messages listlengths 2 2 |
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[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $x \\le 2024$ and $x^2 + x + 1$ is divisible by $13$. 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: Let $N = 2^{10} \\cdot 3^{12} \\cdot 5^{14}$. Find the number of positive divisors of $N^2$ that 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 solv... |
[
{
"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 length of $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"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, let's see. I need to find how many positive integers up to 202... |
[
{
"content": "Question: In a triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $M$ be the midpoint of $BC$. Find the length of the altitude from $B$ to $AM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find t... |
[
{
"content": "Question: Let $P(x) = x^3 - ax^2 + bx - c$ be a polynomial with integer coefficients. We are given that $P(1) = 0$, and the roots of $P(x)$ are $r_1, r_2, r_3$. If $r_1, r_2, r_3$ form an arithmetic progression and $a+b+c = 10$, find the value of $P(2)$.\nPlease reason step by step, and put your f... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2+n+1$ is a divisor of $n^5+n^4+n^3+n^2+n+1$.\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 su... |
[
{
"content": "Question: Triangle $ABC$ has side lengths $AB=5$, $BC=7$, 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 len... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a multiple of 7 and $n$ has exactly three digits, all of which are odd. Find the sum of all integers in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"conten... |
[
{
"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: Two distinct numbers $a$ and $b$ are chosen from the set $\\{1, 2, 3, \\dots, 10\\}$. What is the probability that $a+b$ is an even number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to f... |
[
{
"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 $N+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"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... |
[
{
"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: Let $S$ be the set of all positive integers $n$ such that $n^2 + 20n + 11$ 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 $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ divides $N^2$ and $k$ does not divide $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many pos... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 2$ 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 to find how many positive integ... |
[
{
"content": "Question: Two distinct numbers $a$ and $b$ are chosen from the set $\\{1, 2, 3, \\dots, 20\\}$. What is the probability that $a+b$ is an even number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to f... |
[
{
"content": "Question: A sequence of integers $a_1, a_2, a_3, \\dots$ is defined by $a_1 = 2$ and $a_{n+1} = a_n^2 - a_n + 1$ for $n \\ge 1$. Find the last two digits of $a_{2023}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=2$, $P(1)=3$, and $P(2)=5$. 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: 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: 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 $\\angle BAC$. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOka... |
[
{
"content": "Question: In a certain game, a player starts with 1 dollar. In each round, the player flips a fair coin. If it lands heads, the player doubles their money. If it lands tails, the player loses 1 dollar. The game ends when the player has 0 dollars or 4 dollars. What is the probability that the playe... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2023 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 find how many positive integers n ... |
[
{
"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 with integer coefficients such that $P(1)=2$, $P(2)=3$, and $P(3)=5$. 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 nee... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 3n + 2$ 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 $N$ 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 $N+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 1$. Suppose $r_1$ 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^... |
[
{
"content": "Question: Find the number of ordered pairs of positive integers $(a,b)$ such that $a \\le b$ and $a^b = b^a$.\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 posit... |
[
{
"content": "Question: Let $P(x) = x^3 - 9x^2 + 23x - 15$. The roots of $P(x)$ are $a, b, c$. Find the value of $\\frac{1}{a^2+1} + \\frac{1}{b^2+1} + \\frac{1}{c^2+1}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $N$ be the number of ways to arrange the letters in the word \"MISSISSIPPI\" such that no two I's are adjacent. 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 fi... |
[
{
"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 triangle $ABC$, $AB=7$, $BC=8$, and $CA=9$. 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: Let $P(x) = x^3 - 3x^2 + 2x - 1$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $(r_1+1)(r_2+1)(r_3+1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of $(... |
[
{
"content": "Question: Circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A line through $A$ intersects $\\omega_1$ at $C$ and $\\omega_2$ at $D$. Let $P$ be a point on $\\omega_1$ such that $PB$ is tangent to $\\omega_2$ at $B$. If $AC=3$, $AD=5$, and $PB=6$, find the length of $PC$.\nPlease... |
[
{
"content": "Question: A bag contains 3 red, 4 blue, and 5 green marbles. If three marbles are drawn randomly without replacement, what is the probability that at least two of them are green?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "... |
[
{
"content": "Question: What is the smallest positive integer $n$ such that $n!$ is divisible by $2^{20}$?\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: 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: 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: 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 $a, b, c$ 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: How many ordered pairs of integers $(x, y)$ satisfy $x^2 + y^2 < 100$ and $x \\equiv 2 \\pmod{3}$ and $y \\equiv 1 \\pmod{3}$?\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 sequence is defined by $a_1 = 1$ and $a_{n+1} = \\frac{a_n}{1+na_n}$ for $n \\ge 1$. Find the value of $a_{2023}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this sequenc... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. 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{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to ... |
[
{
"content": "Question: A circle is inscribed in a rhombus with side length 10. If the ratio of the lengths of the diagonals of the rhombus is $3:4$, what is the radius of the inscribed circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"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: 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 ratio of the area of triangle $ABF$ ... |
[
{
"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 $2$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"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 $S$ be the set of all positive integers $n$ for which $n^2 + 2024$ is a perfect square. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all pos... |
[
{
"content": "Question: In a certain game, a player rolls a standard six-sided die. If the die shows a 1 or a 2, the player moves forward 1 space. If it shows a 3 or a 4, the player moves forward 2 spaces. If it shows a 5 or a 6, the player moves forward 3 spaces. What is the probability that after exactly 3 ro... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = 2$, $P(1) = 3$, and $P(2) = 6$. 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: In triangle $ABC$, $AB=5$, $BC=7$, and $AC=6$. 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 $AIB$ 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 n² + 20... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. 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{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to ... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2023 such that $n^2 + 2n + 12$ is a multiple of 11.\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 positiv... |
[
{
"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 $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"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 $E$ be a point on $AC$ such that $BE$ bisects $\\angle ABC$. Find the length of $DE$.\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 the sum of the digits of $N$ is 12. Find the sum of the digits of $N^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(1)=1$, $P(2)=2$, and $P(3)=4$. 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 need t... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 2$ 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 to figure out how many positive... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x < 1000$ such that $x^2 + 7x + 1$ is divisible by $11$. 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: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. 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{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to ... |
[
{
"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 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 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: In a certain game, a player starts with $1$ dollar. In each round, the player flips a fair coin. If it lands heads, the player doubles their money. If it lands tails, the player loses $1$ dollar. The game ends when the player has $0$ dollars or $4$ dollars. What is the probability that t... |
[
{
"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 $N+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, l... |
[
{
"content": "Question: A group of $6$ friends are sitting around a circular table. Each friend is wearing a hat that is either red or blue. How many distinct arrangements of hat colors are there such that no two adjacent friends are wearing red hats?\nPlease reason step by step, and put your final answer withi... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. Let $M$ be the midpoint of $BC$. A point $P$ is chosen on $AM$ such that $BP$ bisects $\\angle ABC$. Find the ratio $AP/PM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"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 $k \\ge 1$ such that $a_k > 2023$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. Let $D$ be a point on $BC$ such that $AD$ is an 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: Circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A line through $A$ intersects $\\omega_1$ at $P$ and $\\omega_2$ at $Q$, where $A$ is between $P$ and $Q$. The tangents to $\\omega_1$ and $\\omega_2$ at $P$ and $Q$ respectively intersect at $T$. If $AP=3$, $AQ=5$, and... |
[
{
"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: 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 $AD$ such that $BE \\perp AD$. Find the length of $DE$.\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 $\\angle BAC$. 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 ratio of the area of triangle $ABF$ ... |
[
{
"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: Regular hexagon $ABCDEF$ has side length 6. A point $P$ is chosen uniformly at random inside the hexagon. What is the probability that the distance from $P$ to vertex $A$ is greater than 3?\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 $n^2$ 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: How many positive integers $n$ less than or equal to 1000 are there such that $n^2 + 2n$ is a multiple of $n+4$?\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: A point $P$ is chosen randomly in the interior of a square $ABCD$ with side length 4. 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 diagonal $AC$?\nPlease reason step by step, and put your final answer withi... |
[
{
"content": "Question: In triangle $ABC$, $AB=10$, $BC=17$, and $CA=21$. 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 $N = 1! + 2! + 3! + \\dots + 2023!$. 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>\nOkay, so I need to find the remainder when N = 1! + 2! + 3! + ... + 202... |
[
{
"content": "Question: Two circles with radii 9 and 16 are externally tangent to each other. A common external tangent touches the circles at points $A$ and $B$. Find the length of the segment $AB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"cont... |
[
{
"content": "Question: Let $N = 2^{2023} \\cdot 3^{2024} \\cdot 5^{2025}$. 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 div... |
[
{
"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: Let $S$ be the set of all positive integers $n$ such that $n$ has exactly three positive divisors. If the sum of these three divisors is $S_n$, find the sum of all $n < 100$ for which $S_n$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 6$. Find the sum of all real values of $x$ such that $f(f(f(x))) = 6$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve this problem where I have a function f(x)... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(1)=1$, $P(2)=2$, 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 nee... |
[
{
"content": "Question: Points $A, B, C, D$ are on a circle such that $AB=3$, $BC=4$, $CD=5$, and $DA=6$. Find the length of the diagonal $AC$.\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 th... |
[
{
"content": "Question: Two circles, $C_1$ and $C_2$, with radii $r_1=5$ and $r_2=3$ respectively, are externally tangent at point $A$. A common external tangent line $L$ touches $C_1$ at $B$ and $C_2$ at $D$. Find the area of triangle $ABD$.\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 $M$ be the midpoint of $BC$. A point $D$ is on $AC$ such that $BD$ bisects $\\angle ABC$. Find the length of $AM^2 - BD^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"co... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 5$. What is the smallest possible positive integer value of $P(5)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<t... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 20$ and $P(2) = 23$. What is the remainder when $P(0)$ is divided by $3$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see... |
[
{
"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: 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 I nee... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=2$, $P(1)=3$, and $P(2)=5$. 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, let me 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 $E$ 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{}.",
... |
[
{
"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... |
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