messages listlengths 2 2 |
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[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = P(x)^2+1$ for all real $x$. If $P(0)=0$, find $P(1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find P(1) where P is a polynomial sati... |
[
{
"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 $n^2+2023n$ 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 posi... |
[
{
"content": "Question: A circle passes through the points $(0,0)$, $(1,7)$, and $(7,1)$. Find the square of the radius of this circle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the square of the radius... |
[
{
"content": "Question: A regular hexagon has area $A$. A new regular hexagon is formed by connecting the midpoints of the sides of the first hexagon. What is the area of the new hexagon in terms of $A$?\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 $P(0)=2$, $P(1)=3$, and $P(2)=10$. 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 ... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ has exactly three positive divisors. Find the sum of all elements in $S$ that are less than 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\... |
[
{
"content": "Question: A circle with center $(0,0)$ and radius $r$ is tangent to the line $y=x+4$ at point $A$. A second circle with center $(5,0)$ and radius $R$ is tangent to the line $y=x+4$ at point $B$. If the distance between $A$ and $B$ is $3\\sqrt{2}$, find the value of $r+R$.\nPlease reason step by st... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, the sum of the digits of $N$ is 12, and $N$ has exactly 12 divisors. Find the value of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<... |
[
{
"content": "Question: Let $f(x)$ be a polynomial with integer coefficients such that $f(0)=2$ and $f(1)=3$. If $f(n)$ is a prime number for four distinct positive integers $n$, find the minimum possible value of $f(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "us... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x + 2$. 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 smallest integer $n$ such that $a_n > 2024$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"conte... |
[
{
"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 $N$ be the smallest positive integer such that $N$ is divisible by 9, $N$ has exactly 9 distinct positive divisors, and the sum of the digits of $N$ is 9. Find the sum of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $AC=6$. Let $M$ be the midpoint of $BC$. A point $D$ is chosen on $AC$ such that $AD=2DC$. Let $P$ be the intersection of $AM$ and $BD$. Find the ratio $\\frac{AP}{PM}$.\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 $n$ is a multiple of 7, and the sum of the digits of $n$ is 7. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOk... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1)=1$, $f(2)=5$, and $f(3)=12$. 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... |
[
{
"content": "Question: Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $M$ be the midpoint of $BC$. Let $P$ be a point on $AD$ such that $MP \\perp AD$. Find the length of $MP$.\nPlease reason step by step, and pu... |
[
{
"content": "Question: Let $f(x) = x^2 + 2x + 1$. Define $f^1(x) = f(x)$ and $f^{n+1}(x) = f(f^n(x))$. Find the sum of all real solutions to the equation $f^2(x) = 0$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need... |
[
{
"content": "Question: A function $f: \\mathbb{Z}^+ \\to \\mathbb{Z}^+$ is defined by $f(1)=1$ and $f(n) = f(n-1) + n$ if $n$ is even, and $f(n) = f(n-1) + 1$ if $n$ is odd. What is the value of $f(2023)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: A sequence is defined by $a_1 = 3$ and $a_n = a_{n-1}^2 - 2a_{n-1} + 2$ for $n \\ge 2$. Find the smallest integer $k$ such that $a_k > 2024$.\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 smallest positive integer such that $N$ is divisible by 9, and the sum of the digits of $N$ is 9. If the last two digits of $N$ are 25, find the sum of the digits of $N/25$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
... |
[
{
"content": "Question: In $\\triangle ABC$, $AB=13$, $BC=14$, and $CA=15$. 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 $BE$ is the median to $AC$. Let $F$ be the intersection of $AD$ and $BE$. Find the area of $\\triangle ABF$.\nPlease ... |
[
{
"content": "Question: How many positive integers $n$ are there such that $n$ is a multiple of 7, and the least common multiple of $n$ and 12 is 84?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how ... |
[
{
"content": "Question: Let $P(x) = x^3 - 2x^2 - 5x + 10$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $(r_1^2 - 1)(r_2^2 - 1)(r_3^2 - 1)$.\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: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 \\equiv 1 \\pmod{15}$. 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 ... |
[
{
"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: A triangle has side lengths $a, b, c$ such that $a^2+b^2+c^2 = 200$. If the area of the triangle is $25\\sqrt{3}$, find the perimeter of the triangle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so ... |
[
{
"content": "Question: A sequence of numbers $a_0, a_1, a_2, \\ldots$ is defined by $a_0 = 1$ and $a_{n+1} = \\frac{a_n}{1+na_n}$ for $n \\ge 0$. Find the smallest positive integer $n$ such that $a_n < \\frac{1}{2024}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "use... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ has exactly 12 positive divisors, and the sum of the divisors of $n$ is $2n$. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"con... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x < 1000$ such that $x$ is a multiple of $3$ and $x$ is a multiple of $5$, but $x$ is not a multiple of $7$. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 7, the sum of the digits of $N$ is 7, and $N$ has exactly 7 distinct positive divisors. Find $N$.\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 $n$ is a multiple of 7, $n$ has exactly three positive divisors, and the sum of the digits of $n$ is 7. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "u... |
[
{
"content": "Question: A circular table has 12 chairs. We want to seat 4 men and 4 women such that no two men sit next to each other. How many distinct seating arrangements are possible? Rotations are considered the same, but reflections are distinct.\nPlease reason step by step, and put your final answer with... |
[
{
"content": "Question: A regular hexagon $ABCDEF$ has side length 6. Let $M$ be the midpoint of $AB$, and let $N$ be the midpoint of $DE$. Let $P$ be a point on $BC$ such that $BP=2$. Let $Q$ be the intersection of $MN$ and $AP$. Find the area of triangle $QPC$.\nPlease reason step by step, and put your final ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $x \\le 2024$ and $x$ is not divisible by 2, 3, or 5. 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 nee... |
[
{
"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 $P(10)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find a polynomial P(x) that s... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $x+y+z=1$. What is the minimum value of $\\frac{x^2+y^2}{z} + \\frac{y^2+z^2}{x} + \\frac{z^2+x^2}{y}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2024 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 the number of... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 \\equiv 1 \\pmod{15}$. 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 ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n$ divides $2024^2$ and $n$ is a multiple of $22$. 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 pos... |
[
{
"content": "Question: Let $f(n)$ be the number of positive integers $k$ such that $1 \\le k \\le n$ and $\\gcd(k, n) = 1$. Find the smallest positive integer $n$ such that $f(n) = 2024$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ is a multiple of $24$ and $n^3$ is a multiple of $108$. Find the sum of all elements in $S$ that are less than $100$.\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 $n^2+1$ 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: A and B are two points on a circle with center O and radius 1. The angle AOB is $60^\\circ$. A second circle is tangent to OA at A and OB at B. What is the radius of the second circle?\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 $n^2 + 15n + 27$ 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: Let $f(x) = x^2 - 2x + 2$. Define a sequence $a_0, a_1, a_2, \\dots$ by $a_0 = 0$ and $a_{n+1} = f(a_n)$ for $n \\ge 0$. Find the smallest integer $n \\ge 1$ such that $a_n > 2023$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 6$. Suppose that $x_1, x_2, \\dots, x_n$ are distinct real numbers such that $f(f(\\dots f(x_i)\\dots)) = 2$ for $i=1, 2, \\dots, n$, where $f$ is applied $k$ times. Find the maximum possible value of $n$.\nPlease reason step by step, and put your final answer with... |
[
{
"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, find 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 fi... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ has exactly three divisors. Find the sum of all elements in $S$ that are less than 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than 1000 such that $x^2 \\equiv 2 \\pmod{7}$. 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 positive integers ... |
[
{
"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 integer value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"content": "Question: A square $ABCD$ with side length 1 is inscribed in a circle. Let $P$ be a point on the minor arc $AB$. Let $PA=x$, $PB=y$, and $PC=z$. Find the value of $x^2+y^2+z^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $1 \\le n \\le 1000$ and $n$ is a multiple of 7 or $n$ is a multiple of 11 (or both). Find the sum of the elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"content": "Question: For a positive integer $n$, let $S_n$ be the sum of the digits of $n$. For example, $S_{123} = 1+2+3 = 6$. Find the smallest positive integer $n$ such that $S_n = 2024$ and $S_{n+1} = 1$.\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 $n!$ ends with exactly 2023 zeros. 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 figure out h... |
[
{
"content": "Question: A function $f: \\mathbb{Z}^+ \\to \\mathbb{Z}^+$ is defined by $f(1)=1$ and $f(n) = f(n-1) + n$ if $n$ is even, and $f(n) = f(n-1) + 1$ if $n$ is odd, for $n > 1$. Find $f(20)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"co... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with $a, b, c$ real numbers. 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 value of $b/a$?\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 $n^2+n+1$ is a prime number. What is the maximum possible number of consecutive integers in $S$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $N = \\overline{abc}$ be a three-digit number such that $N = a! + b! + c!$. 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 a three-digit number N, which is ... |
[
{
"content": "Question: What is the smallest positive integer $n$ such that $n!$ is divisible by $10^{100}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the smallest positive integer n such that n! is div... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2024 are there such that $n^2+n+1$ is 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 positive integ... |
[
{
"content": "Question: Let $n$ be a positive integer. Consider a sequence of $n$ integers $a_1, a_2, \\ldots, a_n$ such that $1 \\le a_1 < a_2 < \\ldots < a_n \\le 2n$. If the sum $a_1 + a_2 + \\ldots + a_n$ is even, what is the smallest possible value of $a_1$?\nPlease reason step by step, and put your final ... |
[
{
"content": "Question: A circle is inscribed in a rhombus ABCD. The circle is tangent to sides AB, BC, CD, and DA at points P, Q, R, and S respectively. If the area of the rhombus is 96 and the length of the diagonal AC is 16, what is the length of the segment PR?\nPlease reason step by step, and put your fina... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $1 \\le x \\le 2023$ and $x^2 \\equiv 1 \\pmod{101}$.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... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a multiple of $3$, $n$ is a multiple of $7$, and $n$ has exactly $12$ positive divisors. What is the sum of the elements in $S$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: Let $f(x) = x^2 + ax + b$ be a quadratic polynomial such that $a$ and $b$ are integers. If the roots of $f(x)$ are $\\alpha$ and $\\beta$, and $\\alpha^3 + \\beta^3 = 2023$, find the value of $a^2b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "... |
[
{
"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 $f(1) = 1$, $f(2) = 4$, what is the minimum possible value of $f(3)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{... |
[
{
"content": "Question: Let $f(x)$ be a polynomial of degree 4 with real coefficients such that $f(x) = f(2-x)$ for all real $x$. If $f(0)=0$, $f(1)=1$, and $f(2)=0$, find $f(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay... |
[
{
"content": "Question: A regular hexagon has side length 6. A smaller regular hexagon is inscribed in the first hexagon by connecting the midpoints of its sides. What is the area of the region between the two hexagons?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user... |
[
{
"content": "Question: A function $f: \\mathbb{Z}^+ \\to \\mathbb{Z}^+$ is defined by $f(1)=1$, $f(2)=2$, and $f(n) = f(n-1) + f(n-2)$ for $n \\ge 3$. How many positive integers $n \\le 2023$ are there such that $f(n)$ is a multiple of 4?\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 $N$ be the number of positive integers $x$ less than 2023 such that $x^2 \\equiv x \\pmod{100}$. 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 in... |
[
{
"content": "Question: A function $f(n)$ is defined for positive integers $n$ as follows: $f(1)=1$, and for $n>1$, $f(n) = f(n-1) + \\frac{1}{f(n-1)}$. Find the smallest integer $n$ such that $f(n) \\ge 6$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with integer coefficients. Suppose that $P(0) = 0$ and $P(1) = 1$. If $P(k) = k$ for some integer $k \\ge 2$, what is the smallest possible value of $a+b+c$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"r... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = P(x)^2+1$ for all real $x$. If $P(0)=2$, find the value of $P(\\sqrt{3})$.\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... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to $2023$ such that $\\lfloor \\sqrt{x} \\rfloor$ divides $x$. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fi... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a divisor of $2024^2$ and $n$ is a multiple of $22$. Find the sum of the elements of $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2024 such that $x^2 \\equiv 25 \\pmod{100}$. 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... |
[
{
"content": "Question: Let $N$ be a positive integer. We are given $N$ points in the plane, no three of which are collinear. Each pair of points is connected by a line segment. Each segment is colored either red or blue. If there are exactly $R$ red segments and $B$ blue segments, and the number of monochromat... |
[
{
"content": "Question: Points $A, B, C$ are on a circle such that $AB=4$ and $BC=6$. Let $M$ be the midpoint of $AC$. If $BM=2$, find 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... |
[
{
"content": "Question: Let $N = \\overline{abc}$ be a three-digit number such that $N$ is divisible by 9. If the digits $a, b, c$ are distinct and chosen from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$, how many such numbers $N$ are there?\nPlease reason step by step, and put your final answer within \\boxed{}.... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with real coefficients. 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 value of $b/a$?\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 $n^2$ divides $2^{n-1}!$. Find the number of elements in $S$ that are less than or equal to 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: A polynomial $P(x)$ has integer coefficients and satisfies $P(1)=1$, $P(2)=2$, and $P(3)=4$. What is the smallest possible value of $|P(0)|$?\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 $2^{2024} + 2^{2023} + \\dots + 2^1 + 2^0$. 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 n... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=2$ and $P(1)=3$. What is the remainder when $P(2)$ is divided by $5$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fi... |
[
{
"content": "Question: A sequence of positive integers $a_1, a_2, a_3, \\dots$ is defined by $a_1 = 1$ and $a_{n+1}^2 = a_n^2 + 2a_n + 2$ for all $n \\ge 1$. Find the value of $a_{10}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2+n+1$ is a prime number. What is the sum of the digits of the largest two-digit number in $S$?\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 $P(x) = (x-1)(x-2)(x-3)(x-4)Q(x) + Rx + S$, where $Q(x)$ is a polynomial and $R, S$ are constants. If $P(1)=1$, $P(2)=3$, $P(3)=7$, and $P(4)=13$, find the value of $R+S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n$ divides $20!$ and $n$ has exactly 20 divisors. Find the remainder when $N$ is divided by $1000$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOk... |
[
{
"content": "Question: Let $N = \\overline{abc}$ be a three-digit number such that $N$, when divided by $a+b+c$, gives a quotient of $k$ and a remainder of $k$. If $k$ is an integer, what is the maximum possible value of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role"... |
[
{
"content": "Question: A square $ABCD$ has side length $12$. Let $P$ be a point on $AB$ such that $AP=4$. Let $Q$ be a point on $BC$ such that $BQ=3$. Let $R$ be a point on $CD$ such that $CR=6$. Let $S$ be a point on $DA$ such that $DS=x$. If the area of quadrilateral $PQRS$ is $100$, what is the value of $x$... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2024 are there such that $n^2+n+1$ is 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 positive integ... |
[
{
"content": "Question: Let $f(x)$ be a polynomial with integer coefficients such that $f(1)=2$, $f(2)=3$, and $f(3)=5$. Find the smallest possible value of $|f(0)|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: Let $N$ be the number of positive integers $k$ such that $1 \\le k \\le 100$ and the polynomial $P(x) = x^2 - x - k$ has integer roots. Find the value of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"content": "Question: Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $AC=6$. Let $M$ be the midpoint of $BC$. A point $P$ is chosen on $AM$ such that $CP$ is perpendicular to $AM$. Find the length of $BP$.\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 $7$ and $n$ has exactly $12$ positive divisors. Find the sum of the digits of the smallest integer in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x + 1$. Define a sequence $a_n$ by $a_1 = 3$ and $a_{n+1} = f(a_n)$ for $n \\ge 1$. Find the product of the first 100 terms of the sequence, $\\prod_{n=1}^{100} a_n$.\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 $P(0) = 1$, $P(1) = 2$, and $P(2) = 4$. 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, so... |
[
{
"content": "Question: A regular hexagon $ABCDEF$ has side length 6. Let $M$ be the midpoint of $AB$. Let $P$ be a point on $DE$ such that $AP$ bisects the area of the hexagon. Find the length of $DP$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"c... |
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