problem stringlengths 7 6.14k | messages listlengths 2 2 |
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Find the minimum value of $\frac{m}{n^2}$ if $-8 < n < m < 7$ and $m, n$ are integers. | [
{
"content": "Find the minimum value of $\\frac{m}{n^2}$ if $-8 < n < m < 7$ and $m, n$ are integers.",
"role": "user"
},
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"content": "<think>\nOkay, so I need to find the minimum value of m divided by n squared, right? The constraints are that both m and n are integers, and it's given that -8 is l... |
## Step 1: Understand the problem | [
{
"content": "## Step 1: Understand the problem",
"role": "user"
},
{
"content": "However, I don't see a problem provided. Please provide the problem you would like help with, and I'll be happy to assist you through the steps to find a solution.",
"role": "assistant"
}
] |
## Step 1: Define the problem | [
{
"content": "## Step 1: Define the problem",
"role": "user"
},
{
"content": "I'm happy to help you with a problem, but you haven't specified one yet. Please provide the problem you'd like assistance with, and I'll guide you through the steps to solve it.",
"role": "assistant"
}
] |
Let \(a_1, a_2, \ldots, a_n\) be a permutation of the numbers \(1, 2, \ldots, n\) with \(n \geq 2\). Determine the largest possible value of the sum \[ S(n) = |a_2 - a_1| + |a_3 - a_2| + \cdots + |a_n - a_{n-1}|. \] | [
{
"content": "Let \\(a_1, a_2, \\ldots, a_n\\) be a permutation of the numbers \\(1, 2, \\ldots, n\\) with \\(n \\geq 2\\). Determine the largest possible value of the sum \\[ S(n) = |a_2 - a_1| + |a_3 - a_2| + \\cdots + |a_n - a_{n-1}|. \\]",
"role": "user"
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"content": "<think>\nOkay, so I need... |
## Step 1: Define the problem | [
{
"content": "## Step 1: Define the problem",
"role": "user"
},
{
"content": "I'm happy to help you with a problem. However, you haven't specified what the problem is yet. Please provide more details or context about the issue you're facing, and I'll do my best to assist you.",
"role": "assistan... |
Find all pairs \((x, y)\) that satisfy the equation \(x^2 - y(x^2 - 1) + 3(xy + x - 1) = 0\). | [
{
"content": "Find all pairs \\((x, y)\\) that satisfy the equation \\(x^2 - y(x^2 - 1) + 3(xy + x - 1) = 0\\).",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to solve this equation: \\(x^2 - y(x^2 - 1) + 3(xy + x - 1) = 0\\). Hmm, first step is probably to simplify the equation ... |
Given n identical sticks lined up in a row, and k of them are chosen:
a) How many ways can k sticks be chosen from n sticks?
b) How many ways can k sticks be chosen such that no two chosen sticks are consecutive?
c) How many ways can k sticks be chosen such that there are at least j sticks between each pair of chosen s... | [
{
"content": "Given n identical sticks lined up in a row, and k of them are chosen:\na) How many ways can k sticks be chosen from n sticks?\nb) How many ways can k sticks be chosen such that no two chosen sticks are consecutive?\nc) How many ways can k sticks be chosen such that there are at least j sticks betw... |
Find the number of triangles with positive area whose vertices are among fifteen distinct points on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; 3 other points on side $\overline{AB}$; 4 other points on side $\overline{BC}$; and 5 other points on side $\overline{CA}$. | [
{
"content": "Find the number of triangles with positive area whose vertices are among fifteen distinct points on $\\triangle ABC$: the 3 vertices $A$, $B$, and $C$; 3 other points on side $\\overline{AB}$; 4 other points on side $\\overline{BC}$; and 5 other points on side $\\overline{CA}$.",
"role": "user... |
The equation $x^3 + ax^2 + bx + c = 0$ has roots $\alpha$, $\beta$, and $\gamma$. If $a = 6$ and $b = -15$, then what is the value of $c$? | [
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"content": "The equation $x^3 + ax^2 + bx + c = 0$ has roots $\\alpha$, $\\beta$, and $\\gamma$. If $a = 6$ and $b = -15$, then what is the value of $c$?",
"role": "user"
},
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"content": "To find the value of c, we can use Vieta's formulas. According to Vieta's formulas, for a cubic equation $x^3 ... |
The graph of a function f is shown below. The graph has a horizontal asymptote at y = 2, and a vertical asymptote at x = -1. | [
{
"content": "The graph of a function f is shown below. The graph has a horizontal asymptote at y = 2, and a vertical asymptote at x = -1.",
"role": "user"
},
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"content": "Based on the information given, we can make some conclusions about the function f, but we cannot determine its exact equation. ... |
## Step 1: Understand the problem | [
{
"content": "## Step 1: Understand the problem",
"role": "user"
},
{
"content": "I'm happy to help with a problem, but you haven't provided one yet. Please share the problem you'd like assistance with, and I'll do my best to guide you through the solution step by step.",
"role": "assistant"
}... |
The points P, Q, R, S are the midpoints of the sides AB, BC, CD, DA of the rectangle ABCD, respectively. Let T be the midpoint of RS. What is the ratio of the area of triangle PQT to the area of rectangle ABCD? | [
{
"content": "The points P, Q, R, S are the midpoints of the sides AB, BC, CD, DA of the rectangle ABCD, respectively. Let T be the midpoint of RS. What is the ratio of the area of triangle PQT to the area of rectangle ABCD?",
"role": "user"
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"content": "<think>\nOkay, so I have this geometry pr... |
Points $P$, $Q$, and $R$ are the midpoints of the medians of $\triangle ABC$. If the area of $\triangle ABC$ is $1024$, compute the area of $\triangle PQR$. | [
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"content": "Points $P$, $Q$, and $R$ are the midpoints of the medians of $\\triangle ABC$. If the area of $\\triangle ABC$ is $1024$, compute the area of $\\triangle PQR$.",
"role": "user"
},
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"content": "<think>\nOkay, so I need to find the area of triangle PQR, where P, Q, and R are the midpoin... |
Find the largest prime factor of $\sum_{n=2}^8 n^3$. | [
{
"content": "Find the largest prime factor of $\\sum_{n=2}^8 n^3$.",
"role": "user"
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"content": "<think>\nOkay, so I need to find the largest prime factor of the sum from n=2 to 8 of n cubed. Let me break this down step by step. First, I should calculate the sum of the cubes of the numbers from... |
Let \(ABC\) be a triangle in the \(xy\)-plane, where \(B\) is at the origin \((0,0)\). Let \(BC\) be extended to \(D\) such that \(BC:CD = 1:1\), \(CA\) be extended to \(E\) such that \(CA:AE = 1:2\), and \(AB\) be extended to \(F\) such that \(AB:BF = 1:3\). Given that \(G(32,24)\) is the centroid of triangle \(ABC\) ... | [
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"content": "Let \\(ABC\\) be a triangle in the \\(xy\\)-plane, where \\(B\\) is at the origin \\((0,0)\\). Let \\(BC\\) be extended to \\(D\\) such that \\(BC:CD = 1:1\\), \\(CA\\) be extended to \\(E\\) such that \\(CA:AE = 1:2\\), and \\(AB\\) be extended to \\(F\\) such that \\(AB:BF = 1:3\\). Given that \... |
Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(0) + 1 = f(1) \) and for any real numbers \( x \) and \( y \),
\[ f(xy - x) + f(x + f(y)) = yf(x) + 3. \] | [
{
"content": "Determine all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that \\( f(0) + 1 = f(1) \\) and for any real numbers \\( x \\) and \\( y \\),\n\\[ f(xy - x) + f(x + f(y)) = yf(x) + 3. \\]",
"role": "user"
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"content": "<think>\nOkay, so I need to solve this functional ... |
Find all functions \( f(x) \) such that
\[
(f(x))^2 f\left( \frac{1-x}{1+x} \right) = 64x \quad \forall x \in \mathbb{R} \setminus \{0, 1, -1\}.
\] | [
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"content": "Find all functions \\( f(x) \\) such that\n\\[\n(f(x))^2 f\\left( \\frac{1-x}{1+x} \\right) = 64x \\quad \\forall x \\in \\mathbb{R} \\setminus \\{0, 1, -1\\}.\n\\]",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all functions f(x) that satisfy the equation (f(x))² t... |
In a simulation experiment, two independent observations \( X_1 \) and \( X_2 \) are generated from the Poisson distribution with mean 1. The experiment is said to be successful if \( X_1 + X_2 \) is odd. What is the expected value of the number of experiments required to obtain the first success? | [
{
"content": "In a simulation experiment, two independent observations \\( X_1 \\) and \\( X_2 \\) are generated from the Poisson distribution with mean 1. The experiment is said to be successful if \\( X_1 + X_2 \\) is odd. What is the expected value of the number of experiments required to obtain the first su... |
## Step 1: Understand the problem | [
{
"content": "## Step 1: Understand the problem",
"role": "user"
},
{
"content": "I'm happy to help you with your problem. However, I don't see a problem statement. Please provide the problem you need help with, and I'll do my best to break it down into steps and provide a clear and concise solutio... |
Find the sum of the series $\frac{1}{2!} + \frac{x}{3!} + \frac{x^2}{4!} + \frac{x^3}{5!} + \cdots$. | [
{
"content": "Find the sum of the series $\\frac{1}{2!} + \\frac{x}{3!} + \\frac{x^2}{4!} + \\frac{x^3}{5!} + \\cdots$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the series 1/(2!) + x/3! + x²/4! + x³/5! + ... Hmm, let's see. Alright, first, I should try to write t... |
Consider all possible natural seven-digit numbers where each digit from $1$ to $7$ is used exactly once. If these numbers are listed in ascending order, what is the $1995$th number in this list? | [
{
"content": "Consider all possible natural seven-digit numbers where each digit from $1$ to $7$ is used exactly once. If these numbers are listed in ascending order, what is the $1995$th number in this list?",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this problem. So,... |
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