data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A circle with diameter $AB$ is drawn, and the point $ P$ is chosen on segment $AB$ so that $\\frac{AP}{AB} =\\frac{1}{42}$ . Two new circles $a$ and $b$ are drawn with diameters $AP$ and $PB$ respectively. The perpendicular line to $AB$ passing through $ P$ ... | math | {
"style": "rule",
"ground_truth": "205/1681"
} | {
"split": "train",
"index": 23249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.) Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 23250,
"solution": "\nTo solve the problem, we need to determine the sum of the digits of \\( B \\), which is derived from processing the large number \\( 4444^{4444} \\).\n\n### Step 1: Determine the sum of the digits of \\( 4444^{4444} \\).\n\nThe first step is to find \\( A \\), th... |
deepscaler | [
{
"role": "user",
"content": "Positive numbers $x$, $y$, and $z$ satisfy $xyz = 10^{81}$ and $(\\log_{10}x)(\\log_{10} yz) + (\\log_{10}y) (\\log_{10}z) = 468$. Find $\\sqrt {(\\log_{10}x)^2 + (\\log_{10}y)^2 + (\\log_{10}z)^2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 23251,
"solution": "Let $a=\\log_{10}x$, $b=\\log_{10}y,$ and $c=\\log_{10}z$.\n We have $a+b+c=81$ and $a(b+c)+bc=ab+ac+bc=468$. Since these two equations look a lot like Vieta's for a cubic, create the polynomial $x^3-81x^2+468x=0$ (leave the constant term as $0$ to make things easy... |
deepscaler | [
{
"role": "user",
"content": "Given a regular 14-gon, where each vertex is connected to every other vertex by a segment, three distinct segments are chosen at random. What is the probability that the lengths of these three segments refer to a triangle with positive area?\nA) $\\frac{73}{91}$\nB) $\\frac{74}... | math | {
"style": "rule",
"ground_truth": "\\frac{77}{91}"
} | {
"split": "train",
"index": 23252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are several positive numbers written on the board. The sum of the five largest numbers is 0.29 of the sum of all the numbers, and the sum of the five smallest numbers is 0.26 of the sum of all the numbers. How many numbers in total are written on the board? Let's think st... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 23253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the sequences consisting of five letters from the set $\\{T, U, R, N, I, P\\}$ (with repetitions allowed) are arranged in alphabetical order in a dictionary. Two sequences are called \"anagrams\" of each other if one can be obtained by rearranging the letters of the other. ... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23254,
"solution": "Convert each letter to a digit in base $6: I \\mapsto 0, N \\mapsto 1, P \\mapsto 2, R \\mapsto 3, T \\mapsto 4, U \\mapsto 5$. Then the dictionary simply consists of all base-6 integers from $00000_{6}$ to $555555_{6}$ in numerical order. If one number can be obta... |
deepscaler | [
{
"role": "user",
"content": "Humpty Dumpty walks on a straight line, taking either 37 steps to the left or 47 steps to the right per minute.\n\nWhat is the minimum time it will take for him to be one step to the right of the starting point? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 23255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \\dotsm$ plus 2, the result in the form of \"$\\sqrt[a]{b}$ plus $c$\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[9]{81} + 2"
} | {
"split": "train",
"index": 23256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a rectangle such that $A B=20$ and $A D=24$. Point $P$ lies inside $A B C D$ such that triangles $P A C$ and $P B D$ have areas 20 and 24, respectively. Compute all possible areas of triangle $P A B$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "98, 118, 122, 142"
} | {
"split": "train",
"index": 23257,
"solution": "There are four possible locations of $P$ as shown in the diagram. Let $O$ be the center. Then, $[P A O]=10$ and $[P B O]=12$. Thus, $[P A B]=[A O B] \\pm[P A O] \\pm[P B O]=120 \\pm 10 \\pm 12$, giving the four values $98,118,122$, and 142."
} |
deepscaler | [
{
"role": "user",
"content": "$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:\r\n\\[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "162\\sqrt{3}"
} | {
"split": "train",
"index": 23258,
"solution": "\nGiven that \\( x \\), \\( y \\), and \\( z \\) are positive reals such that \\( x + y + z = xyz \\), we aim to find the minimum value of:\n\\[\nx^7(yz-1) + y^7(zx-1) + z^7(xy-1).\n\\]\n\nFirst, we use the given condition \\( x + y + z = xyz \\). By the AM-GM ineq... |
deepscaler | [
{
"role": "user",
"content": "A list of five positive integers has a median of 3 and a mean of 11. 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": "47"
} | {
"split": "train",
"index": 23259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can a President and a Vice-President be chosen from a group of 5 people (assuming that the President and the Vice-President cannot be the same person)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill the nine numbers $1, 2, \\cdots, 9$ into a $3 \\times 3$ grid, placing one number in each cell, such that the numbers in each row increase from left to right and the numbers in each column decrease from top to bottom. How many different ways are there to achieve this arran... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 23261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the greatest integer less than or equal to $\\frac{5^{98} + 2^{104}}{5^{95} + 2^{101}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 23262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( \\cos \\left( \\frac {\\pi}{2}+\\alpha \\right)=3\\sin \\left(\\alpha+ \\frac {7\\pi}{6}\\right) \\), find the value of \\( \\tan \\left( \\frac {\\pi}{12}+\\alpha \\right) = \\) ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt {3} - 4"
} | {
"split": "train",
"index": 23263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the prime factorization of $1007021035035021007001$ . (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\\ldots p_k^{e_k}$ where $p_1,\\ldots,p_k$ are distinct prime numbers and $e_1,\\ldots,e_k$ are positive integers.) Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "7^7 \\times 11^7 \\times 13^7"
} | {
"split": "train",
"index": 23264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 23265,
"solution": "1. **Calculate the area of both pizzas**:\n - The area of a circle is given by the formula $A = \\pi r^2$.\n - For the pizza with radius $4$ inches, the area is:\n \\[\n A_1 = \\pi \\times 4^2 = 16\\pi \\text{ square inches}\n \\]\n - For the pizz... |
deepscaler | [
{
"role": "user",
"content": "What is the maximum value of $-4z^2+20z-6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 23266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $i^2 = -1$, then the sum $\\cos{45^\\circ} + i\\cos{135^\\circ} + \\cdots + i^n\\cos{(45 + 90n)^\\circ} + \\cdots + i^{40}\\cos{3645^\\circ}$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}(21 - 20i)"
} | {
"split": "train",
"index": 23267,
"solution": "1. **Identify the pattern and simplify each term:**\n - The given sum involves terms of the form $i^n \\cos((45 + 90n)^\\circ)$ for $n = 0$ to $40$.\n - We know that $i^2 = -1$, so $i^3 = -i$, and $i^4 = 1$. This implies that powers of $i$ repeat every four ter... |
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": 23268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the product \\[\n\\left(\\frac{1\\cdot3}{2\\cdot2}\\right)\\left(\\frac{2\\cdot4}{3\\cdot3}\\right)\\left(\\frac{3\\cdot5}{4\\cdot4}\\right)\\cdots\\left(\\frac{97\\cdot99}{98\\cdot98}\\right)\\left(\\frac{98\\cdot100}{99\\cdot99}\\right)?\n\\] Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{50}{99}"
} | {
"split": "train",
"index": 23269,
"solution": "1. **Rewrite the Product**: We start by rewriting the given product:\n \\[\n \\left(\\frac{1\\cdot3}{2\\cdot2}\\right)\\left(\\frac{2\\cdot4}{3\\cdot3}\\right)\\left(\\frac{3\\cdot5}{4\\cdot4}\\right)\\cdots\\left(\\frac{97\\cdot99}{98\\cdot98}\\right)\\left(\\... |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, side $AB$ measures $6$ units and side $BC$ measures $3$ units, as shown. Points $F$ and $G$ are on side $CD$ with segment $DF$ measuring $1$ unit and segment $GC$ measuring $2$ units, and lines $AF$ and $BG$ intersect at $E$. What is the area of triangle $A... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 23270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer. Let's think ste... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 23271,
"solution": "Let the two points $P$ and $Q$ be defined with coordinates; $P=(x_1,y_1)$ and $Q=(x_2,y_2)$\n We can calculate the area of the parallelogram with the determinant of the matrix of the coordinates of the two points(shoelace theorem).\n$\\det \\left(\\begin{array}{c} ... |
deepscaler | [
{
"role": "user",
"content": "Add $81.76$ to $34.587$ and round your answer to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "116.3"
} | {
"split": "train",
"index": 23272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hyperbola \\[\\frac{(x-3)^2}{5^2} - \\frac{(y-17)^2}{12^2} = 1\\]has two foci, which have different $x-$coordinates. Find the coordinates of the one with the larger $x-$coordinate. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(16,17)"
} | {
"split": "train",
"index": 23273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a randomly selected number $x$ in the interval $[0,\\pi]$, determine the probability of the event \"$-1 \\leqslant \\tan x \\leqslant \\sqrt {3}$\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{12}"
} | {
"split": "train",
"index": 23274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x^2- 2x = 0$ and $x \\neq 0$, 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": "2"
} | {
"split": "train",
"index": 23275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One day, a group of young people came to the Platonic Academy located in the outskirts of Athens. The academy's gate was closed, and above the gate a sign read: \"No one ignorant of geometry may enter!\" Next to the sign was a diagram with four small rectangles of areas $20, 40... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 23276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|m|=3$, $|n|=2$, and $m<n$, find the value of $m^2+mn+n^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 23277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a book, the pages are numbered from 1 through $n$. When summing the page numbers, one page number was mistakenly added three times instead of once, resulting in an incorrect total sum of $2046$. Identify the page number that was added three times. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 23278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = 5,$ then find\n\\[\\begin{vmatrix} a - c & b - d \\\\ c & d \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 23279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $xOy$, a pole coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\\rho\\cos^2\\theta=2a\\sin\\theta$ ($a>0$... | math | {
"style": "rule",
"ground_truth": "\\frac {3+ \\sqrt {10}}{2}"
} | {
"split": "train",
"index": 23280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 20 candies: 4 chocolate, 6 mint, and 10 butterscotch. What is the minimum number of candies that must be removed to be certain that at least two candies of each flavor have been eaten? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 23281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kelvin the Frog is trying to hop across a river. The river has 10 lilypads on it, and he must hop on them in a specific order (the order is unknown to Kelvin). If Kelvin hops to the wrong lilypad at any point, he will be thrown back to the wrong side of the river and will have ... | math | {
"style": "rule",
"ground_truth": "176"
} | {
"split": "train",
"index": 23282,
"solution": "Kelvin needs (at most) $i(10-i)$ hops to determine the $i$ th lilypad he should jump to, then an additional 11 hops to actually get across the river. Thus he requires $\\sum_{i=1}^{10} i(10-i)+11=176$ hops to guarantee success."
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$ and $a_{n+1}= \\frac {a_n}{a_n+2}$ $(n\\in\\mathbb{N}^*)$, find the value of $a_{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{1023}"
} | {
"split": "train",
"index": 23283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Several consecutive natural numbers are written on the board. It is known that \\(48\\%\\) of them are even, and \\(36\\%\\) of them are less than 30. Find the smallest of the written numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the region enclosed by the graph of $|x-60|+|y|=\\left|\\frac{x}{4}\\right|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 23285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 23286,
"solution": "\nGiven the equation:\n\n\\[\nabc + a + b + c = ab + bc + ca + 5\n\\]\n\nwe seek to find the minimum possible value of \\(a^2 + b^2 + c^2\\) where \\(a\\), \\(b\\), and \\(c\\) are real numbers.\n\nRearrange the given equation:\n\n\\[\nabc + a + b + c - ab - bc - c... |
deepscaler | [
{
"role": "user",
"content": "We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich ex... | math | {
"style": "rule",
"ground_truth": "2n - 1"
} | {
"split": "train",
"index": 23287,
"solution": "\nGiven are \\( n \\) coins of different weights and \\( n \\) balances, where \\( n > 2 \\). One of these balances is faulty and provides arbitrary results on each turn. Our goal is to determine the smallest number of turns required to find the heaviest coin.\n\n#... |
deepscaler | [
{
"role": "user",
"content": "In a class, each student has either 5 or 6 friends (friendship is mutual), and any two friends have a different number of friends. What is the minimum number of students, greater than 0, that can be in the class? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 23288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC, AB = 10, BC = 8, CA = 7$ and side $BC$ is extended to a point $P$ such that $\\triangle PAB$ is similar to $\\triangle PCA$. The length of $PC$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{3}"
} | {
"split": "train",
"index": 23289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gregor divides 2015 successively by 1, 2, 3, and so on up to and including 1000. He writes down the remainder for each division. What is the largest remainder he writes down? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "671"
} | {
"split": "train",
"index": 23290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of $N$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{4}"
} | {
"split": "train",
"index": 23291,
"solution": "If the first die is odd, which has $\\frac{1}{2}$ probability, then $N$ can be any of $0,1,2,3,4,5,6,7$ with equal probability, because multiplying each element of $\\{0, \\ldots, 7\\}$ with an odd number and taking modulo 8 results in the same numbers, as all odd ... |
deepscaler | [
{
"role": "user",
"content": "In a physical education class, students line up in four rows to do exercises. One particular class has over 30 students, with three rows having the same number of students and one row having one more student than the other three rows. What is the smallest possible class size fo... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 23292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of the parabola $y^{2}=-4x$, and the equation of the line $l$ as $2x+y-4=0$. There is a moving point $A$ on the parabola. The distance from point $A$ to the $y$-axis is $m$, and the distance from point $A$ to the line $l$ is $n$. Find the minimum value of $m+... | math | {
"style": "rule",
"ground_truth": "\\frac{6 \\sqrt{5}}{5}-1"
} | {
"split": "train",
"index": 23293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$, $b$, and $c$ are natural numbers, and $a < b$, $a + b = 719$, $c - a = 915$, then the largest possible value of $a + b + c$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1993"
} | {
"split": "train",
"index": 23294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two boxes, each containing the chips numbered $1$, $2$, $4$, $5$, a chip is drawn randomly from each box. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. 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": 23295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( a_{n} \\) denotes the last two digits of \\( 7^{n} \\). For example, \\( a_{2} = 49 \\), \\( a_{3} = 43 \\). Find the value of \\( a_{1} + a_{2} + a_{3} + \\cdots + a_{2007} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50199"
} | {
"split": "train",
"index": 23296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an arithmetic sequence {a\\_n} with a non-zero common difference. Given that a\\_3 = 7 and a\\_1 - 1, a\\_2 - 1, a\\_4 - 1 form a geometric sequence, find the value of a\\_10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all positive factors of 6000. What is the probability of a random quadruple $(a, b, c, d) \\in S^{4}$ satisfies $$\\operatorname{lcm}(\\operatorname{gcd}(a, b), \\operatorname{gcd}(c, d))=\\operatorname{gcd}(\\operatorname{lcm}(a, b), \\operatorname{lcm}(c... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{512}"
} | {
"split": "train",
"index": 23298,
"solution": "For each prime factor, let the greatest power that divides $a, b, c, d$ be $p, q, r, s$. WLOG assume that $p \\leq q$ and $r \\leq s$, and further WLOG assume that $p \\leq r$. Then we need $r=\\min (q, s)$. If $q=r$ then we have $p \\leq q=r \\leq s$, and if $r=s$... |
deepscaler | [
{
"role": "user",
"content": "The last two digits of the decimal representation of the square of a natural number are the same and are not zero. What are these digits? Find all solutions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 23299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) be a function satisfying \\( f(m+n) \\geq f(m) + f(f(n)) - 1 \\) for all \\( m, n \\in \\mathbb{N} \\). What values can \\( f(2019) \\) take? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"split": "train",
"index": 23300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\triangle ABC\\) be an isosceles right triangle with \\(AB=AC=10\\). Let \\(M\\) be the midpoint of \\(BC\\) and \\(N\\) the midpoint of \\(BM\\). Let \\(AN\\) hit the circumcircle of \\(\\triangle ABC\\) again at \\(T\\). Compute the area of \\(\\triangle TBC\\). Let's... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 23301,
"solution": "Note that since quadrilateral \\(BACT\\) is cyclic, we have \\(\\angle BTA=\\angle BCA=45^{\\circ}=\\angle CBA=\\angle CTA\\). Hence, \\(TA\\) bisects \\(\\angle BTC\\), and \\(\\angle BTC=90^{\\circ}\\). By the angle bisector theorem, we then have \\(\\frac{BT}{TC... |
deepscaler | [
{
"role": "user",
"content": "After viewing the John Harvard statue, a group of tourists decides to estimate the distances of nearby locations on a map by drawing a circle, centered at the statue, of radius $\\sqrt{n}$ inches for each integer $2020 \\leq n \\leq 10000$, so that they draw 7981 circles altoge... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 23302,
"solution": "Consider a coordinate system on any line $\\ell$ where 0 is placed at the foot from $(0,0)$ to $\\ell$. Then, by the Pythagorean theorem, a point $(x, y)$ on $\\ell$ is assigned a coordinate $u$ for which $x^{2}+y^{2}=u^{2}+a$ for some fixed $a$ (dependent only on ... |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\(a, b, c\\) are chosen such that \\(a < b < c\\). It is also known that the system of equations \\(2x + y = 2037\\) and \\(y = |x-a| + |x-b| + |x-c|\\) has exactly one solution. Find the minimum possible value of \\(c\\). Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "1019"
} | {
"split": "train",
"index": 23303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \\frac{1}{3}$, and\n\\[c_n = \\frac{2 - c_{n-1}}{3c_{n-2}}\\] for all $n \\ge 3$. Find $c_{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 23304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c$, respectively, and $a=1, A=\\frac{\\pi}{6}$.\n\n(Ⅰ) When $b=\\sqrt{3}$, find the magnitude of angle $C$; \n(Ⅱ) Find the maximum area of $\\triangle ABC$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{2+ \\sqrt{3}}{4}"
} | {
"split": "train",
"index": 23305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of a pyramid is a parallelogram with sides measuring 10 cm and 18 cm, and an area of 90 cm². The height of the pyramid passes through the intersection point of the diagonals of the base and is 6 cm. Determine the lateral surface area of the pyramid. Let's think step by... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 23306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2a\\sin B= \\sqrt{3}b$ and $\\cos C = \\frac{5}{13}$: \n(1) Find the value of $\\sin A$; \n(2) Find the value of $\\cos B$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{12\\sqrt{3} - 5}{26}"
} | {
"split": "train",
"index": 23307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different i... | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 23308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**Sarah has misplaced her friend Layla's phone number. Sarah recalls that the first four digits are either 3086, 3089, or 3098. The remaining digits are 0, 1, 2, and 5, but she is not sure about their order. If Sarah randomly calls a seven-digit number based on these criteria, ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{72}"
} | {
"split": "train",
"index": 23309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\\{m, m+4, m+10, n+1, n+2, 2n\\}$ are equal to $n$. What is $m+n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23310,
"solution": "1. **Identify the median**: The median of the set $\\{m, m+4, m+10, n+1, n+2, 2n\\}$ is the average of the third and fourth smallest numbers. Given that the median is $n$, we have:\n \\[\n \\frac{(m+10) + (n+1)}{2} = n\n \\]\n Simplifying this equation:\n ... |
deepscaler | [
{
"role": "user",
"content": "Michael is playing basketball. He makes $10 \\%$ of his shots, and gets the ball back after $90 \\%$ of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually makes a shot? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{19}"
} | {
"split": "train",
"index": 23311,
"solution": "We find the probability Michael never makes a shot. We do casework on the number of shots Michael takes. He takes only one shot with probability $\\frac{9}{10} \\cdot \\frac{1}{10}$ (he misses with probability $\\frac{9}{10}$ and does not get the ball back with pro... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=a^{2}\\sin 2x+(a-2)\\cos 2x$, if its graph is symmetric about the line $x=-\\frac{\\pi}{8}$, determine the maximum value of $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": 23312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 23313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function $f(x) = x^2 +2\\sqrt{x}$. Evaluate $2f(2) - f(8)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-56"
} | {
"split": "train",
"index": 23314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a $4 \\times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a blac... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{65536}"
} | {
"split": "train",
"index": 23315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a fixed circle \\\\(F_{1}:(x+2)^{2}+y^{2}=24\\\\) and a moving circle \\\\(N\\\\) passing through point \\\\(F_{2}(2,0)\\\\) and tangent to circle \\\\(F_{1}\\\\), denote the locus of the center of circle \\\\(N\\\\) as \\\\(E\\\\).\n\n\\\\((I)\\\\) Find the equation of t... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{9}"
} | {
"split": "train",
"index": 23316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( ABCD \\) be a square with side length \\( 5 \\), and \\( E \\) be a point on \\( BC \\) such that \\( BE = 3 \\) and \\( EC = 2 \\). Let \\( P \\) be a variable point on the diagonal \\( BD \\). Determine the length of \\( PB \\) if \\( PE + PC \\) is minimized. Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{2}}{8}"
} | {
"split": "train",
"index": 23317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each permutation $a_1,a_2,a_3,\\cdots,a_{10}$ of the integers $1,2,3,\\cdots,10$, form the sum\n\\[|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|.\\]\n The average value of all such sums can be written in the form $\\dfrac{p}{q}$, where $p$ and $q$ are relatively prim... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 23318,
"solution": "Similar to Solution 1, we can find the average value of $|a_2 - a_1|$, and multiply this by 5 due to symmetry. And again due to symmetry, we can arbitrarily choose $a_2 > a_1$. Thus there are $\\binom{10}{2} = 45$ ways to pick the two values of $a_2$ and $a_1$ from... |
deepscaler | [
{
"role": "user",
"content": "The energy stored by a pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges are initially placed at the corners of a square with each side length $d$. This configuration stores ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, given the parametric equations of circle $C_{1}$ as $\\left\\{\\begin{array}{l}x=2+2\\cos\\alpha\\\\ y=1+2\\sin\\alpha\\end{array}\\right.$ ($\\alpha$ is the parameter), establish a polar coordinate system with the coordinate origin as ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"split": "train",
"index": 23320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular triangle $EFG$ with a side length of $a$ covers a square $ABCD$ with a side length of 1. Find the minimum value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\frac{2}{\\sqrt{3}}"
} | {
"split": "train",
"index": 23321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $(-2)^{23} + 2^{(2^4+5^2-7^2)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8388607.99609375"
} | {
"split": "train",
"index": 23322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is the center $O$ of a sphere with a radius of 10. The sphere intersects the face $A A_{1} D_{1} D$ by a circle with a radius of 1, the face $A_{1} B_{1} C_{1} D_{1}$ by a circle with a radius of 1, and the face $C D D_{1} C_{1}... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 23323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ passing through the point $E(\\sqrt{3}, 1)$, with an eccentricity of $\\frac{\\sqrt{6}}{3}$, and $O$ as the coordinate origin.\n(I) Find the equation of the ellipse $C$;\n(II) If point $P$ is a mo... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 23324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the point \\( P \\) inside the triangle \\( \\triangle ABC \\), satisfying \\( \\overrightarrow{AP} = \\frac{1}{3} \\overrightarrow{AB} + \\frac{1}{4} \\overrightarrow{AC} \\), let the areas of triangles \\( \\triangle PBC \\), \\( \\triangle PCA \\), and \\( \\triangle P... | math | {
"style": "rule",
"ground_truth": "5:4:3"
} | {
"split": "train",
"index": 23325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the measure, in degrees, of one interior angle of a regular hexagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 23326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 20 positive divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 23327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\\textsc{am}$, and the sunset as $8:15\\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set? Let... | math | {
"style": "rule",
"ground_truth": "5:21\\textsc{pm}"
} | {
"split": "train",
"index": 23328,
"solution": "1. **Convert the length of daylight into a time format**: The length of daylight is given as 10 hours and 24 minutes. This can be represented as $10:24$.\n\n2. **Convert the sunrise time into a 24-hour format**: The sunrise time is given as $6:57\\textsc{am}$. In 2... |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ and $y$ are distinct nonzero real numbers such that $x+\\frac{2}{x} = y + \\frac{2}{y}$, what is $xy$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 23329,
"solution": "1. **Identify the Equation**: Given that $x$ and $y$ are distinct nonzero real numbers such that \n \\[\n x + \\frac{2}{x} = y + \\frac{2}{y},\n \\]\n we denote this common value by $A$. Thus, we have\n \\[\n A = x + \\frac{2}{x} = y + \\frac{2}{y}.\n ... |
deepscaler | [
{
"role": "user",
"content": "Let $n>1$ be an integer. For each numbers $(x_1, x_2,\\dots, x_n)$ with $x_1^2+x_2^2+x_3^2+\\dots +x_n^2=1$, denote\n$m=\\min\\{|x_i-x_j|, 0<i<j<n+1\\}$\nFind the maximum value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "{m \\leq \\sqrt{\\frac{12}{n(n-1)(n+1)}}}"
} | {
"split": "train",
"index": 23330,
"solution": "\nLet \\( n > 1 \\) be an integer. For any set of numbers \\((x_1, x_2, \\ldots, x_n)\\) such that the condition \\( x_1^2 + x_2^2 + x_3^2 + \\cdots + x_n^2 = 1 \\) holds, we need to determine the maximum possible value of \\( m \\), where:\n\n\\[\nm = \\min\\{|x_i... |
deepscaler | [
{
"role": "user",
"content": "Calculate the volume of the tetrahedron with vertices at points \\( A_{1}, A_{2}, A_{3}, A_{4} \\). Additionally, find its height dropped from vertex \\( A_{4} \\) onto the face \\( A_{1} A_{2} A_{3} \\).\n\nVertices:\n- \\( A_{1}(-1, 2, 4) \\)\n- \\( A_{2}(-1, -2, -4) \\)\n- \... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 23331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x) = x^k$ where $k < 0$, what is the range of $f(x)$ on the interval $[1, \\infty)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,1]"
} | {
"split": "train",
"index": 23332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $x^{2}+2 x=i$ has two complex solutions. Determine the product of their real parts. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1-\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 23333,
"solution": "The solutions are $x+1= \\pm e^{\\frac{i \\pi}{8}} \\sqrt[4]{2}$. The desired product is then $$\\left(-1+\\cos \\left(\\frac{\\pi}{8}\\right) \\sqrt[4]{2}\\right)\\left(-1-\\cos \\left(\\frac{\\pi}{8}\\right) \\sqrt[4]{2}\\right)=1-\\cos ^{2}\\left(\\frac{\\pi}{8}... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system \\(xOy\\), there is a point \\(P(0, \\sqrt{3})\\) and a line \\(l\\) with the parametric equations \\(\\begin{cases} x = \\dfrac{1}{2}t \\\\ y = \\sqrt{3} + \\dfrac{\\sqrt{3}}{2}t \\end{cases}\\) (where \\(t\\) is the parameter). Using the ori... | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"split": "train",
"index": 23334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all real values of $x$ with $0 < x < \\frac{\\pi}{2}$ such that $\\sin x$, $\\cos x$, and $\\tan x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\\tan^2 x$ over all $x$ in $S$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 23335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\\frac{C}{B} = \\frac{7}{3},$ what is the smallest possible value of $A + B + C + D$? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 23336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the series:\n\\[\n\\sum_{n=1}^\\infty \\frac{3^n}{1 + 3^n + 3^{n+1} + 3^{2n+1}}.\n\\] 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": 23337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=A C=\\frac{25}{14} B C$. Let $M$ denote the midpoint of $\\overline{B C}$ and let $X$ and $Y$ denote the projections of $M$ onto $\\overline{A B}$ and $\\overline{A C}$, respectively. If the areas of triangle $A B C$ and quadrilateral $A X M ... | math | {
"style": "rule",
"ground_truth": "1201"
} | {
"split": "train",
"index": 23338,
"solution": "By similar triangles, one can show that $[A X M Y]=2 \\cdot[A M X]=\\left(\\frac{24}{25}\\right)^{2} \\cdot 2[A B M]=\\left(\\frac{24}{25}\\right)^{2} \\cdot[A B C]$. Thus the answer is $25^{2}+24^{2}=1201$."
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the series $$\\frac{2^1}{8^1 - 1} + \\frac{2^2}{8^2 - 1} + \\frac{2^3}{8^3 - 1} + \\frac{2^4}{8^4 - 1} + \\cdots.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 23339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $4kx-4y-k=0$ intersects the parabola $y^{2}=x$ at points $A$ and $B$. If the distance between $A$ and $B$ is $|AB|=4$, determine the distance from the midpoint of chord $AB$ to the line $x+\\frac{1}{2}=0$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 23340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square \\(ABCD\\), an isosceles triangle \\(AEF\\) is inscribed such that point \\(E\\) lies on side \\(BC\\) and point \\(F\\) lies on side \\(CD\\), and \\(AE = AF\\). The tangent of angle \\(AEF\\) is 3. Find the cosine of angle \\(FAD\\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 23341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The task is given a finite increasing sequence \\( a_{1}, a_{2}, \\ldots, a_{n} \\) (\\(n \\geq 3\\)) of natural numbers, and for all \\( k \\leq n-2 \\), the equality \\( a_{k+2}=3 a_{k+1}-2 a_{k}-1 \\) holds. The sequence must necessarily contain the term \\( a_{k}=2021 \\). ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum possible value of $\\left(x^{2}+6 x+2\\right)^{2}$ over all real numbers $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23343,
"solution": "This is $\\left((x+3)^{2}-7\\right)^{2} \\geq 0$, with equality at $x+3= \\pm \\sqrt{7}$."
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the series $\\frac{3}{4} + \\frac{5}{8} + \\frac{9}{16} + \\frac{17}{32} + \\frac{33}{64} + \\frac{65}{128} - 3.5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1}{128}"
} | {
"split": "train",
"index": 23344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a natural number $N$, if at least eight out of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called an \"Eight Immortals Number.\" What is the smallest \"Eight Immortals Number\" greater than $2000$? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 23345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\log_{10}\\sin x + \\log_{10}\\cos x= -2$ and that $\\log_{10}(\\sin x+\\cos x)=\\frac{1}{2}(\\log_{10}m-2)$, find $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 23346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exists a polynomial $P$ of degree 5 with the following property: if $z$ is a complex number such that $z^{5}+2004 z=1$, then $P(z^{2})=0$. Calculate the quotient $P(1) / P(-1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2010012 / 2010013"
} | {
"split": "train",
"index": 23347,
"solution": "Let $z_{1}, \\ldots, z_{5}$ be the roots of $Q(z)=z^{5}+2004 z-1$. We can check these are distinct (by using the fact that there's one in a small neighborhood of each root of $z^{5}+2004 z$, or by noting that $Q(z)$ is relatively prime to its derivative). And certa... |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb {Z}\\to\\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,\n$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ or } f(x) = x"
} | {
"split": "train",
"index": 23348,
"solution": "\nWe are given the functional equation for \\( f: \\mathbb{Z} \\to \\mathbb{Z} \\):\n\\[\n2f(a^2 + b^2 + c^2) - 2f(ab + bc + ca) = f(a - b)^2 + f(b - c)^2 + f(c - a)^2\n\\]\nfor any integers \\( a, b, \\) and \\( c \\).\n\nTo find all such functions \\( f \\), we p... |
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