messages listlengths 2 2 |
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[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ divides $N^2$ and $k$ does not divide $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many pos... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 2$, $f(2) = 5$, and $f(3) = 10$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the va... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $f(x) = \\frac{x}{1-x}$. For a positive integer $n$, define $f^n(x)$ as $f(f(...f(x)...))$ where $f$ is applied $n$ times. Find $f^{2023}(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 6$. Find the sum of all roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: A 5-digit number is chosen at random. What is the probability that the product of its digits is odd?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the probability that a randomly cho... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: How many positive integers $n$ are there such that $n$ divides $n^2+3n+20$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integers $ n $ satisfy the condition... |
[
{
"content": "Question: Let $f(x)$ be a polynomial such that $f(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of all coefficients of $f(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: A circle with center $(3,4)$ is tangent to the line $3x+4y=10$. What is the radius of the circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the radius of a circle that has its ce... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k \\le N$ and $k$ divides $N^2$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integer... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the sum of the coe... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 6$. Find the sum of all roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k \\le N$ and $k$ divides $N^2$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integer... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 7$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"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": "<think>\nOkay, so I need to find the sum of all coefficients ... |
[
{
"content": "Question: How many positive integers $n$ are there such that $n$ is a divisor of $10^{10}$ and $n$ is also a divisor of $20^{20}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 7$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $N = 2^{2024}$. How many integers $k$ are there such that $1 \\le k \\le N$ and $k$ is a divisor of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many integers k t... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ is a divisor of $2^{2024}$. How many elements are in $S$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this pr... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of all roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out how to solve this p... |
[
{
"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": "<think>\nOkay, let me try to figure out this problem. So, we ... |
[
{
"content": "Question: A triangle has sides of length 7, 8, and 9. Find the area of the triangle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the area of a triangle with sides of length 7, 8, and 9. Hmm,... |
[
{
"content": "Question: A circle is inscribed in a square of side length 2. Four smaller circles are then inscribed in the four corners, each tangent to two sides of the square and to the large inscribed circle. What is the radius of one of these smaller circles?\nPlease reason step by step, and put your final ... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ divides $N$ and $k+1$ divides $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive in... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of all roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: A triangle has side lengths $a, b, c$. If $a^2+b^2+c^2 = ab+bc+ca$, what type of triangle is it?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out what type of triangle has side le... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: A function $f: \\mathbb{Z}^+ \\to \\mathbb{Z}^+$ satisfies $f(1)=1$ and $f(n) = \\sum_{k=1}^{n-1} f(k)f(n-k)$ for $n \\ge 2$. Find the value of $f(6)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so... |
[
{
"content": "Question: A rectangle with side lengths $a$ and $b$ is inscribed in a circle. Another circle is inscribed in this rectangle. If the area of the region between the outer circle and the inner circle is $25\\pi$, what is the perimeter of the rectangle?\nPlease reason step by step, and put your final ... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius $R$ is tangent to the lines $y=x+10$ and $y=x-10$. Find the value of $R^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of R squar... |
[
{
"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": "<think>\nOkay, let me try to figure out this problem. So, we have a polynomial P(x) such... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ for which the quadratic equation $x^2 + nx + 12 = 0$ has two distinct integer roots. Find the sum of the absolute values of the elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ divides $N$ and $k+1$ divides $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many positive integers... |
[
{
"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": "<think>\nOkay, so I need to find all posit... |
[
{
"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": "<think>\nOkay, so I need to find all posit... |
[
{
"content": "Question: Let $P(x) = x^3 - 3x^2 + ax + b$ be a polynomial such that its roots form an arithmetic progression. If the sum of the squares of the roots is 14, find the value of $a+b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"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 all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all posit... |
[
{
"content": "Question: A function $f(x)$ satisfies $f(x) + x f(1-x) = x^2$ for all real numbers $x$. Find the value of $f(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of f(2) for a function... |
[
{
"content": "Question: How many positive integers $n$ are there such that $n^2 + 2023$ is a perfect square?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integers n make n² + 2023 a... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2) = x^4 - 2x^2 + 1$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the sum of all coeff... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, and the sum of the digits of $N$ is 12. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's try to figure out this... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius $R$ is tangent to the lines $y=x+4$ and $y=x-4$. Find $R^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the radius squared of a circle cent... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2+2024$ is a perfect square. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all posit... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2024$ is a perfect square. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all pos... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ is a multiple of $20!$. Find the sum of the distinct prime factors of the smallest integer in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think... |
[
{
"content": "Question: Let $f(x)$ be a polynomial of degree 3 such that $f(0)=1$, $f(1)=2$, $f(2)=4$, and $f(3)=8$. Find $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of f(4) where f(x) i... |
[
{
"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 all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all posit... |
[
{
"content": "Question: Let $f(x)$ be a polynomial such that $f(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all roots of $f(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: A circle is inscribed in a rhombus. The diagonals of the rhombus are 10 and 24. What is the radius of the inscribed circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the radius o... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a multiple of 7 and the sum of the digits of $n$ is 10. Find the smallest integer in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius $r$ is tangent to the line $y = x+4$ at point $A$ and to the line $y = -x+4$ at point $B$. Find the length of the major arc $AB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2024 such that $x^2 + x + 1$ is divisible by 7. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many ... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ divides $N$ and $k+1$ divides $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive in... |
[
{
"content": "Question: Let $f(x)$ be a polynomial such that $f(x^2+1) = f(x)^2+1$ for all real numbers $x$. If $f(0)=0$ and $f(1)=1$, find $f(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius $R$ is intersected by the line $y=x+k$ at two distinct points $A$ and $B$. If the length of the chord $AB$ is $L$, express $k^2$ in terms of $R$ and $L$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$ and $P(1)=2$. If $P(n)=7$ for some integer $n$, what is the sum of all possible values of $n$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<t... |
[
{
"content": "Question: Let $f(x)$ be a polynomial of degree 3 such that $f(0)=1$, $f(1)=2$, $f(2)=4$, and $f(3)=8$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of f(4)... |
[
{
"content": "Question: A sequence is defined by $a_1 = 3$, $a_2 = 5$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the remainder when $a_{2024}$ is divided by 7.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me tr... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $1 \\le x \\le 1000$ and $x$ is not divisible by 3, 5, or 7. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the numb... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a multiple of 7 and the sum of the digits of $n$ is 7. Find the sum of the three smallest elements of $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the sum of all coe... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = P(x)^2+1$ for all real numbers $x$. If $P(0)=0$ and $P(1)=1$, find the value of $P(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's try to solv... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the sum of the coe... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 2$. Define a sequence $a_0 = 1$ and $a_{n+1} = f(a_n)$ for $n \\ge 0$. Find the smallest integer $n \\ge 0$ such that $a_n > 2023$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $N = 2^{15} \\cdot 3^{10} \\cdot 5^5$. How many positive divisors of $N^2$ are not divisors of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive divisors... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2) = x^4 + 5x^2 + 6$. Find the sum of all possible values of $P(1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all possible v... |
[
{
"content": "Question: A sequence of positive integers $a_1, a_2, \\ldots$ is defined by $a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. How many terms in the first 2024 terms of this sequence are divisible by 4?\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"content": "Question: Let $N = 2^{2024}$. How many positive integers $k$ are there such that $k$ divides $N$ and $k+1$ divides $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive in... |
[
{
"content": "Question: Let $N = 1 \\cdot 2 \\cdot 3 \\cdot \\ldots \\cdot 2023$. Find the largest integer $k$ such that $2^k$ divides $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the largest integer ... |
[
{
"content": "Question: Let $N = 1! + 2! + 3! + \\dots + 2023!$. Find the remainder when $N$ is divided by 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the remainder when N = 1! + 2! + 3! + ... + 2023... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius $r$ is intersected by the line $y=2x+5$ at two distinct points $A$ and $B$. If the length of the segment $AB$ is $2\\sqrt{5}$, what is the value of $r$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"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 all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all posit... |
[
{
"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": "<think>\nOkay, so I need to find the sum of all coefficients ... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = \\frac{a_{n-1}^2 + 1}{a_{n-2}}$ for $n \\ge 3$. Find the value of $a_8$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the v... |
[
{
"content": "Question: In $\\triangle ABC$, $AB=AC=10$ and $BC=12$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE \\perp AC$. Find the length of $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = 1$ and $P(1) = 2$.Suppose that $P(n) = 0$ for some integer $n$. Find the sum of all possible values of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"c... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with real coefficients. If $f(x) \\ge 0$ for all real $x$, and $f(0)=1$, $f(1)=2$, find the minimum possible value of $f(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"c... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius 5 intersects the line $y = x+1$ at two distinct points $A$ and $B$. Let $O$ be the origin. Find the area of triangle $OAB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thin... |
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