data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb{R}$ be the set of real numbers. Determine all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that, for any real numbers $x$ and $y$, \\[ f(f(x)f(y)) + f(x+y) = f(xy). \\]\n\n[i] Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "f(x) = 0f(x) = 1 - xf(x) = x - 1"
} | {
"split": "train",
"index": 25549,
"solution": "\nLet \\(\\mathbb{R}\\) be the set of real numbers. We are tasked with finding all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that for any real numbers \\( x \\) and \\( y \\), the functional equation:\n\n\\[\nf(f(x)f(y)) + f(x+y) = f(xy)\n\\]\n... |
deepscaler | [
{
"role": "user",
"content": "Three of the edges of a cube are $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ and $\\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR... | math | {
"style": "rule",
"ground_truth": "525"
} | {
"split": "train",
"index": 25550,
"solution": "\"For non-asymptote version of image, see Image:1998_AIME-11.png\"\n[asy] import three; size(280); defaultpen(linewidth(0.6)+fontsize(9)); currentprojection=perspective(30,-60,40); triple A=(0,0,0),B=(20,0,0),C=(20,0,20),D=(20,20,20); triple P=(5,0,0),Q=(20,0,15),R... |
deepscaler | [
{
"role": "user",
"content": "*How many odd numbers between 200 and 999 have distinct digits, and no digit greater than 7?* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 25551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle and a circle intersect so that each side of the triangle contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the triangle to the area of the circle? Express your answer as a common fraction in term... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4\\pi}"
} | {
"split": "train",
"index": 25552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a > 0 \\), if \\( f(g(h(a))) = 17 \\), where \\( f(x) = x^2 + 5 \\), \\( g(x) = x^2 - 3 \\), and \\( h(x) = 2x + 1 \\), what is the value of \\( a \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{3 + 2\\sqrt{3}}}{2}"
} | {
"split": "train",
"index": 25553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Any six points are taken inside or on an equilateral triangle with side length 1. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. T... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 25554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ if the three numbers $2, x$, and 10 have an average 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": 25555,
"solution": "Since the average of $2, x$ and 10 is $x$, then $\\frac{2 + x + 10}{3} = x$. Multiplying by 3, we obtain $2 + x + 10 = 3x$. Re-arranging, we obtain $x + 12 = 3x$ and then $2x = 12$ which gives $x = 6$."
} |
deepscaler | [
{
"role": "user",
"content": "Given two parallel lines \\\\(l_{1}\\\\) and \\\\(l_{2}\\\\) passing through points \\\\(P_{1}(1,0)\\\\) and \\\\(P_{2}(0,5)\\\\) respectively, and the distance between \\\\(l_{1}\\\\) and \\\\(l_{2}\\\\) is \\\\(5\\\\), then the slope of line \\\\(l_{1}\\\\) is \\_\\_\\_\\_\\_... | math | {
"style": "rule",
"ground_truth": "\\dfrac {5}{12}"
} | {
"split": "train",
"index": 25556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence where each term is either 1 or 2, starting with 1, and where between the \\(k\\)-th 1 and the \\((k+1)\\)-th 1 there are \\(2^{k-1}\\) 2's (i.e., the sequence is 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, ...), determine the sum of the first 19... | math | {
"style": "rule",
"ground_truth": "3986"
} | {
"split": "train",
"index": 25557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes af... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 25558,
"solution": "1. **Calculate Rachel's running details:**\n - Rachel completes a lap every 90 seconds.\n - In 10 minutes (600 seconds), Rachel completes $\\frac{600}{90} = 6\\frac{2}{3}$ laps. This means she completes 6 full laps and is $\\frac{2}{3}$ of a lap into her sevent... |
deepscaler | [
{
"role": "user",
"content": "Does there exist an integer \\( n \\) such that \\( 21n \\equiv 1 \\mod 74 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 25559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $64^5 = 32^x$, what is the value of $2^{-x}$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{64}"
} | {
"split": "train",
"index": 25560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that line $l$ intersects the hyperbola $\\frac{x^2}{9} - \\frac{y^2}{4} = 1$ at points $A$ and $B$, and the midpoint of segment $AB$ has coordinates $(6, 2)$, determine the slope of line $l$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 25561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"split": "train",
"index": 25562,
"solution": "1. **Identify the Triangle and Circle**: Triangle $ABC$ is equilateral with side length $6$. We need to find the area of the circle passing through points $A$, $O$ (the incenter of $\\triangle ABC$), and $C$.\n\n2. **Properties of the Incenter**: In an equilateral ... |
deepscaler | [
{
"role": "user",
"content": "Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen ... | math | {
"style": "rule",
"ground_truth": "\\frac{n}{2n+1}"
} | {
"split": "train",
"index": 25563,
"solution": "1. **Counting Total Ways to Choose Vertices**: \n The total number of ways to choose three vertices from a regular polygon with $(2n+1)$ sides is given by the combination formula:\n \\[\n \\binom{2n+1}{3} = \\frac{(2n+1)(2n)(2n-1)}{6}\n \\]\n\n2. **Countin... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\begin{cases} \\log_{2}x,x > 0 \\\\ x^{2}+4x+1,x\\leqslant 0\\\\ \\end{cases}$, if the real number $a$ satisfies $f(f(a))=1$, calculate the sum of all possible values of the real number $a$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "-\\frac{15}{16} - \\sqrt{5}"
} | {
"split": "train",
"index": 25564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 5$, $AC = 5$, and $BC = 6$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{68}{15}"
} | {
"split": "train",
"index": 25565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to color every integer either red or blue such that \\(n\\) and \\(n+7\\) are the same color for all integers \\(n\\), and there does not exist an integer \\(k\\) such that \\(k, k+1\\), and \\(2k\\) are all the same color? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 25566,
"solution": "It suffices to color the integers from 0 through 6 and do all arithmetic mod 7. WLOG, say that 0 is red (we'll multiply by 2 in the end). Then 1 must be blue because \\((0,0,1)\\) can't be monochromatic. 2 must be red because \\((1,2,2)\\) can't be monochromatic. T... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $D,$ $E,$ and $F$ are points on sides $\\overline{BC},$ $\\overline{AC},$ and $\\overline{AB},$ respectively, so that $BD:DC = CE:EA = AF:FB = 1:2.$\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, D, E, F, P, Q, R;\n\nA = (2,5);\nB = (0,0);\nC = (7,0);\nD = inte... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 25567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive numbers \\( x, y, \\) and \\( z \\) are such that \\( x + y + z = 5 \\). What is the minimum value of the expression \\( x^{2} + y^{2} + 2z^{2} - x^{2} y^{2} z \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 25568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance from home to work is $s = 6$ km. At the moment Ivan left work, his favorite dog dashed out of the house and ran to meet him. They met at a distance of one-third of the total route from work. The dog immediately turned back and ran home. Upon reaching home, the dog ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius \\( r \\) are externally tangent to each other and internally tangent to the ellipse \\( x^2 + 4y^2 = 8 \\). Find \\( r \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 25570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=|2x-a|+|x+ \\frac {2}{a}|$ \n$(1)$ When $a=2$, solve the inequality $f(x)\\geqslant 1$; \n$(2)$ Find the minimum value of the function $g(x)=f(x)+f(-x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt {2}"
} | {
"split": "train",
"index": 25571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $k$ be the answer to this problem. The probability that an integer chosen uniformly at random from $\\{1,2, \\ldots, k\\}$ is a multiple of 11 can be written as $\\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 25572,
"solution": "We write $k=11 q+r$ for integers $q, r$ with $0 \\leq r<11$. There are $q$ multiples of 11 from 1 to $k$, inclusive, so our probability is $\\frac{a}{b}=\\frac{q}{11 q+r}$. Let $d=\\operatorname{gcd}(q, r)=\\operatorname{gcd}(q, 11 q+r)$, so that the fraction $\\fr... |
deepscaler | [
{
"role": "user",
"content": "Among the four-digit numbers composed of the digits $0$, $1$, $2$, $3$, $4$, $5$ without repetition, there are a total of \\_\\_\\_\\_\\_ numbers that are not divisible by $5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 25573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square quilt block shown is made from nine unit squares, some of which have been divided in half to form triangles. What fraction of the square quilt is shaded? Express your answer as a common fraction.\n\n[asy]size(75);\nfill(scale(3)*unitsquare,gray(.6));\npath[] interior... | math | {
"style": "rule",
"ground_truth": "\\frac13"
} | {
"split": "train",
"index": 25574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system xOy, the parametric equations of line l are given by: $$\\begin{cases}x=1+t\\cos α \\\\ y=2+t\\sin α\\end{cases}$$ (t is a parameter, 0≤a<π). Establish a polar coordinate system with O as the pole and the positive semi-axis of the x-axis as th... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 25575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the arithmetic series\n\\[20+20\\frac{1}{5}+20\\frac{2}{5}+\\cdots+40\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3030"
} | {
"split": "train",
"index": 25576,
"solution": "1. **Identify the series type and formula**: The given series is an arithmetic series. The sum of an arithmetic series where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms can be calculated using the formula:\n \\[\n S = \\frac{... |
deepscaler | [
{
"role": "user",
"content": "When the expression $(2x^4+3x^3+x-14)(3x^{10}-9x^7+9x^4+30)-(x^2+5)^7$ is expanded, what is the degree of the resulting polynomial? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 25577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We call a number antitriangular if it can be expressed in the form \\(\\frac{2}{n(n+1)}\\) for some natural number \\(n\\). For how many numbers \\(k\\) (where \\(1000 \\leq k \\leq 2000\\)) can the number 1 be expressed as the sum of \\(k\\) antitriangular numbers (not necessa... | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 25578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(1, -2)$ and $\\overrightarrow{b}=(3, 4)$, the projection of vector $\\overrightarrow{a}$ in the direction of vector $\\overrightarrow{b}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 25579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 20 points, each pair of adjacent points are equally spaced. By connecting four points with straight lines, you can form a square. Using this method, you can form _ squares. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 25580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, if $a_3 + a_5 + a_{12} - a_2 = 12$, calculate the value of $a_7 + a_{11}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$. Then Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120 < n \\leq 130"
} | {
"split": "train",
"index": 25582,
"solution": "1. **Identify the conditions**: We need to find the smallest nonprime integer $n$ greater than $1$ that has no prime factors less than $10$. This means all prime factors of $n$ must be $10$ or greater.\n\n2. **Prime factors greater than $10$**: The smallest prime n... |
deepscaler | [
{
"role": "user",
"content": "Solve in integers the system of equations: $$x^2-y^2=z$$\n $$3xy+(x-y)z=z^2$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "{(2, 1, 3), (1, 2, -3), (1, 0, 1), (0, 1, -1), (0, 0, 0)}"
} | {
"split": "train",
"index": 25583,
"solution": "\nTo solve the system of equations in integers:\n\\[\nx^2 - y^2 = z\n\\]\n\\[\n3xy + (x-y)z = z^2\n\\]\n\nwe begin by analyzing the first equation. The expression \\( x^2 - y^2 = z \\) can be factored as:\n\\[\n(x+y)(x-y) = z\n\\]\nSubstituting this into the second... |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations \\begin{align*} a + b &= -3, \\\\ ab + bc + ca &= -4, \\\\ abc + bcd + cda + dab &= 14, \\\\ abcd &= 30. \\end{align*} There exist relatively prime positive integers $m$ and $n$ such that \\[a^2 + b^2 +... | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 25584,
"solution": "From the fourth equation we get $d=\\frac{30}{abc}.$ substitute this into the third equation and you get $abc + \\frac{30(ab + bc + ca)}{abc} = abc - \\frac{120}{abc} = 14$. Hence $(abc)^2 - 14(abc)-120 = 0$. Solving we get $abc = -6$ or $abc = 20$. From the first ... |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions:\n[list]\n[*] no two consecutive digits are equal, and\n[*] the last digit is a prime.\n[/list] Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5} \\cdot 9^n - \\frac{2}{5} \\cdot (-1)^n"
} | {
"split": "train",
"index": 25585,
"solution": "\nTo solve this problem, we need to determine the number of \\( n \\)-digit positive integers that meet two criteria:\n1. No two consecutive digits are equal.\n2. The last digit is a prime number.\n\n### Step 1: Count All \\( n \\)-Digit Numbers\n\nThe total number... |
deepscaler | [
{
"role": "user",
"content": "Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he disco... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 25586,
"solution": "Let's denote the amount of money Fox has at the beginning as $x$ coins. We will analyze the changes in the amount of money Fox has after each crossing and payment.\n\n1. **First Crossing:**\n - Before crossing: Fox has $x$ coins.\n - After doubling: $2x$ coins.... |
deepscaler | [
{
"role": "user",
"content": "The sale ad read: \"Buy three tires at the regular price and get the fourth tire for 3 dollars.\" Sam paid 240 dollars for a set of four tires at the sale. What was the regular price of one tire? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 25587,
"solution": "1. **Define the variable**: Let $x$ be the regular price of one tire.\n\n2. **Set up the equation**: According to the sale ad, Sam buys three tires at the regular price and gets the fourth tire for $3. Therefore, the total cost for four tires is:\n \\[\n 3x + 3... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive number \\( c \\) with the following property: For any integer \\( n \\geqslant 4 \\) and any set \\( A \\subseteq \\{1, 2, \\ldots, n\\} \\), if \\( |A| > c n \\), then there exists a function \\( f: A \\rightarrow \\{1, -1\\} \\) such that \\( \\left... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 25588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 25589,
"solution": "1. **Define Variables:**\n Let $m$ represent the total amount of money Brianna has. Let $c$ represent the total cost of all the CDs she wants to buy.\n\n2. **Establish Relationships:**\n According to the problem, Brianna uses one fifth of her money to buy one t... |
deepscaler | [
{
"role": "user",
"content": "Let p, q, r, s, and t be distinct integers such that (9-p)(9-q)(9-r)(9-s)(9-t) = -120. Calculate the value of p+q+r+s+t. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 25590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Segment $AB$ has midpoint $C$, and segment $BC$ has midpoint $D$. Semi-circles are constructed with diameters $\\overline{AB}$ and $\\overline{BC}$ to form the entire region shown. Segment $CP$ splits the region into two sections of equal area. What is the degree measure of ang... | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"split": "train",
"index": 25591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers \\( n \\) that satisfy\n\\[ 15 < n^2 < 120. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 25592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 25593,
"solution": "1. **Identify the curves and their equations:**\n - The first curve is a circle given by \\(x^2 + y^2 = 25\\).\n - The second curve is an ellipse given by \\((x-4)^2 + 9y^2 = 81\\).\n\n2. **Substitute \\(y^2\\) from the circle's equation into the ellipse's equa... |
deepscaler | [
{
"role": "user",
"content": "Let $A=\\{a_{1}, b_{1}, a_{2}, b_{2}, \\ldots, a_{10}, b_{10}\\}$, and consider the 2-configuration $C$ consisting of \\( \\{a_{i}, b_{i}\\} \\) for all \\( 1 \\leq i \\leq 10, \\{a_{i}, a_{i+1}\\} \\) for all \\( 1 \\leq i \\leq 9 \\), and \\( \\{b_{i}, b_{i+1}\\} \\) for all ... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 25594,
"solution": "Let \\( A_{n}=\\{a_{1}, b_{1}, a_{2}, b_{2}, \\ldots, a_{n}, b_{n}\\} \\) for \\( n \\geq 1 \\), and consider the 2-configuration \\( C_{n} \\) consisting of \\( \\{a_{i}, b_{i}\\} \\) for all \\( 1 \\leq i \\leq n, \\{a_{i}, a_{i+1}\\} \\) for all \\( 1 \\leq i \\... |
deepscaler | [
{
"role": "user",
"content": "How many solutions in natural numbers \\( x, y \\) does the system of equations have\n\n\\[\n\\left\\{\\begin{array}{l}\n\\gcd(x, y)=20! \\\\\n\\text{lcm}(x, y)=30!\n\\end{array} \\quad (\\text{where } n! = 1 \\cdot 2 \\cdot 3 \\cdot \\ldots \\cdot n) ?\\right.\n\\]\n Let's thi... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 25595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest positive integer $n$ that satisfies $n^{200}<3^{500}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 25596,
"solution": "Note that $n^{200}=(n^{2})^{100}$ and $3^{500}=(3^{5})^{100}$. Since $n$ is a positive integer, then $n^{200}<3^{500}$ is equivalent to $n^{2}<3^{5}=243$. Note that $15^{2}=225,16^{2}=256$ and if $n \\geq 16$, then $n^{2} \\geq 256$. Therefore, the largest possible... |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\begin{pmatrix} 0 & c & -b \\\\ -c & 0 & a \\\\ b & -a & 0 \\end{pmatrix} \\begin{pmatrix} a^2 & ab & ac \\\\ ab & b^2 & bc \\\\ ac & bc & c^2 \\end{pmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}"
} | {
"split": "train",
"index": 25597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\\pi$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{7}"
} | {
"split": "train",
"index": 25598,
"solution": "1. **Identify the areas of the circles**: \n - The largest circle has radius 3, so its area is $9\\pi$.\n - The middle circle has radius 2, so its area is $4\\pi$.\n - The smallest circle has radius 1, so its area is $\\pi$.\n\n2. **Set up the equations for s... |
deepscaler | [
{
"role": "user",
"content": "Given that at the beginning of the year, 40% of students answered \"Yes\", 40% answered \"No\", and 20% were undecided, while at the end of the year, 60% answered \"Yes\", 30% answered \"No\", and 10% remained undecided, find the difference between the maximum and minimum possi... | math | {
"style": "rule",
"ground_truth": "60\\%"
} | {
"split": "train",
"index": 25599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all the roots of\n\\[z^8 - z^6 + z^4 - z^2 + 1 = 0,\\]the maximum imaginary part of a root can be expressed as $\\sin \\theta,$ where $-90^\\circ \\le \\theta \\le 90^\\circ.$ Find $\\theta.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54^\\circ"
} | {
"split": "train",
"index": 25600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction. Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 25601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\\mathrm {P}=r(\\cos{\\theta^{\\circ}}+i\\sin{\\theta^{\\circ}})$, where $0<r$ and $0\\leq \\theta <360$. Find $\\theta$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "276"
} | {
"split": "train",
"index": 25602,
"solution": "\\begin{eqnarray*} 0 &=& z^6 - z + z^4 + z^3 + z^2 + z + 1 = z(z^5 - 1) + \\frac{z^5-1}{z-1}\\\\ 0 &=& \\frac{(z^5 - 1)(z(z-1)+1)}{z-1} = \\frac{(z^2-z+1)(z^5-1)}{z-1} \\end{eqnarray*}\n Thus $z^5 = 1, z \\neq 1 \\Longrightarrow z = \\mathrm{cis}\\ 72, 144, 216, 28... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $BC=4$, $\\cos B= \\frac{1}{4}$, then $\\sin B=$ _______, the minimum value of $\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$ is: _______. 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": 25603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=\\sin\\left(2x+\\frac{\\pi}{3}\\right)+\\sqrt{3}\\sin^2x-\\sqrt{3}\\cos^2x-\\frac{1}{2}$. \n$(1)$ Find the smallest positive period and the interval of monotonicity of $f(x)$; \n$(2)$ If $x_0\\in\\left[\\frac{5\\pi}{12},\\frac{2\\pi}{3}\\right]$ and $f(x_{0})=\\frac... | math | {
"style": "rule",
"ground_truth": "-\\frac{3+\\sqrt{6}}{6}"
} | {
"split": "train",
"index": 25604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(n) \\) be the sum of the squares of the digits of positive integer \\( n \\) (in decimal). For example, \\( f(123) = 1^{2} + 2^{2} + 3^{2} = 14 \\). Define \\( f_{1}(n) = f(n) \\), and \\( f_{k+1}(n) = f\\left(f_{k}(n)\\right) \\) for \\( k = 1, 2, 3, \\ldots \\). Wha... | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 25605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the following equations:\n\na) $\\log _{1 / 5} \\frac{2+x}{10}=\\log _{1 / 5} \\frac{2}{x+1}$;\n\nb) $\\log _{3}\\left(x^{2}-4 x+3\\right)=\\log _{3}(3 x+21)$;\n\nc) $\\log _{1 / 10} \\frac{2 x^{2}-54}{x+3}=\\log _{1 / 10}(x-4)$;\n\nd) $\\log _{(5+x) / 3} 3=\\log _{-1 /(x... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 25606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\sin (\\omega x+\\frac{\\pi }{3})+2$, its graph shifts to the right by $\\frac{4\\pi }{3}$ units and coincides with the original graph. Find the minimum value of $|\\omega|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3}{2}"
} | {
"split": "train",
"index": 25607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Denote $\\phi=\\frac{1+\\sqrt{5}}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a \"base-$\\phi$ \" value $p(S)$. For example, $p(1101)=\\phi^{3}+\\phi^{2}+1$. For any positive integer $n$, let $f(n)$ be the number of such str... | math | {
"style": "rule",
"ground_truth": "\\frac{25+3 \\sqrt{69}}{2}"
} | {
"split": "train",
"index": 25608,
"solution": "We write everything in base $\\phi$. Notice that $$\\frac{\\phi^{48 n}-1}{\\phi^{48}-1}=10 \\ldots 010 \\ldots 01 \\ldots 10 \\ldots 01$$ where there are $n-1$ blocks of 47 zeros each. We can prove that every valid base-$\\phi$ representation comes from replacing a... |
deepscaler | [
{
"role": "user",
"content": "Let $d_n$ be the determinant of the $n \\times n$ matrix whose entries, from\nleft to right and then from top to bottom, are $\\cos 1, \\cos 2, \\dots, \\cos\nn^2$. Evaluate\n$\\lim_{n\\to\\infty} d_n$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 25609,
"solution": "The limit is $0$; we will show this by checking that $d_n = 0$ for all $n \\geq 3$.\nStarting from the given matrix, add the third column to the first column; this does not change the\ndeterminant. However, thanks to the identity\n$\\cos x + \\cos y = 2 \\cos \\fra... |
deepscaler | [
{
"role": "user",
"content": "A standard 52-card deck contains cards of 4 suits and 13 numbers, with exactly one card for each pairing of suit and number. If Maya draws two cards with replacement from this deck, what is the probability that the two cards have the same suit or have the same number, but not b... | math | {
"style": "rule",
"ground_truth": "15/52"
} | {
"split": "train",
"index": 25610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $ABC$ is a straight line. What is the value of $y$?\n\n[asy]\ndraw((-2,0)--(8,0),linewidth(0.7)); draw((8,0)--(5,-5.5)--(0,0),linewidth(0.7));\nlabel(\"$A$\",(-2,0),W); label(\"$B$\",(0,0),N); label(\"$C$\",(8,0),E); label(\"$D$\",(5,-5.5),S);\n\nlabel(\"$148^\\... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 25611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The inclination angle of the line given by the parametric equations \\[\n\\begin{cases} \nx=1+t \\\\\ny=1-t\n\\end{cases}\n\\]\ncalculate the inclination angle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3\\pi }{4}"
} | {
"split": "train",
"index": 25612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$ Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "4944"
} | {
"split": "train",
"index": 25613,
"solution": "1. **Divisibility by 4**: A number is divisible by 4 if its last two digits form a number that is divisible by 4. Since the number consists only of the digits 4 and 9, the last two digits must be 44 to ensure divisibility by 4, as 44 is divisible by 4.\n\n2. **Divi... |
deepscaler | [
{
"role": "user",
"content": "Given that the distance from point $P(x,y)$ to $A(0,4)$ and $B(-2,0)$ is equal, the minimum value of ${2}^{x}+{4}^{y}$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 25614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place the terms of the sequence $\\{2n-1\\}$ ($n\\in\\mathbb{N}^+$) into brackets according to the following pattern: the first bracket contains the first term, the second bracket contains the second and third terms, the third bracket contains the fourth, fifth, and sixth terms... | math | {
"style": "rule",
"ground_truth": "1251"
} | {
"split": "train",
"index": 25615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangle \\( \\triangle ABC \\), \\( \\angle C = 90^{\\circ} \\), and \\( CB > CA \\). Point \\( D \\) is on \\( BC \\) such that \\( \\angle CAD = 2 \\angle DAB \\). If \\( \\frac{AC}{AD} = \\frac{2}{3} \\) and \\( \\frac{CD}{BD} = \\frac{m}{n} \\) where \\( m \\) and \... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 25616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $ABCD$, $\\overrightarrow{AB}=(1,1)$, $\\overrightarrow{DC}=(1,1)$, $\\frac{\\overrightarrow{BA}}{|\\overrightarrow{BA}|}+\\frac{\\overrightarrow{BC}}{|\\overrightarrow{BC}|}=\\frac{\\sqrt{3}\\overrightarrow{BD}}{|\\overrightarrow{BD}|}$, calculate the area of ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 25617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 2006 integers $a_1, a_2, \\ldots a_{2006}$ satisfy the following conditions: $a_1=0$, $|a_2|=|a_1+2|$, $|a_3|=|a_2+2|$, $\\ldots$, $|a_{2006}|=|a_{2005}+2|$, then the minimum value of $a_1+a_2+\\ldots+a_{2005}$ is. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "-2004"
} | {
"split": "train",
"index": 25618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n \\ge 3$ be an integer. Rowan and Colin play a game on an $n \\times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid, and Colin is allowed to permute the columns of the grid. A grid coloring is $orderly... | math | {
"style": "rule",
"ground_truth": "2 \\cdot n! + 2"
} | {
"split": "train",
"index": 25619,
"solution": "We focus on the leftmost column for simplicity. Let $m$ be the number of red squares in this column. We then have five cases:\n 1. $m=1$ \nWhen Rowan permutes the rows of the coloring, we consider only the first column, which by the above contains $m=1$ red colors,... |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular storage with length 20 feet, width 15 feet, and height 10 feet, and with the floor and each of the four walls being two feet thick, calculate the total number of one-foot cubical blocks needed for the construction. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "1592"
} | {
"split": "train",
"index": 25620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "987^2+1597^2"
} | {
"split": "train",
"index": 25621,
"solution": "\nWe are tasked with finding the maximum value of \\( m^2 + n^2 \\), where \\( m \\) and \\( n \\) are integers within the range \\( 1, 2, \\ldots, 1981 \\), satisfying the equation:\n\n\\[\n(n^2 - mn - m^2)^2 = 1.\n\\]\n\n### Step 1: Analyze the Equation\n\nThe eq... |
deepscaler | [
{
"role": "user",
"content": "A line connects points $(2,1)$ and $(7,3)$ on a square that has vertices at $(2,1)$, $(7,1)$, $(7,6)$, and $(2,6)$. What fraction of the area of the square is above this line? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 25622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: (1) $(\\sqrt[3]{1.5} \\times \\sqrt[6]{12})^{2} + 8{1}^{0.75} - (-\\frac{1}{4})^{-2} - 5 \\times 0.12{5}^{0}$; (2) $\\lg 25 + \\lg 2 \\cdot \\lg 50 + (\\lg 2)^{2} - e^{3\\ln 2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 25623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S = (1+i)^{17} - (1-i)^{17}$, where $i=\\sqrt{-1}$. Find $|S|$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 25624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $B$ has its center at $(-6, 2)$ and a radius of $10$ units. What is the sum of the $y$-coordinates of the two points on circle $B$ that are also on the $y$-axis? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 25625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the variables $a, b, c, d$, and $e$ represents a positive integer with the properties that $b+d>a+d$, $c+e>b+e$, $b+d=c$, $a+c=b+e$. Which of the variables has the greatest value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "c"
} | {
"split": "train",
"index": 25626,
"solution": "Since $b+d>a+d$, then $b>a$. This means that $a$ does not have the greatest value. Since $c+e>b+e$, then $c>b$. This means that $b$ does not have the greatest value. Since $b+d=c$ and each of $b, c, d$ is positive, then $d<c$, which means that $d$ does not have the... |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be an equilateral triangle with side length $s$. Point $P$ is an internal point of the triangle such that $AP=1$, $BP=\\sqrt{3}$, and $CP=2$. Find the value of $s$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 25627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all positive integers $n$ such that $\\text{lcm}(2n, n^2) = 14n - 24$ ?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 25628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "675"
} | {
"split": "train",
"index": 25629,
"solution": "Let the numbers be $a,a+1,a+2,a+3,a+4.$ When then know $3a+6$ is a perfect cube and $5a+10$ is perfect cube. Since $5a+10$ is divisible by $5$ we know that $5a+10 = (5k)^3$ since otherwise we get a contradiction. This means $a = 25k^3 - 2$ in which plugging into th... |
deepscaler | [
{
"role": "user",
"content": "Which of the following expressions is equal to an odd integer for every integer $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2017+2n"
} | {
"split": "train",
"index": 25630,
"solution": "When $n=1$, the values of the five expressions are 2014, 2018, 2017, 2018, 2019. \n When $n=2$, the values of the five expressions are 2011, 2019, 4034, 2021, 2021. \n Only the fifth expression $(2017+2n)$ is odd for both of these choices of $n$, so this must be th... |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?\n[asy] size(0,50); draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle); dot((-1,1)); dot(... | math | {
"style": "rule",
"ground_truth": "\\frac{4-\\pi}{\\pi}"
} | {
"split": "train",
"index": 25631,
"solution": "1. **Identify the Geometry of the Problem**: The problem states that a circle of radius $2$ is divided into four congruent arcs, which are then rearranged to form a star figure. The original circle has a radius of $2$, so its diameter is $4$.\n\n2. **Calculate the ... |
deepscaler | [
{
"role": "user",
"content": "What is the cube of the square of the second smallest prime number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 25632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the whole-number factors of 24? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 25633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide each natural number by the sum of the squares of its digits (for single-digit numbers, divide by the square of the number). Is there a smallest quotient among the obtained quotients, and if so, which one is it? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "1/9"
} | {
"split": "train",
"index": 25634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area, in square units, of a regular hexagon inscribed in a circle whose area is $324\\pi$ square units? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "486 \\sqrt{3}"
} | {
"split": "train",
"index": 25635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer \\( n \\) such that all the roots of \\( z^6 - z^3 + 1 = 0 \\) are \\( n^{\\text{th}} \\) roots of unity? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 25636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cubical cake with edge length 3 inches is iced on the sides and the top. It is cut vertically into four pieces, such that one of the cut starts at the midpoint of the top edge and ends at a corner on the opposite edge. The piece whose top is triangular contains an area $A$ an... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 25637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid right prism $PQRSTU$ has a height of 20. Its bases are equilateral triangles with side length 10. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RT$, respectively. Calculate the perimeter of triangle $MNO$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "5 + 10\\sqrt{5}"
} | {
"split": "train",
"index": 25638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$ Find the minimum value of\n\\[x^2 + 4xy + 4y^2 + 2z^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 25639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \\ln x$, with $0 < x_1 < x_2$. The points $C$ and $D$ trisect $\\overline{AB}$, with $AC < CB$. Through $C$ a horizontal line is drawn to cut the curve at $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 25640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given are real numbers $x, y$. For any pair of real numbers $a_{0}, a_{1}$, define a sequence by $a_{n+2}=x a_{n+1}+y a_{n}$ for $n \\geq 0$. Suppose that there exists a fixed nonnegative integer $m$ such that, for every choice of $a_{0}$ and $a_{1}$, the numbers $a_{m}, a_{m+1... | math | {
"style": "rule",
"ground_truth": "y = 0, 1, \\frac{1 + \\sqrt{5}}{2}, \\frac{1 - \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 25641,
"solution": "Note that $x=1$ (or $x=0$ ), $y=0$ gives a constant sequence, so it will always have the desired property. Thus, $y=0$ is one possibility. For the rest of the proof, assume $y \\neq 0$. We will prove that $a_{m}$ and $a_{m+1}$ may take on any pair of values, for an... |
deepscaler | [
{
"role": "user",
"content": "An ordered pair $(a, b)$ of positive integers is called spicy if $\\operatorname{gcd}(a+b, ab+1)=1$. Compute the probability that both $(99, n)$ and $(101, n)$ are spicy when $n$ is chosen from $\\{1,2, \\ldots, 2024\\}$ uniformly at random. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{96}{595}"
} | {
"split": "train",
"index": 25642,
"solution": "We claim that $(a, b)$ is spicy if and only if both $\\operatorname{gcd}(a+1, b-1)=1$ and $\\operatorname{gcd}(a-1, b+1)=1$. To prove the claim, we note that $$\\operatorname{gcd}(a+b, ab+1)=\\operatorname{gcd}(a+b, b(-b)+1)=\\operatorname{gcd}(a+b, b^{2}-1)$$ Henc... |
deepscaler | [
{
"role": "user",
"content": "Given that the line $y=kx+b$ is a common tangent to the curves $y=\\ln \\left(1+x\\right)$ and $y=2+\\ln x$, find the value of $k+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3-\\ln 2"
} | {
"split": "train",
"index": 25643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $5 \\cdot \\frac{12}{7} \\cdot \\frac{49}{-60}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 25644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the decimal $0.7\\overline{56}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{749}{990}"
} | {
"split": "train",
"index": 25645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We label the sides of a dodecagon as \\(C_{1}, C_{2}, \\ldots, C_{12}\\). In how many ways can we color the sides of a dodecagon with four colors such that two adjacent sides are always colored differently? (Two colorings are considered different if any one side \\(C_{i}\\) is ... | math | {
"style": "rule",
"ground_truth": "531444"
} | {
"split": "train",
"index": 25646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area, in square units, of a trapezoid bounded by the lines $y = x + 2$, $y = 12$, $y = 3$, and the $y$-axis? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49.5"
} | {
"split": "train",
"index": 25647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A list of five positive integers has a median of 4 and a mean of 15. What is the maximum possible value of the list's largest element? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 25648,
"solution": ""
} |
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