messages listlengths 2 2 |
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[
{
"content": "Question: Let $f(n)$ be the number of positive integers $k$ such that $1 \\le k \\le n$ and $k^2$ divides $n$. Find the value of $f(2024!)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the n... |
[
{
"content": "Question: A rectangular box has integer side lengths. The sum of the areas of its three distinct faces is 100 square units. What is the maximum possible volume of the box?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: A sequence of positive integers $a_1, a_2, \\dots, a_n, \\dots$ is defined by $a_1 = 1$ and $a_{n+1} = a_n + \\lfloor \\sqrt{a_n} \\rfloor$ for all $n \\ge 1$. Find the smallest integer $k$ such that $a_k \\ge 2024$.\nPlease reason step by step, and put your final answer within \\boxed{}... |
[
{
"content": "Question: A rectangular box has integer side lengths $a, b, c$ such that $1 \\le a \\le b \\le c$. The sum of the lengths of all its edges is 20. The surface area of the box is 18. What is the volume of the box?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role":... |
[
{
"content": "Question: A circle is inscribed in a rhombus ABCD. The circle is tangent to side AB at point P. If AP = 3 and PB = 12, what is the area of the rhombus?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n$ divides $2^{2023} + 2^{2022} + \\dots + 2^1 + 2^0$. Find the number of prime factors of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, $P(1)=2$, and $P(2)=4$. Find the smallest possible value of $|P(3)|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: A circular table has 12 chairs. We want to seat 4 men and 4 women such that no two people of the same gender sit next to each other. How many distinct seating arrangements are possible? Rotations are considered distinct if the people in corresponding seats are different.\nPlease reason s... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius 1 is inscribed in an isosceles trapezoid. The longer base of the trapezoid lies on the x-axis. If the area of the trapezoid is $16$, what is the perimeter of the trapezoid?\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ is a divisor of $(n-1)!$. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all posit... |
[
{
"content": "Question: Let $N$ be the number of ordered pairs of integers $(x,y)$ such that $x^2 + xy + y^2 = 2023$. Find the sum of the absolute values of all such $x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x$. Find the number of distinct real solutions to the equation $f(f(f(x))) = 3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of distinct real solution... |
[
{
"content": "Question: A circle is inscribed in an equilateral triangle. A smaller equilateral triangle is inscribed in the circle. What is the ratio of the area of the larger triangle to the area of the smaller triangle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "u... |
[
{
"content": "Question: Let $f(x)$ be a polynomial of degree 3 such that $f(0)=1$, $f(1)=3$, $f(2)=7$, and $f(3)=13$. 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) for a cubic... |
[
{
"content": "Question: Let $f(x)$ be a polynomial of degree 3 with real coefficients 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... |
[
{
"content": "Question: Let $f$ be a function such that $f(xy) = f(x) + f(y)$ for all positive real numbers $x$ and $y$. If $f(2) = 1$, what is $f(8)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find ... |
[
{
"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 $N$ be the smallest positive integer such that $N$ is a multiple of $17$ and the sum of the digits of $N$ is $17$. Find the number of digits in $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: A circle passes through the points $(0,0)$, $(1,0)$, and $(0,1)$. 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 passes t... |
[
{
"content": "Question: Let $N = 2^{2024}$. How many positive integers $k$ are there such that $k$ divides $N$ and $k+2$ 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: A circle is inscribed in a rhombus. The diagonals of the rhombus are 10 and 24. Find 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 of a... |
[
{
"content": "Question: Let $f(x)$ be a polynomial of degree 3 such that $f(1)=1$, $f(2)=2$, $f(3)=3$, and $f(4)=5$. Find the value of $f(5)$.\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(5)... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n$ divides $(n-1)! + 1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all positive integers n such that n divides (n-1)! + 1. Hmm... |
[
{
"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 all elements in $S$ that are less than $1000$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
... |
[
{
"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 $P(x) = x^3 - 2x^2 + ax - 6$ be a polynomial where $a$ is a real number. If the roots of $P(x)$ are $r_1, r_2, r_3$ and they satisfy $r_1^2 + r_2^2 + r_3^2 = 10$, what is the value of $a$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: A triangle has sides of length $a, b, c$. The medians to sides $a$ and $b$ are perpendicular. If $a=6$ and $b=8$, find the length of side $c$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"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 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 pos... |
[
{
"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 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 pos... |
[
{
"content": "Question: In $\\triangle ABC$, let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE \\perp AC$. If $AB=10$, $AC=12$, and $BC=14$, find the length of $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}."... |
[
{
"content": "Question: Let $N$ be the number of ordered pairs of integers $(x,y)$ such that $x^2 + y^2 = 2024$ and $x \\ge y$. Find the value of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the numbe... |
[
{
"content": "Question: A function $f: \\mathbb{R} \\to \\mathbb{R}$ satisfies $f(x+y) = f(x) + f(y) + 2xy$ for all real numbers $x, y$. If $f(1) = 2$, what is the value of $f(3)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOka... |
[
{
"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 $a, b, c$ such that $a^2+b^2+c^2 = 83$ and $ab+bc+ca = 71$. What is the perimeter of the triangle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out th... |
[
{
"content": "Question: A circle with center $(3,4)$ and radius 5 intersects the $x$-axis at points $A$ and $B$. Let $O$ be the origin $(0,0)$. Find the area of triangle $OAB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + \\frac{a_{n-2}}{n}$ for $n \\ge 3$. Find the value of $a_5$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the val... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2+1$ 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": "<think>\nOkay, so I need to find all positi... |
[
{
"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: Let $N$ be the number of positive integers $n$ such that $n^2+2023$ is a perfect square. 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 number of positive integers $ n $... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = P(x)^2+1$ for all real numbers $x$. If $P(0)=0$, 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 solve this proble... |
[
{
"content": "Question: Let $f(x) = x^2 + ax + b$. If $f(x)$ has roots $r_1$ and $r_2$, and $f(x+1)$ has roots $r_1-1$ and $r_2-1$, what is the sum of all possible values of $a$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ for which $n^2 + 20n - 19$ 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... |
[
{
"content": "Question: A circle with center $(3,4)$ is tangent to the line $3x+4y=0$. What is the area 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 area of a circle that has its center ... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with real coefficients. If $f(x) = x$ has no real solutions, and $f(f(x)) = x$ has exactly one real solution, find the value of $b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{... |
[
{
"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 all elements in $S$ that are less than 1000.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"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 $P(x-1)$.\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 roots of t... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2024n$ 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 po... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$, $a_2 = x$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. If $a_7 = 100$, find the value of $x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the ... |
[
{
"content": "Question: A triangle has side lengths 13, 14, and 15. A circle is inscribed in this triangle. A second circle is tangent to the inscribed circle and to two sides of the triangle. What is the radius of this second circle?\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 $k$ is a divisor of $N^2$ and $k \\cdot N$ 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... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x - 1$. Define a sequence by $a_0 = 1$ and $a_{n+1} = f(a_n)$ for $n \\ge 0$. Find the value of $a_{2023}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value... |
[
{
"content": "Question: Let $f(x) = x^2 + ax + b$ be a quadratic polynomial with integer coefficients $a$ and $b$. If $f(1) = 3$ and $f(2)$ is a prime number, find the sum of all possible values of $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"c... |
[
{
"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 $N$ be the number of positive integers $x$ less than 1000 such that $x^2 \\equiv 25 \\pmod{100}$. 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 number of positive i... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $1 \\le n \\le 100$ and $n^2+n+1$ is a prime number. Find the number of elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ has exactly 8 divisors (including 1 and $n$), and the sum of its divisors is three times $n$. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn randomly without replacement. What is the probability that at least one of the drawn balls is blue?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than 1000 such that $x$ can be expressed in the form $a^2 + b^2$ for some integers $a$ and $b$, and $x$ is not divisible by 3. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12 and $N$ has exactly 12 divisors. Find the sum of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"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 $f(x) = x^2 - 4x + 6$. If $r_1$ and $r_2$ are the roots of $f(x) = 0$, find the value of $r_1^3 + r_2^3$.\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_1^3 + r_2... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ has exactly 15 positive divisors. 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 a... |
[
{
"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 $f(1)=1$, $f(2)=2$, and $f(n) = f(n-1) + f(n-2)$ for $n \\ge 3$. Find the remainder when $f(2023)$ is divided by $10$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"co... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2+6n+5$ is a multiple of $n+1$. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all po... |
[
{
"content": "Question: A circle with center $(3,4)$ and radius $5$ is intersected by the line $y=mx$. If the chord formed by the intersection points has length $2\\sqrt{5}$, find the sum of all possible values of $m$.\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 $P(n)=0$ for some integer $n$, what is the smallest possible positive value of $|n|$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"conte... |
[
{
"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 $N = 1 \\cdot 2 \\cdot 3 \\cdots 2023$. Find the largest integer $k$ such that $N$ is divisible by $2^k$.\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 $ k $ ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=2023$, $P(1)=2024$, and $P(2)=2025$. What is the smallest possible value of $|P(3)|$?\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 intersected by the line $y=x+1$ at two distinct points $A$ and $B$. If the length of the chord $AB$ is $2\\sqrt{7}$, what is the value of $r^2$?\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 $n$ is a multiple of 3, and the sum of the digits of $n$ is 9. Find the number of elements in $S$ that are less than 1000.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $1 \\le n \\le 2023$ and $n$ is divisible by 3 or 5, but not 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 th... |
[
{
"content": "Question: A sequence of positive integers $a_1, a_2, \\dots, a_n$ is called \"geometric-like\" if for any $k \\ge 2$, $a_k^2 = a_{k-1}a_{k+1}$. If $a_1 = 2$ and $a_5 = 162$, find the sum of all possible values of $a_3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2} + 1$ for $n \\ge 3$. Find the remainder when $a_{2024}$ is divided by 5.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius $R$ is intersected by the line $y=x+a$ at two distinct points $A$ and $B$. If the length of the chord $AB$ is $2\\sqrt{7}$, and the distance from the origin to the line is $\\sqrt{2}$, find the value of $R^2$.\nPlease reason step by step, and put y... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n^2+2n+2$ is a divisor of $n^4+4n^2+16$. 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 number of positive inte... |
[
{
"content": "Question: A triangle has side lengths $a, b, c$ such that $a^2 + b^2 = 2024$ and $c = 10$. Let $A, B, C$ be the angles opposite to sides $a, b, c$ respectively. Find the maximum possible value of $\\sin C$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "use... |
[
{
"content": "Question: Consider a sequence of integers $a_1, a_2, \\ldots, a_{2024}$ such that $a_1 = 1$, $a_2 = 2$, and for $n \\ge 3$, $a_n = a_{n-1} + a_{n-2}$. Find the remainder when $a_{2024}$ is divided by 5.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"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: 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 $f(x) = x^2 - 2x + 4$. For how many integers $n$ is $f(f(n))$ a prime number?\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 n make f(f(n)) a prime number,... |
[
{
"content": "Question: A function $f: \\mathbb{Z}^+ \\to \\mathbb{Z}^+$ is defined by $f(1)=1$ and for $n>1$, $f(n)$ is the smallest prime factor of $n$. Let $S_k = \\sum_{n=1}^k f(n)$. Find the smallest integer $k$ such that $S_k > 100$.\nPlease reason step by step, and put your final answer within \\boxed{}.... |
[
{
"content": "Question: A circle with center $(3,4)$ and radius $5$ intersects the $x$-axis at points $A$ and $B$. Let $O$ be the origin $(0,0)$. Find the area of triangle $AOB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = -2$, $P(1) = 3$, and $P(2) = 14$. Find the smallest possible value of $|P(3)|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so ... |
[
{
"content": "Question: A regular octagon has side length $s$. A square is inscribed in the octagon such that its vertices alternate between the sides of the octagon. What is the area of the inscribed square in terms of $s$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": ... |
[
{
"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, so I need to find... |
[
{
"content": "Question: Let $f(x)$ be a polynomial of degree 3 such that $f(1)=1$, $f(2)=4$, $f(3)=9$, and $f(4)=15$. Find the value of $f(5)$.\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(5... |
[
{
"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, let's try to figure out how many positive integer... |
[
{
"content": "Question: Let $S$ be the set of points $(x,y)$ in the coordinate plane such that $x$ and $y$ are integers and $x^2 + y^2 \\le 25$. If a point is chosen uniformly at random from $S$, what is the probability that its coordinates $(x,y)$ satisfy $y \\ge x$?\nPlease reason step by step, and put your f... |
[
{
"content": "Question: Let $N = 1 \\cdot 2 \\cdot 3 \\cdot \\ldots \\cdot 2023$. Find the number of integers $k$ such that $1 \\le k \\le 2023$ and $\\gcd(k, N/k) = 1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I nee... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = P(x)^2 + 1$ and $P(0)=0$. 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 co... |
[
{
"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 $P(x^2+1) = P(x)^2+1$ 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, so I need to find... |
[
{
"content": "Question: A circle with center $(0,0)$ passes through the points $(3,4)$ and $(a,b)$. If the tangent line to the circle at $(3,4)$ is parallel to the line $ax + by = 10$, what is the radius of the circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"... |
[
{
"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 $P(x^2+1) = x^4 + 4x^2 + 5$. Find $P(x^2-1)$.\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) su... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 6$. Define a sequence $a_0, a_1, a_2, \\dots$ by $a_0 = 1$ and $a_{n+1} = f(a_n)$ for $n \\ge 0$. Find the number of digits in $a_{10}$ when written in base 10.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 \\equiv 1 \\pmod{105}$. 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 posi... |
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