messages listlengths 2 2 |
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[
{
"role": "user",
"content": "In a right triangle with legs of length 8 cm and 15 cm, a rectangle is inscribed such that its sides are parallel to the sides of the triangle. If the perimeter of the rectangle is 50 cm, what is the length of the hypotenuse of the triangle?"
},
{
"role": "assistant",
... |
[
{
"role": "user",
"content": "The sum of the squares of the first $n$ positive odd integers is equal to the sum of the squares of three consecutive odd integers. Find the smallest possible value of the middle integer.\n\n(Note: I've created a problem that requires a different approach to solve, as it involv... |
[
{
"role": "user",
"content": "Let $f(n)$ be a function that rounds the square root of $n$ up to the nearest integer. Find the sum of all numbers $n$ in the range $1\\leq n \\leq 1000$ for which $f(n)$ is a perfect square."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find th... |
[
{
"role": "user",
"content": "A fair coin is flipped 10 times, and the number of consecutive heads that appear at any point during the 10 flips is recorded. What is the probability that the longest sequence of consecutive heads has a length of exactly 4? Express your answer as a common fraction."
},
{
... |
[{"role":"user","content":"Let $a_0 = 1,$ $b_0 = 2,$ and\n\\[a_{n + 1} = a_n + \\frac{1}{b_n} \\quad(...TRUNCATED) |
[{"role":"user","content":"How many integers are there such that their squares are the sum of two or(...TRUNCATED) |
[{"role":"user","content":"What is the smallest positive integer $y$ such that when $99y$ is divided(...TRUNCATED) |
[{"role":"user","content":"Let \\(a\\) and \\(b\\) be the real roots of the equation \\(x^4 -4x-1 = (...TRUNCATED) |
[{"role":"user","content":"A certain number $m$ can be expressed as the product of two positive inte(...TRUNCATED) |
[{"role":"user","content":"In the triangle $ABC$ the midpoint $M$ of side $BC$ is equidistant to the(...TRUNCATED) |
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