messages listlengths 2 2 |
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[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial 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, let... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 2$ and $P(2) = 7$. What is the remainder when $P(2023)$ is divided by $2021$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x + 2$. Define a sequence by $a_0 = 0$ and $a_{n+1} = f(a_n)$ for $n \\ge 0$. Find the smallest integer $N$ such that $a_N > 2023$.\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 $P(1)=2$, $P(2)=3$, and $P(3)=5$. Find the remainder when $P(x)$ is divided by $(x-1)(x-2)(x-3)$.\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 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: 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: Let $P(x)$ be a polynomial with integer coefficients such that $P(1)=1$, $P(2)=2$, and $P(3)=4$. If $P(4)=k$, find the smallest possible value of $|k|$ if $k$ is an integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"conten... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 100$ 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 + 100... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 2024$ 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 $N = 2^{10} \\cdot 3^{10}$. 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: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: A point $P$ is chosen uniformly at random from the interior of a square with vertices at $(0,0)$, $(1,0)$, $(1,1)$, 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 nearest vertex of the square?\nP... |
[
{
"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: 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 $a_k > 2024$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with real coefficients such that $P(x^2+1) = (P(x))^2+1$ for all real $x$. If $P(0) = 0$, find the value of $P(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. ... |
[
{
"content": "Question: Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. Suppose that $P(1)=10$, $P(2)=20$, and $P(3)=30$. Find the value of $P(12) + P(-8)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"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, let me try ... |
[
{
"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: An arithmetic sequence has first term $a_1 = 7$ and common difference $d = 3$. A geometric sequence has first term $b_1 = 4$ and common ratio $r = 2$. Let $S_n = a_n + b_n$. Find the smallest positive integer $n$ such that $S_n > 1000$.\nPlease reason step by step, and put your final ans... |
[
{
"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 bag contains 5 red balls and 3 blue balls. Two balls are drawn randomly without replacement. What is the probability that at least one of the balls drawn is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thin... |
[
{
"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 less than o... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^3$ has exactly 10 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 all positi... |
[
{
"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 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 $\\frac{AF}{FD}$.\nPlease... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $x^2 + 2023x + 2024$ is a perfect square. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of positive inte... |
[
{
"content": "Question: Let $P(x) = x^3 - 10x^2 + 29x - k$ for some integer $k$. If the three roots of $P(x)$ are integers that form an arithmetic progression, find the value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"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 $x^2 + y^2 \\le 1$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"content": "Question: A point $P$ is chosen uniformly at random from the interior of a square with vertices at $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$. What is the probability that the distance from $P$ to the nearest side of the square is less than or equal to $1/4$?\nPlease reason step by step, and put your ... |
[
{
"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 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 $N = 2^{10} \\cdot 3^{10}$. 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 N² are perf... |
[
{
"content": "Question: Let $P(x) = x^3 - 9x^2 + 23x - 15$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $(r_1+r_2)^2 + (r_2+r_3)^2 + (r_3+r_1)^2$.\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 $ABC$, $AB=13$, $BC=14$, 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: An ant starts at the origin $(0,0)$ and makes a sequence of moves. Each move consists of moving one unit right or one unit up. The ant stops when it first reaches a point $(x,y)$ such that $y=2x$. What is the probability that the ant stops at the point $(2,4)$?\nPlease reason step by ste... |
[
{
"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 $S$ be the set of all positive integers $n$ such that $n$ has exactly three positive divisors. If $n_1 < n_2 < \\dots < n_k$ are the elements of $S$ less than 100, find the value of $n_1 + n_2 + \\dots + n_k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2023$ 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: How many positive integers $n$ less than or equal to 1000 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: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. 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: 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: 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 bag contains 5 red balls and 3 blue balls. Two balls are drawn randomly without replacement. What is the probability that at least one of the drawn balls is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thin... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this problem. So, w... |
[
{
"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: 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, let's see. The problem is asking how m... |
[
{
"content": "Question: How many positive integers less than 1000 have the property that the sum of their digits is a multiple of 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 int... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2+n+1$ is divisible 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 integ... |
[
{
"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 pos... |
[
{
"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: 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 $n$ less than or equal to 2023 are there such that $n^2 + 2n + 2$ is divisible by 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: 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 $a+b+c=1$. What is the maximum possible value of $ab+bc+ca-abc$? Express your answer as a simplified fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\n... |
[
{
"content": "Question: Let $P(x) = x^3 - 3x^2 + 2x - 1$. If $r_1, r_2, r_3$ are the roots of $P(x)=0$, 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: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all coefficient... |
[
{
"content": "Question: 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 find how many positive intege... |
[
{
"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 of numbers $a_1, a_2, a_3, \\dots$ is defined by $a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the sum of all $n$ such that $1 \\le n \\le 100$ and $a_n$ is a multiple of $3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",... |
[
{
"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: 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 $7$?\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 $n^2 + 2n + 2$ 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 $S$ be the set of all positive integers $n$ such that $n^2 + 2024$ is a perfect square. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all pos... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ has exactly 9 positive divisors. Find the sum of all prime numbers $p$ such that $p$ is a divisor of at least one number in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"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: 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: In triangle $ABC$, $AB=3$, $BC=4$, and $CA=5$. Let $D$ be a point on $BC$ such that $AD$ bisects angle $A$. 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": "<th... |
[
{
"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 $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\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 value of P(3) where P is a polyn... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n$ is a divisor of $20!$ and $n$ has exactly 20 divisors.\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: Points $A=(0,0)$, $B=(4,0)$, and $C=(0,3)$ form a triangle. A line $L$ passes through the incenter of triangle $ABC$ and is parallel to $BC$. Find the length of the segment of $L$ that lies inside triangle $ABC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"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 a certain game, a player starts with $100. If the player rolls a standard six-sided die, they add the number rolled to their score. If they flip a fair coin, they multiply their score by 2 if it's heads, or divide by 2 if it's tails. What is the probability that after one die roll and... |
[
{
"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 side $AC$. Let $F$ be the intersection of $AD$ and $BE$. Find the ratio of the area of triangle $... |
[
{
"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 $\\frac{AF}{FD}$.\nPlease reas... |
[
{
"content": "Question: How many ordered pairs of integers $(x, y)$ satisfy $x^2 + y^2 < 25$?\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 ordered pairs of integers (x, y) satisfy the inequa... |
[
{
"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... |
[
{
"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: What is the smallest positive integer $n$ 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 smallest positive integer n such that n² + 202... |
[
{
"content": "Question: In a certain game, a player starts with $N$ points. In each round, the player flips a fair coin. If it lands heads, the player's score is doubled. If it lands tails, the player's score is halved. What is the probability that after 4 rounds, the player's score is exactly $N$?\nPlease reas... |
[
{
"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 $P(x)$ be a polynomial such that $P(x^2) = x^4 + 5x^2 + 6$ for all real numbers $x$. 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... |
[
{
"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 $N$ 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 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: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 1$ is divisible by $n+1$ and $n+2$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many p... |
[
{
"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 this expression?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"cont... |
[
{
"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, so I nee... |
[
{
"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 positive integers $n$ such that $n^2 + 10n + 25$ is a perfect square. Find the sum of all prime numbers $p$ such that $p$ divides at least one element of $S$.\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: If $x$ and $y$ are positive real numbers such that $x^2 + y^2 = 1$ and $x^4 + y^4 = \\frac{1}{2}$, find the value of $x^6 + y^6$.\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 $x, y, z$ 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 $n!$ 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 find the num... |
[
{
"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: Find the number of positive integers $n$ less than or equal to 2023 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 find the number of positive... |
[
{
"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: Two distinct numbers are chosen randomly and independently 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... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $E$ be the foot of the altitude from $B$ to $AC$. Let $F$ be the intersection of $AD$ and $BE$. Find the area of triangle $ABF$.\nPlease reason step by step, and put your final... |
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