index-completion_cnt stringlengths 3 6 | conversations listlengths 2 2 | answer stringlengths 1 23 |
|---|---|---|
2-0 | [
{
"from": "human",
"value": "Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b = 0$ also have a common real root. Compute the sum $a + b + c$.\nPlease reason step by step, and put... | -3 |
3-0 | [
{
"from": "human",
"value": "There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $1$, the second in box $2$, and so forth up to the seventh card which is placed in box $7$. You then sta... | 3 |
4-0 | [
{
"from": "human",
"value": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum\n\\[ \\sum_{(a,b,c) \\in S} \\frac{1}{2^a 3^b 5^c} \\]\ncan be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.\n... | 37 |
5-0 | [
{
"from": "human",
"value": "For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are to determine **for which integers $ n $** the expression\n\n$$\nn^4 + 6n^3 + 11n^2 + 3n + ... | 10 |
6-0 | [
{
"from": "human",
"value": "Four positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest possible value of $d$ such that the arithmetic mean of the numbers $a, b, c$ is half of the arithmetic mean of the numbers $a, b, c, d$.\nPlease reason step by step, and put your fi... | 10 |
7-0 | [
{
"from": "human",
"value": "$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?\nPlease reason... | 2 |
8-0 | [
{
"from": "human",
"value": "Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are asked to find the **smallest prime** $ p $ such that:\... | 509 |
9-0 | [
{
"from": "human",
"value": "Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$.\nPlea... | 3 |
10-0 | [
{
"from": "human",
"value": "There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other pl... | 1007 |
12-0 | [
{
"from": "human",
"value": "Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "W... | 5 |
14-0 | [
{
"from": "human",
"value": "Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\\frac{a\\pi+b}{c}$. ... | 1 |
15-0 | [
{
"from": "human",
"value": "Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM, and each papaya costs $5$ RM. He bought at least one of each fruit and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought?\nP... | 15 |
17-0 | [
{
"from": "human",
"value": "Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\nn(S(... | 402 |
19-0 | [
{
"from": "human",
"value": "Find all integers $k$ such that all roots of the following polynomial are also integers:\n\n$$f(x) = x^3 - (k-3)x^2 - 11x + (4k-8).$$\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the polynomial:\... | 5 |
20-0 | [
{
"from": "human",
"value": "A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and thos... | 300 |
21-0 | [
{
"from": "human",
"value": "If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given two cubic equations:\n\n1. $ a^3 + 12... | 1 |
22-0 | [
{
"from": "human",
"value": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:\n\\[\n\\sum_{(a,b,c)\\in S}\\frac{1}{2^a3^b5^c}\n\\]\nThis sum can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers... | 37 |
23-0 | [
{
"from": "human",
"value": "Some positive integers are initially written on a board, where each 2 of them are different. Each time we can do the following moves: \n1. If there are 2 numbers (written on the board) in the form $n, n+1$, we can erase them and write down $n-2$. \n2. If there are 2 numbers (w... | -3 |
24-0 | [
{
"from": "human",
"value": "Find all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of $n^4 + 6$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\... | 3 |
26-0 | [
{
"from": "human",
"value": "The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player remov... | 14 |
27-0 | [
{
"from": "human",
"value": "Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ a... | 7 |
29-0 | [
{
"from": "human",
"value": "Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given tw... | 30 |
32-0 | [
{
"from": "human",
"value": "Find the sum of all possible values of $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the expressio... | 9 |
33-0 | [
{
"from": "human",
"value": "Let $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of $M$ is $300$. Determine $n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given:\n\... | 10 |
34-0 | [
{
"from": "human",
"value": "Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pays him a ruble and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solut... | 4 |
35-0 | [
{
"from": "human",
"value": "Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n3^{15} + 3^{11} + 3^6 + 1\n$$\n\nWe are ... | 64 |
36-0 | [
{
"from": "human",
"value": "On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, the remainder of the sum of the erased numbers divided by 13 is written on the board. After a few repetitions of this process, only three numbers are left, of which two are 9 and... | 8 |
37-0 | [
{
"from": "human",
"value": "For positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, which value of $n$ maximizes the ratio $\\frac{s(n)}{n}$?\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"... | 1099 |
38-0 | [
{
"from": "human",
"value": "Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of... | 21 |
40-0 | [
{
"from": "human",
"value": "Find a necessary and sufficient condition on the positive integer $n$ such that the equation $x^n + (2 + x)^n + (2 - x)^n = 0$ has a rational root.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given th... | 1 |
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