messages listlengths 2 2 |
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[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $1 \\le n \\le 100$. For how many values of $n \\in S$ is the number $n^2 - n + 41$ a prime number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, $P(1)=2$, and $P(2)=4$. If $P(3)=k$, find the smallest possible value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so ... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $x+y+z=1$. Find the minimum value of $\\frac{x^2}{y} + \\frac{y^2}{z} + \\frac{z^2}{x}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: How many positive integers less than or equal to 2023 are divisible by 3 or 5 but not by 7?\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 up to 2023 ... |
[
{
"content": "Question: Find the number of positive integer solutions $(x, y)$ to the equation $2x + 3y = 100$.\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 integer solutions (x, y) ... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE \\perp AD$. Find the length of $AE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of $\\frac{a^2}{b} + \\frac{b^2}{c} + \\frac{c^2}{a}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: A sequence of integers $a_1, a_2, a_3, \\dots$ is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the remainder when $a_{2023}$ is divided by $7$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: How many ordered pairs of positive integers $(x,y)$ satisfy $x^2 - y^2 = 2023$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many ordered pairs of positive integers (x, y) satis... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 = 2025$ and $x \\ge y$.\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 ordered pairs of intege... |
[
{
"content": "Question: Let $P(x) = x^3 - 6x^2 + 11x - 6$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $r_1^2 + r_2^2 + r_3^2$.\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: Let $N$ be the number of positive integers $x$ such that $x \\le 2023$ and $x^2 + 2x + 2$ is divisible by $5$. 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 ho... |
[
{
"content": "Question: A point $(x,y)$ is chosen uniformly at random from the square region defined by $0 \\le x \\le 2$ and $0 \\le y \\le 2$. What is the probability that $x^2 + y^2 \\le 1$ or $x+y \\ge 3$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 10n + 24$ 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 find the number of positive integers n such that n² ... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\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 $P(1) = 2$, $P(2) = 3$, and $P(3) = 5$. Find 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 ... |
[
{
"content": "Question: Two distinct numbers are chosen randomly from the set $\\{1, 2, 3, \\dots, 10\\}$. What is the probability that their sum 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 ... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $AC=8$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. 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: A point $P$ is chosen uniformly at random from the interior of a square with vertices $(0,0), (1,0), (1,1), (0,1)$. What is the probability that the distance from $P$ to the nearest side of the square is less than its distance to the nearest vertex of the square?\nPlease reason step by s... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of $\\frac{a^2}{b} + \\frac{b^2}{c} + \\frac{c^2}{a}$.\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 integers $n$ such that $1 \\le n \\le 100$ and $n^2+n+1$ is divisible by 7. 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 fi... |
[
{
"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: How many positive integers $n$ less than 1000 are there such that $n$ is a multiple of 3, and the sum of the digits of $n$ is a multiple of 5?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"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 $M$ be the midpoint of $BC$. Find the length of $DM$.\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 the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\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 $n^2 + 2023$ 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 pos... |
[
{
"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 $N = 1 \times 2 \times 3 \\times \\dots \\times 2023$. Find the largest integer $k$ such that $2023^k$ 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 inte... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of $\\frac{a^2}{b} + \\frac{b^2}{c} + \\frac{c^2}{a}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: In a triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $M$ be the midpoint of $BC$. Find the length of $DM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"conte... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the 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 the roots of the polyn... |
[
{
"content": "Question: Let $N = 2^{10} \\cdot 3^{10}$. How many positive integer divisors of $N^2$ are perfect squares?\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 divisor... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Prove that $\\frac{a^2}{b} + \\frac{b^2}{c} + \\frac{c^2}{a} \\ge 1$. What is the minimum value of $\\frac{a^2}{b} + \\frac{b^2}{c} + \\frac{c^2}{a}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"content": "Question: A point $P$ is chosen uniformly at random from the interior of a square with vertices $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$. What is the probability that the distance from $P$ to the nearest side of the square is less than the distance from $P$ to the center of the square?\nPlease reaso... |
[
{
"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 $BE \\perp AC$. Find the area of triangle $ADE$.\nPlease reason step by step, and put your final answer within \\boxed{... |
[
{
"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 $\\angle A$. Let $E$ be a point on $AC$ such that $BE$ is the angle bisector of $\\angle B$. Let $I$ be the incenter of $\\triangle ABC$. Find the ratio of the area of $\\... |
[
{
"content": "Question: How many positive integers less than or equal to 2023 are not divisible by 3, 5, or 7?\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 up to 2023 aren'... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 1$ is divisible by $n+1$?\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 posit... |
[
{
"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 minimum 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 to... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2023$ is divisible by $n+1$?\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 positiv... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: Find the number of ordered pairs of positive integers $(a, b)$ such that $a^2b = 10!$.\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 ordered pairs of positive integers ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $x \\le 2023$ and $x^2 + 2x + 2$ is divisible by $5$. 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 positi... |
[
{
"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 sequence is defined by $a_1 = 1$, $a_2 = 3$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the remainder when $a_{2023}$ is divided by 7.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 13$ is divisible by $11$?\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 posit... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE$ bisects $\\angle ABC$. If $AD$ and $BE$ intersect at $I$, find the ratio of the area of triangle $ABI$ to the area of triangle $ABC... |
[
{
"content": "Question: Find the number of positive integers $n$ 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 find the number of positive integers $ n $ such that $ n^2... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2023 such that $n^2 + 2n + 1$ is divisible by $n+1$.\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 in... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $M$ be the midpoint of $AD$. If $BM$ intersects $AC$ at $E$, find the ratio $\\frac{AE}{EC}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"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 the sum of the squares of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<th... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $M$ be the midpoint of $BC$. Find the length of $DM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content... |
[
{
"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 committee of 5 people is to be chosen from a group of 6 men and 4 women. What is the probability that the committee has more men than women, given that it has at least one woman?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: Find the number of ordered pairs of positive integers $(a, b)$ such that $a^2 - b^2 = 2023$.\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 ordered pairs of positive int... |
[
{
"content": "Question: What is the smallest positive integer $n$ such that $n$ has exactly 12 positive divisors?\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 that has exact... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2) = x^4 + 5x^2 + 6$. Find the sum of the roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the roots of the polynom... |
[
{
"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: What is the smallest positive integer $n$ such that $n!$ is divisible by $7^{77}$?\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! $ ... |
[
{
"content": "Question: Let $P(x) = x^3 - 10x^2 + 29x - 20$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $(r_1+r_2)(r_2+r_3)(r_3+r_1)$.\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 with integer coefficients such that $P(0)=2023$ and $P(1)=2024$. What is the remainder when $P(2)$ is divided by 3?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=8$. Let $M$ be the midpoint of $BC$. Let $D$ be a point on $AC$ such that $AD=3$. Find the length of $MD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2024 such that $x^2 + 2x + 2$ is divisible by 5. 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 $S$ 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 pos... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $AC=15$. 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 side $AC$. Let $F$ be the intersection of $AD$ and $BE$. Find the ratio of the area of triangle $... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2+20n+12$ 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: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $I$ be the incenter of $\\triangle ABC$. A line through $I$ parallel to $BC$ intersects $AB$ at $D$ and $AC$ at $E$. Find the length of segment $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=8$. Let $M$ be the midpoint of $BC$. Find the length of the median $AM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the length of the me... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 1$ is a perfect square?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positiv... |
[
{
"content": "Question: Let $N = 2^{10} \\cdot 3^{10}$. How many positive divisors of $N^2$ are perfect squares?\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 of $ N^2 $ are... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 1$ is a perfect square?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positiv... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x + 2$. Define a sequence $a_0, a_1, a_2, \\dots$ such that $a_0 = 0$ and $a_{n+1} = f(a_n)$ for all $n \\ge 0$. Find the smallest positive integer $n$ such that $a_n > 2023$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"... |
[
{
"content": "Question: Consider a regular hexagon $ABCDEF$ with side length 6. Let $P$ be a point in the interior of the hexagon such that the sum of the distances from $P$ to the sides of the hexagon is $9\\sqrt{3}$. Find the area of the smallest triangle formed by three vertices of the hexagon.\nPlease reaso... |
[
{
"content": "Question: Let $N = 2^{10} \\cdot 3^{10}$. How many positive divisors of $N^2$ are perfect squares?\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 problem. The question ... |
[
{
"content": "Question: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with integer coefficients. Suppose $P(1) = 2$, $P(2) = 5$, and $P(3) = 10$. Find the value of $P(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so ... |
[
{
"content": "Question: Two circles, $C_1$ and $C_2$, with radii $r_1=5$ and $r_2=3$ respectively, are externally tangent. A common external tangent line to $C_1$ and $C_2$ is drawn. Let $A$ and $B$ be the points of tangency on $C_1$ and $C_2$ respectively. Find the length of segment $AB$.\nPlease reason step b... |
[
{
"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 the sum of the digits of the number $N/12$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"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 sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the last two digits of $a_{2024}$.\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: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 1$ is a perfect square, and $n+1$ is a perfect cube?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's try t... |
[
{
"content": "Question: In a certain game, a player rolls a fair six-sided die. If the die shows a 1, 2, or 3, the player earns 2 points. If the die shows a 4 or 5, the player earns 3 points. If the die shows a 6, the player earns 5 points. What is the expected value of the total points earned after rolling the... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n$ divides $n^2+n+1$.\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 such that n divides n² + n +... |
[
{
"content": "Question: A sequence is defined by $a_1=1$, $a_2=2$, and $a_n = a_{n-1} + \frac{1}{a_{n-2}}$ for $n \neq 1, 2$. Find the smallest positive integer $k$ such that $a_k > 20$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$ and $P(1)=2$. If $P(k)=0$ for some integer $k$, find the value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $x+y+z=1$. Find the minimum value of $\\frac{x^2}{y} + \\frac{y^2}{z} + \\frac{z^2}{x}$.\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 $n^2$ has exactly three positive divisors. If $n \\le 100$, what is the sum of all such $n$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=0$ and $P(1)=1$. If $P(x)$ is divided by $x(x-1)$, the remainder is $ax+b$. Find the value of $a+b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"content": "Question: Let $N = 2^{2023}$. Find the number of positive integers $k$ 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 find the number of posi... |
[
{
"content": "Question: In triangle $ABC$, $AB=8$, $BC=10$, and $AC=12$. Let $D$ be a point on $AC$ such that $BD$ bisects $\\angle B$. Let $E$ be a point on $BC$ such that $AE \\perp BD$. Find the length of $DE$.\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 the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: Two distinct numbers are chosen randomly from the set $\\{1, 2, 3, \\dots, 10\\}$. What is the probability that their product is an even number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need... |
[
{
"content": "Question: How many positive integers $n \\le 200$ are there such that $n^2 + 7n + 10$ is a multiple of $n+2$?\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 integers n ≤... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 1$ is a perfect cube?\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 ... |
[
{
"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 $M$ be the midpoint of $BC$. Find the length of $DM$.\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 the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"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: Consider a sequence defined by $a_1 = 1$, $a_2 = 3$, and $a_n = 3a_{n-1} - 2a_{n-2}$ for $n \\ge 3$. Find the smallest integer $k$ such that $a_k > 2023$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
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