2}-1}$, where $x=2\\sqrt{5}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{10}"
} | {
"split": "train",
"index": 6515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to $1000$ is $(\\sin t + i \\cos t)^n = \\sin nt + i \\cos nt$ true for all real $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 6516,
"solution": "Let $t=0$. Then, we have $i^n=i$ which means $n\\equiv 1\\pmod{4}$. Thus, the answer is $\\boxed{250}$."
} |
deepscaler | [
{
"role": "user",
"content": "If the line $ax - by + 2 = 0$ $(a > 0, b > 0)$ passes through the center of the circle ${x}^{2} + {y}^{2} + 4x - 4y - 1 = 0$, find the minimum value of $\\frac{2}{a} + \\frac{3}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 + 2 \\sqrt{6}"
} | {
"split": "train",
"index": 6517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane c... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 6518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$r(x)$ has domain $\\{-1,0,1,2\\}$ and range $\\{0,2,4,6\\}$. $s(x)$ has domain $\\{1,2,3,4\\}$ and is defined by $s(x)=x+1$. What is the sum of all possible values of $s(r(x))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 6519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(ABCD\\) is a parallelogram. \\(E\\) is on side \\(AB\\), and \\(F\\) is on side \\(DC\\). \\(G\\) is the intersection point of \\(AF\\) and \\(DE\\), and \\(H\\) is the intersection point of \\(CE\\) and \\(BF\\). Given that the area of parallelogram \\(ABCD\... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{92}"
} | {
"split": "train",
"index": 6520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 6521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of\n\\[2ab \\sqrt{2} + 2bc.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 6522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 6523,
"solution": "1. **Identify the games where the team lost by one run**: The problem states that the team lost by one run in exactly five games. Since they cannot score twice as many runs as their opponents and still lose by one run, these games must be the ones where they scored ... |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system, the symmetric point of point $A(-2,1,3)$ with respect to the $x$-axis is point $B$. It is also known that $C(x,0,-2)$, and $|BC|=3 \\sqrt{2}$. Find the value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 6524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, it is known that $\\sqrt{3}\\sin{2B} = 1 - \\cos{2B}$.\n\n(1) Find the value of angle $B$; \n(2) If $BC = 2$ and $A = \\frac{\\pi}{4}$, find the area of triangle $ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 6525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\left(z_{n}\\right)$ of complex numbers satisfies the following properties: $z_{1}$ and $z_{2}$ are not real. $z_{n+2}=z_{n+1}^{2} z_{n}$ for all integers $n \\geq 1$. $\\frac{z_{n+3}}{z_{n}^{2}}$ is real for all integers $n \\geq 1$. $\\left|\\frac{z_{3}}{z_{4}}... | math | {
"style": "rule",
"ground_truth": "65536"
} | {
"split": "train",
"index": 6526,
"solution": "All complex numbers can be expressed as $r(\\cos \\theta+i \\sin \\theta)=r e^{i \\theta}$. Let $z_{n}$ be $r_{n} e^{i \\theta_{n}}$. $\\frac{z_{n+3}}{z_{n}^{2}}=\\frac{z_{n+2}^{2} z_{n+1}}{z_{n}^{2}}=\\frac{z_{n+1}^{5} z_{n}^{2}}{z_{n}^{2}}=z_{n+1}^{5}$ is real for... |
deepscaler | [
{
"role": "user",
"content": "Suppose $\\cos Q = 0.4$ in the diagram below. What is $QR$?\n\n\n\n[asy]\npair P,Q,R;\nP = (0,0);\nQ = (6,0);\nR = (0,6*tan(acos(0.4)));\ndraw(P--Q--R--P);\ndraw(rightanglemark(Q,P,R,18));\nlabel(\"$P$\",P,SW);\nlabel(\"$Q$\",Q,SE);\nlabel(\"$R$\",R,N);\nlabel(\"$12$\",Q/2,S);... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 6527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C$: $y^{2}=4x$ with focus $F$, a line $l$ passing through $F$ intersects $C$ at points $A$ and $B$. Let $M$ be the midpoint of segment $AB$, and $O$ be the origin. The extensions of $AO$ and $BO$ intersect the line $x=-4$ at points $P$ and $Q$ respectively. ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 6528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA polynomial $P(x)$ with integer coefficients possesses the properties\n\n$$\nP(1)=2019, \\quad P(2019)=1, \\quad P(k)=k,\n$$\n\nwhere $k$ is an integer. Find this integer $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 6529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit prime numbers have a units digit of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 6530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 6531,
"solution": "\nLet's denote the distance Mr. Bird needs to travel to work as $d$ miles and the time he should ideally take to reach on time as $t$ hours. \n\n#### Step 1: Establish equations based on given conditions\n- If he drives at 40 mph and is 3 minutes late, the time take... |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\cos 20^\\circ - \\cos 40^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{4}"
} | {
"split": "train",
"index": 6532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\\cdots a_k$ is an integer? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 6533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 3.$ Find the minimum value of\n\\[\\frac{a + b}{abc}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 6534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{H}$ be a regular hexagon with side length one. Peter picks a point $P$ uniformly and at random within $\\mathcal{H}$, then draws the largest circle with center $P$ that is contained in $\\mathcal{H}$. What is this probability that the radius of this circle is les... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3}-1}{3}"
} | {
"split": "train",
"index": 6535,
"solution": "We first cut the regular hexagon $\\mathcal{H}$ by segments connecting its center to each vertex into six different equilateral triangles with side lengths 1. Therefore, each point inside $\\mathcal{H}$ is contained in some equilateral triangle. We first see that fo... |
deepscaler | [
{
"role": "user",
"content": "500 × 3986 × 0.3986 × 5 = ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.25 \\times 3986^2"
} | {
"split": "train",
"index": 6536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the equation with respect to \\( x \\), \\(\\frac{x \\lg^2 a - 1}{x + \\lg a} = x\\), has a solution set that contains only one element, then \\( a \\) equals \\(\\quad\\) . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $4$ cards, marked with $0$, $1$, $2$, $3$ respectively. If two cards are randomly drawn from these $4$ cards to form a two-digit number, what is the probability that this number is even? 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": 6538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive angle $\\theta,$ in degrees, for which\n\\[\\cos \\theta = \\sin 60^\\circ + \\cos 42^\\circ - \\sin 12^\\circ - \\cos 6^\\circ.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66^\\circ"
} | {
"split": "train",
"index": 6539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $y = (m^2 + 2m - 2)x^{\\frac{1}{m-1}}$ is a power function, find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 6540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6541,
"solution": "1. **Identify the factors of 128**: We know that the product of the ages of Kiana and her twin brothers is 128. Since 128 can be expressed as $128 = 2^7$, the ages must be factors of 128.\n\n2. **List possible ages for the twins**: The twins are older than Kiana and... |
deepscaler | [
{
"role": "user",
"content": "Given the polynomial $f(x)=x^{6}-5x^{5}+6x^{4}+x^{2}+0.3x+2$, use Horner's method to calculate $f(-2)$ and find the value of $v_{1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 6542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the radius of the circumscribed circle around an isosceles trapezoid with bases 2 and 14 and a lateral side of 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"split": "train",
"index": 6543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer solution to $\\tan{19x^{\\circ}}=\\dfrac{\\cos{96^{\\circ}}+\\sin{96^{\\circ}}}{\\cos{96^{\\circ}}-\\sin{96^{\\circ}}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "159"
} | {
"split": "train",
"index": 6544,
"solution": "$\\dfrac{\\cos{96^{\\circ}}+\\sin{96^{\\circ}}}{\\cos{96^{\\circ}}-\\sin{96^{\\circ}}} = \\dfrac{1 + \\tan{96^{\\circ}}}{1-\\tan{96^{\\circ}}}$ which is the same as $\\dfrac{\\tan{45^{\\circ}} + \\tan{96^{\\circ}}}{1-\\tan{45^{\\circ}}\\tan{96^{\\circ}}} = \\tan{141... |
deepscaler | [
{
"role": "user",
"content": "Define a function $g(x),$ for positive integer values of $x,$ by \n\\[\ng(x) = \\left\\{\n\\begin{aligned} \n\\log_3 x & \\quad \\text{if } \\log_3 x \\text{is an integer} \\\\\n1 + g(x + 1) & \\quad \\text{otherwise}. \n\\end{aligned} \n\\right.\n\\]\nCompute $g(50)$. Let's th... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 6545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. The smallest value that the range (largest observation minus smallest) can assume for such a sample is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 6546,
"solution": "To find the smallest possible range of a set of five observations with an arithmetic mean of $10$ and a median of $12$, we need to consider the properties of the mean and median, and how they affect the arrangement of the numbers in the set.\n\n1. **Understanding th... |
deepscaler | [
{
"role": "user",
"content": "Let \\( p \\) and \\( q \\) be positive integers such that \\( \\frac{5}{8} < \\frac{p}{q} < \\frac{7}{8} \\). What is the smallest value of \\( p \\) so that \\( p + q = 2005 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "772"
} | {
"split": "train",
"index": 6547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the two-digit integer \\( XX \\), with equal digits, is multiplied by the one-digit integer \\( X \\), the result is the three-digit integer \\( PXQ \\). What is the greatest possible value of \\( PXQ \\) if \\( PXQ \\) must start with \\( P \\) and end with \\( X \\)? Let... | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 6548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral pyramid \\( S A B C D \\) is given, with the base being the parallelogram \\( A B C D \\). A plane is drawn through the midpoint of edge \\( A B \\) that is parallel to the lines \\( A C \\) and \\( S D \\). In what ratio does this plane divide edge \\( S B \\)?... | math | {
"style": "rule",
"ground_truth": "1 : 3"
} | {
"split": "train",
"index": 6549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the lateral side \\( C D \\) of the trapezoid \\( A B C D \\) (\\( A D \\parallel B C \\)), a point \\( M \\) is marked. From the vertex \\( A \\), a perpendicular \\( A H \\) is dropped onto the segment \\( B M \\). It turns out that \\( A D = H D \\). Find the length of th... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin(2x+\\frac{π}{6})+2\\sin^2x$. \n$(1)$ Find the center of symmetry and the interval of monotonic decrease of the function $f(x)$; \n$(2)$ If the graph of $f(x)$ is shifted to the right by $\\frac{π}{12}$ units, resulting in the graph of the functi... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}+1"
} | {
"split": "train",
"index": 6551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a real number \\( x \\), let \\( [x] \\) denote the greatest integer less than or equal to \\( x \\). Find the positive integer \\( n \\) such that \\(\\left[\\log _{2} 1\\right] + \\left[\\log _{2} 2\\right] + \\left[\\log _{2} 3\\right] + \\cdots + \\left[\\log _{2} n\\ri... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 6552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\\frac {m}{n}$, where $m$ and $n$ are relatively prime p... | math | {
"style": "rule",
"ground_truth": "929"
} | {
"split": "train",
"index": 6553,
"solution": "We consider how many ways we can have 2*2 grid\n$(1)$: All the girds are red--$1$ case\n$(2)$: One unit square is blue--The blue lies on the center of the bigger square, makes no 2*2 grid $9-1=8$ cases\n$(3)$: Two unit squares are blue--one of the squares lies in th... |
deepscaler | [
{
"role": "user",
"content": "Find the phase shift of the graph of $y = \\sin (3x - \\pi).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 6554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is the value of $b?$ Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 6555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"monkey\"' are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 6556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The odd numbers from 5 to 21 are used to build a 3 by 3 magic square. If 5, 9 and 17 are placed as shown, 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": "11"
} | {
"split": "train",
"index": 6557,
"solution": "The sum of the odd numbers from 5 to 21 is $5+7+9+11+13+15+17+19+21=117$. Therefore, the sum of the numbers in any row is one-third of this total, or 39. This means as well that the sum of the numbers in any column or diagonal is also 39. Since the numbers in the mi... |
deepscaler | [
{
"role": "user",
"content": "A number $x$ is randomly selected from the interval $\\left[ -\\frac{\\pi}{6}, \\frac{\\pi}{2} \\right]$. Calculate the probability that $\\sin x + \\cos x \\in [1, \\sqrt{2}]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 6558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a > 3$. Determine the value of $a$ given that $f(g(a)) = 16$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\sqrt{6} + 6}"
} | {
"split": "train",
"index": 6559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ i... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 6561,
"solution": "1. **Identify the total number of points and the position of $P$:**\n The square contains $81$ grid points, arranged in a $9 \\times 9$ grid. Point $P$, being at the center, is located at the center grid point.\n\n2. **Determine the number of points excluding $P$:... |
deepscaler | [
{
"role": "user",
"content": "Eleven is the fifth prime number. What is the tenth prime number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 6562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 2\\sin\\omega x \\cdot \\cos(\\omega x) + (\\omega > 0)$ has the smallest positive period of $4\\pi$.\n\n(1) Find the value of the positive real number $\\omega$; \n(2) In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$,... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 6563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $o$ contains the circles $m$ , $p$ and $r$ , such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the c... | math | {
"style": "rule",
"ground_truth": "8/9"
} | {
"split": "train",
"index": 6564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$ ?\n<details><summary>Clarifications</summary>\n\n- $S$ is the ``set of 10 distinct integers'' from th... | math | {
"style": "rule",
"ground_truth": "511"
} | {
"split": "train",
"index": 6565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Connie has a number of gold bars, all of different weights. She gives the 24 lightest bars, which weigh $45 \\%$ of the total weight, to Brennan. She gives the 13 heaviest bars, which weigh $26 \\%$ of the total weight, to Maya. How many bars did Blair receive? Let's think step... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6566,
"solution": "Connie gives 24 bars that account for $45 \\%$ of the total weight to Brennan. Thus, each of these 24 bars accounts for an average of $\\frac{45}{24} \\%=\\frac{15}{8} \\%=1.875 \\%$ of the total weight. Connie gives 13 bars that account for $26 \\%$ of the total we... |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equations of curve $C_1$ are $$\\begin{cases} x=2\\cos\\theta \\\\ y= \\sqrt {3}\\sin\\theta\\end{cases}$$ (where $\\theta$ is the parameter), with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, establishing a p... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {7}"
} | {
"split": "train",
"index": 6567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A Tim number is a five-digit positive integer with the following properties:\n1. It is a multiple of 15.\n2. Its hundreds digit is 3.\n3. Its tens digit is equal to the sum of its first three (leftmost) digits.\n\nHow many Tim numbers are there? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $98 \\times 102$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9996"
} | {
"split": "train",
"index": 6569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the positive real numbers \\(a_{1}, a_{2}, a_{3}, a_{4}\\) satisfy the conditions \\(a_{1} \\geqslant a_{2} a_{3}^{2}, a_{2} \\geqslant a_{3} a_{4}^{2}, a_{3} \\geqslant a_{4} a_{1}^{2}, a_{4} \\geqslant a_{1} a_{2}^{2}\\), find the maximum value of \\(a_{1} a_{2} a_... | math | {
"style": "rule",
"ground_truth": "1/256"
} | {
"split": "train",
"index": 6570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "New this year at HMNT: the exciting game of $R N G$ baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses ... | math | {
"style": "rule",
"ground_truth": "\\frac{409}{125}"
} | {
"split": "train",
"index": 6571,
"solution": "For $i=0,1,2,3$, let $P_{i}$ be the probability that a player on the $i$-th base scores a point before strikeout (with zeroth base being the home base). We have the following equations: $$\\begin{aligned} P_{0} & =\\frac{1}{5}\\left(P_{1}+P_{2}+P_{3}+1\\right) \\\\ ... |
deepscaler | [
{
"role": "user",
"content": "What is the area of the polygon with vertices at $(2, 1)$, $(4, 3)$, $(6, 1)$, $(5, -2)$, and $(3, -2)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\triangle GHI$ is inscribed inside $\\triangle XYZ$ such that $G, H, I$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\\triangle GZC, \\triangle HYD, \\triangle IXF$ have centers $O_1, O_2, O_3$, respectively. Also, $XY = 26, YZ = 28, XZ = 27$, and $\\stackrel{\\fr... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 6573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the following expression: $$ 0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000 $$ The terms with minus signs are exactly the powers of two.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1996906"
} | {
"split": "train",
"index": 6574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{36}"
} | {
"split": "train",
"index": 6575,
"solution": "1. **Understanding the Problem**: Jason rolls three fair six-sided dice and can choose to reroll any subset of them. He wins if the sum of the dice after rerolling is exactly 7. We need to find the probability that he chooses to reroll exactly two dice to optimize h... |
deepscaler | [
{
"role": "user",
"content": "The diagram shows an \\(n \\times (n+1)\\) rectangle tiled with \\(k \\times (k+1)\\) rectangles, where \\(n\\) and \\(k\\) are integers and \\(k\\) takes each value from 1 to 8 inclusive. What is the value of \\(n\\)? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is ${-\\frac{1}{2} \\choose 100} \\div {\\frac{1}{2} \\choose 100}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-199"
} | {
"split": "train",
"index": 6578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers \\( x, y, z \\) satisfy \\( x \\geq y \\geq z \\geq 0 \\) and \\( 6x + 5y + 4z = 120 \\). Find the sum of the maximum and minimum values of \\( x + y + z \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 6579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $xOy$, curve $C_1$ passes through point $P(a, 1)$ with parametric equations $$\\begin{cases} x=a+ \\frac { \\sqrt {2}}{2}t \\\\ y=1+ \\frac { \\sqrt {2}}{2}t\\end{cases}$$ where $t$ is a parameter and $a \\in \\mathbb{R}$. In the polar coord... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{36}"
} | {
"split": "train",
"index": 6580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a community gathering there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\\frac{3}{7}$. Calculate the fraction of the people in the gathering who are married men. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"split": "train",
"index": 6581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 6582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $g$ is defined by $g(z) = (3 + 2i) z^2 + \\beta z + \\delta$ for all complex numbers $z$, where $\\beta$ and $\\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\\beta| + |\\delta|$? Le... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 6583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Descartes's Blackjack: How many integer lattice points (points of the form $(m, n)$ for integers $m$ and $n$) lie inside or on the boundary of the disk of radius 2009 centered at the origin? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12679605"
} | {
"split": "train",
"index": 6584,
"solution": "The number of lattice points inside or on the boundary of a circle with radius $r$ centered at the origin can be approximated using the formula $\\pi r^2 + \\text{error term}$. For a circle with radius 2009, this results in approximately 12679605 lattice points."
} |
deepscaler | [
{
"role": "user",
"content": "After the appearance of purple sand flowerpots in the Ming and Qing dynasties, their development momentum was soaring, gradually becoming the collection target of collectors. With the development of pot-making technology, purple sand flowerpots have been integrated into the dai... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 6585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times does the digit 9 appear in the list of all integers from 1 to 1000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 6586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonals of a rhombus are $10$ inches and $24$ inches. What is the perimeter of the rhombus, in inches? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 6587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C:\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1(a>0,b>0)$ with the right focus $F$, the upper endpoint of the imaginary axis $B$, points $P$ and $Q$ on the hyperbola, and point $M(-2,1)$ as the midpoint of segment $PQ$, where $PQ$ is parallel to $BF$. Find $e^{2}$. L... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}+1}{2}"
} | {
"split": "train",
"index": 6588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the sides \\( AB, BC \\), and \\( AC \\) of triangle \\( ABC \\), points \\( M, N, \\) and \\( K \\) are taken respectively so that \\( AM:MB = 2:3 \\), \\( AK:KC = 2:1 \\), and \\( BN:NC = 1:2 \\). In what ratio does the line \\( MK \\) divide the segment \\( AN \\)? Let's ... | math | {
"style": "rule",
"ground_truth": "6:7"
} | {
"split": "train",
"index": 6589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three squares, with side-lengths 2, are placed together edge-to-edge to make an L-shape. The L-shape is placed inside a rectangle so that all five vertices of the L-shape lie on the rectangle, one of them at the midpoint of an edge, as shown.\nWhat is the area of the rectangle?... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 6590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x)=\\begin{cases} 2^{x}+\\int_{0}^{1}{3(\\sqrt{x}-x^{2})dx} & (x\\geqslant 4) \\\\ f(x+2) & (x < 4) \\end{cases}$. Evaluate the expression $f(\\log_{2}3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 6591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The shortest distance from a moving point P on circle C: $\\rho = -4\\sin\\theta$ to the line $l: \\rho\\sin(\\theta + \\frac{\\pi}{4}) = \\sqrt{2}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 2"
} | {
"split": "train",
"index": 6592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $min\\{ a,b\\}$ represents the smaller value between the real numbers $a$ and $b$, and the vectors $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ satisfy $(\\vert\\overrightarrow{a}\\vert=1,\\vert\\overrightarrow{b}\\vert=2,\\overrightarrow{a}\\cdot\... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 6593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the product $n_1\\cdot n_2.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "418"
} | {
"split": "train",
"index": 6594,
"solution": "We define $Q(x)=P(x)-x+7$, noting that it has roots at $17$ and $24$. Hence $P(x)-x+7=A(x-17)(x-24)$. In particular, this means that $P(x)-x-3=A(x-17)(x-24)-10$. Therefore, $x=n_1,n_2$ satisfy $A(x-17)(x-24)=10$, where $A$, $(x-17)$, and $(x-24)$ are integers. This ... |
deepscaler | [
{
"role": "user",
"content": "Using 6 different colors to paint the 4 cells in the picture, with each cell painted in one color, and requiring that at most 3 colors are used and no two adjacent cells have the same color, how many different coloring methods are there? (Answer using a number.) Let's think ste... | math | {
"style": "rule",
"ground_truth": "390"
} | {
"split": "train",
"index": 6595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)$ be a polynomial of degree at most 3 such that $P(x)=\\frac{1}{1+x+x^{2}}$ for $x=1,2,3,4$. What is $P(5) ?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-3}{91}"
} | {
"split": "train",
"index": 6596,
"solution": "The forward difference of a polynomial $P$ is $\\Delta P(x)=P(x+1)-P(x)$, which is a new polynomial with degree reduced by one. Therefore, if we apply this operation three times we'll get a constant function, and we can work back up to get a value of $P(5)$. Practic... |
deepscaler | [
{
"role": "user",
"content": "Victor has a drawer with 6 socks of 3 different types: 2 complex socks, 2 synthetic socks, and 2 trigonometric socks. He repeatedly draws 2 socks at a time from the drawer at random, and stops if the socks are of the same type. However, Victor is 'synthetic-complex type-blind',... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 6597,
"solution": "Let the socks be $C_{1}, C_{2}, S_{1}, S_{2}, T_{1}, T_{2}$, where $C, S$ and $T$ stand for complex, synthetic and trigonometric respectively. The possible stopping points consist of three pairs of socks of the same type plus four different complex-synthetic $(C-S)$... |
deepscaler | [
{
"role": "user",
"content": "Let \\(p\\) be a prime number and \\(\\mathbb{F}_{p}\\) be the field of residues modulo \\(p\\). Let \\(W\\) be the smallest set of polynomials with coefficients in \\(\\mathbb{F}_{p}\\) such that the polynomials \\(x+1\\) and \\(x^{p-2}+x^{p-3}+\\cdots+x^{2}+2x+1\\) are in \\(... | math | {
"style": "rule",
"ground_truth": "p!"
} | {
"split": "train",
"index": 6598,
"solution": "Note that both of our polynomials are bijective functions on \\(\\mathbb{F}_{p}\\): \\(f_{1}(x)=x+1\\) is the cycle \\(0 \\rightarrow 1 \\rightarrow 2 \\rightarrow \\cdots \\rightarrow(p-1) \\rightarrow 0\\) and \\(f_{2}(x)=x^{p-2}+x^{p-3}+\\cdots+x^{2}+2x+1\\) is t... |
deepscaler | [
{
"role": "user",
"content": "Let sets $X$ and $Y$ have $30$ and $25$ elements, respectively, and there are at least $10$ elements in both sets. Find the smallest possible number of elements in $X \\cup Y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 6599,
"solution": ""
} |
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