original_problem stringlengths 12 7.47k | original_reasoning_trace stringlengths 71 99.6k | conversations listlengths 2 2 |
|---|---|---|
Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$ | <think>
Okay, so I need to find all prime numbers p and positive integers m such that the equation 2p² + p + 9 = m² holds. Hmm, let's start by understanding the problem. I have to find primes p and positive integers m where this quadratic in p becomes a perfect square.
First, since p is a prime number, possible value... | [
{
"from": "human",
"value": "Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find all prime numbers p and positive integers m such that the equation 2p² + p + 9 = m² holds. Hmm, let's start by underst... |
27. Find the digit that has the property that if it is appended to the end of any natural number, the resulting number is equal to the sum of three terms, the first of which is the original number, the second is the number represented by the sought digit, and the third is the product of the first two terms. | <think>
Okay, so I need to find a digit (let's call it d) such that if I append it to the end of any natural number, the resulting number is equal to the sum of three things: the original number, the digit d itself, and the product of the original number and d. Hmm, let me try to parse this step by step.
Let me take a... | [
{
"from": "human",
"value": "27. Find the digit that has the property that if it is appended to the end of any natural number, the resulting number is equal to the sum of three terms, the first of which is the original number, the second is the number represented by the sought digit, and the third is the pr... |
1. In one lyceum, $76 \%$ of the students have at least once not done their homework, and $\frac{5}{37}$ sometimes forget their second pair of shoes. Find the number of students in the lyceum, if it is more than 1000 but less than 2000. | <think>
Alright, let me try to figure out this problem. So, we have a lyceum (which is like a high school, I think) where 76% of the students have at least once not done their homework. Then, 5/37 of the students sometimes forget their second pair of shoes. We need to find the number of students in the lyceum, given th... | [
{
"from": "human",
"value": "1. In one lyceum, $76 \\%$ of the students have at least once not done their homework, and $\\frac{5}{37}$ sometimes forget their second pair of shoes. Find the number of students in the lyceum, if it is more than 1000 but less than 2000."
},
{
"from": "gpt",
"value"... |
Problem 3. Find the smallest positive integer $n$, such that there exist $n$ integers $x_{1}, x_{2}, \ldots, x_{n}$ (not necessarily different), with $1 \leq x_{k} \leq n, 1 \leq k \leq n$, and such that
$$
x_{1}+x_{2}+\cdots+x_{n}=\frac{n(n+1)}{2}, \quad \text { and } \quad x_{1} x_{2} \cdots x_{n}=n!
$$
but $\left... | <think>
Okay, let's try to figure out this problem. We need to find the smallest positive integer n such that there exist n integers x₁, x₂, ..., xₙ (not necessarily distinct) where each xₖ is between 1 and n inclusive. The sum of these integers should be n(n+1)/2, which is the same as the sum of the first n natural nu... | [
{
"from": "human",
"value": "\nProblem 3. Find the smallest positive integer $n$, such that there exist $n$ integers $x_{1}, x_{2}, \\ldots, x_{n}$ (not necessarily different), with $1 \\leq x_{k} \\leq n, 1 \\leq k \\leq n$, and such that\n\n$$\nx_{1}+x_{2}+\\cdots+x_{n}=\\frac{n(n+1)}{2}, \\quad \\text { ... |
"If we divide a square into parts with three lines, then among the resulting polygons, there is alwa(...TRUNCATED) | "<think>\nOkay, so the problem is about dividing a square into parts with three lines, and there's a(...TRUNCATED) | [{"from":"human","value":"If we divide a square into parts with three lines, then among the resultin(...TRUNCATED) |
"Example 7 Real numbers $a$ and $b$ satisfy the equation $\\frac{a^{2} b^{2}}{a^{4}-2 b^{4}}=$\n1. T(...TRUNCATED) | "<think>\nOkay, let's see. The problem says that real numbers a and b satisfy the equation (a²b²)/(...TRUNCATED) | [{"from":"human","value":"Example 7 Real numbers $a$ and $b$ satisfy the equation $\\frac{a^{2} b^{2(...TRUNCATED) |
"10. (12 points) 1 kilogram of soybeans can be made into 3 kilograms of tofu, and 1 kilogram of soyb(...TRUNCATED) | "<think>\nOkay, let's see. I need to figure out how many kilograms of soybeans were used to make soy(...TRUNCATED) | [{"from":"human","value":"10. (12 points) 1 kilogram of soybeans can be made into 3 kilograms of tof(...TRUNCATED) |
"The area of the smallest equilateral triangle with one vertex on each of the sides of the right tri(...TRUNCATED) | "<think>\nOkay, so I need to find the area of the smallest equilateral triangle that has one vertex (...TRUNCATED) | [{"from":"human","value":"The area of the smallest equilateral triangle with one vertex on each of t(...TRUNCATED) |
"G10.2 The medians $A L, B M, C N$ of $\\triangle A B C$ meet at $G$. If the area of $\\triangle A B(...TRUNCATED) | "<think>\nOkay, so I need to find the area of triangle ANG in the given problem. The triangle ABC ha(...TRUNCATED) | [{"from":"human","value":"G10.2 The medians $A L, B M, C N$ of $\\triangle A B C$ meet at $G$. If th(...TRUNCATED) |
"12. Let $f(x)=|x-p|+|x-15|+|x-p-15|$, where $0<p<15$, if $p \\leqslant x \\leqslant 15$, then the m(...TRUNCATED) | "<think>\nOkay, so I need to find the minimum value of the function f(x) = |x - p| + |x - 15| + |x -(...TRUNCATED) | [{"from":"human","value":"12. Let $f(x)=|x-p|+|x-15|+|x-p-15|$, where $0<p<15$, if $p \\leqslant x \(...TRUNCATED) |
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