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[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What is the probability that at least one of the balls drawn is red? Express your answer as a common fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"r... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2) = x^4 + 5x^2 + 6$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see. I need to find the sum of all coefficients o... |
[
{
"content": "Question: Consider a sequence defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the sum of the digits of $a_{10}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the su... |
[
{
"content": "Question: Given that $x, y, z$ are positive real numbers such that $xy=24$, $yz=36$, and $xz=6$. Find the value of $xyz$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see. I need to find the value of xyz given th... |
[
{
"content": "Question: An equilateral triangle is inscribed in a circle of radius 6. A smaller circle is inscribed in the equilateral triangle. What is the area of the region inside the equilateral triangle but outside the smaller circle?\nPlease reason step by step, and put your final answer within \\boxed{}.... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 100 such that $n^2 + 2n + 1$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positive integers n ≤ 1... |
[
{
"content": "Question: Let $f(x) = x^2 - 6x + c$. If the roots of $f(x)$ are $r_1$ and $r_2$, and $r_1^2 + r_2^2 = 20$, find the value of $c$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the value of $ c $ in the... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the value of P(3) where P is a polynomial such that P... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2) = x^4 + 5x^2 + 6$. Find the sum of the roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the sum of the roots of the polynomial P(x),... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2) = x^4 + 5x^2 + 6$. Find the sum of the roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the sum of the roots of the polynomial P(x),... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n$ divides $2024^2$ and $n$ is a multiple of $22$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see. The problem is asking for the number of pos... |
[
{
"content": "Question: A circle is inscribed in a square. A smaller square is inscribed in the circle. What is the ratio of the area of the larger square to the area of the smaller square?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Oka... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 1$. Suppose $r_1$ and $r_2$ are the roots of the equation $f(x) = 0$. Find the value of $(r_1^2 + 1)(r_2^2 + 1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the valu... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $1 \neq n$ and $n^2+1$ is divisible by $n-1$. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find all integers ... |
[
{
"content": "Question: An arithmetic progression has first term $a_1 = 1$ and common difference $d$. A geometric progression has first term $b_1 = 1$ and common ratio $r$. If $a_2 = b_2$ and $a_4 = b_4$, and $d \\ne 0$, find the value of $r$.\nPlease reason step by step, and put your final answer within \\boxe... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $1 \\le n \\le 2023$ and $n^2 + 2n + 2$ is a multiple of $5$. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positiv... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find $P(x+1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let me try to figure out this problem. So, we have a polynomial P(x) such that P(x... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let me try to figure out this problem. So, we have a polynomial P(x) su... |
[
{
"content": "Question: Let $P(x) = x^3 - 3x^2 + 2x - 1$. If $r_1, r_2, r_3$ are the roots of $P(x)=0$, find the value of $(r_1+r_2)^2 + (r_2+r_3)^2 + (r_3+r_1)^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 7x + 10$ is a multiple of $x+2$. Find the value of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to fin... |
[
{
"content": "Question: A sequence of integers $a_0, a_1, a_2, \\dots$ is defined by $a_0 = 1$ and $a_{n+1} = a_n^2 + a_n$ for $n \\ge 0$. Find the number of integers $n$ such that $n \\le 2024$ and $a_n$ is a multiple of 3.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": ... |
[
{
"content": "Question: Let $f(x) = x^3 - 3x^2 + 2x - 1$. If $r_1, r_2, r_3$ are the roots of $f(x)=0$, find the value of $(r_1+1)(r_2+1)(r_3+1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the value of $(r_1 + 1... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2023$ is a perfect square. Find the sum of the elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find all positive int... |
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