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 $P(x) = x^3 - 3x^2 + ax - 4$ be a polynomial. If the roots of $P(x)$ are $r_1, r_2, r_3$ and they satisfy $r_1^2 + r_2^2 + r_3^2 = 11$, find the value of $a$.\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 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 find the value of P(3) where P is a polynomial su... |
[
{
"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 $ABC$, $AB=7$, $BC=8$, 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 $a, b, c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the ... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ is a divisor of $N^2$ and $k$ is not a divisor of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out ... |
[
{
"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: Two distinct numbers are chosen randomly and independently from the set $\\{1, 2, 3, \\dots, 10\\}$. What is the probability that their product is a multiple of 3?\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's see. I need to find the polynomial P(x... |
[
{
"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's see. The problem is asking for the number of positive divis... |
[
{
"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: 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 the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $N = 1! + 2! + 3! + \\dots + 2023!$. Find the remainder when $N$ is divided by 12.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the remainder when N = 1! + 2! + 3! + ... + 2023!... |
[
{
"content": "Question: Let $N = 2^{10} \\cdot 3^{10}$. 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: Two distinct numbers are chosen randomly and independently from the set $\\{1, 2, 3, \\dots, 10\\}$. What is the probability that their product is a multiple of 3?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thin... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ divides $N$ and $k+1$ divides $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive in... |
[
{
"content": "Question: Let $P(x)$ 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, so I need to find the value of P(3) where P is a polynomial su... |
[
{
"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 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 $N = 2^{10} \\cdot 3^{10}$. 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's see. The problem is asking for the number of positive divis... |
[
{
"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 $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ is a divisor of $N^2$ and $k$ 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 h... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. 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... |
[
{
"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 $P(x) = x^3 - 9x^2 + 23x - 15$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $(r_1+1)(r_2+1)(r_3+1)$.\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 $S$ 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 sequences of length 5, consisting of digits from {1, 2, 3, 4} are there such that no two adjacent digits are the same, and the first digit is not 1?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOk... |
[
{
"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 $P(x)$ 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 $P(0)=2023$ and $P(1)=2024$. What is the remainder when $P(10)$ is divided by 9?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see.... |
[
{
"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 $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle A$. 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 $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. 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: Let $N$ be the number of positive integers $x$ such that $1 \\le x \\le 2023$ 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... |
[
{
"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 $M$ be the midpoint of $BC$. Find the length of the altitude from $M$ to $AC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a multiple of 7, and the sum of the digits of $n$ is 10. Find the smallest element in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay... |
[
{
"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: 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 $P(x)$ 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 a triangle $ABC$, $AB=5$, $BC=7$, and $AC=8$. 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 ... |
[
{
"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 $3:4$, find the area of the circle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: How many positive integers less than 1000 are multiples of 3 or 5 but not multiples of 15?\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 10... |
[
{
"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 $BC$. Find the length of $DM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: How many ordered pairs of positive integers $(a,b)$ are there such that $a+b=100$ and $\\text{gcd}(a,b) = 5$?\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 ordered pai... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 1$. Suppose $r_1, r_2, r_3, r_4$ are distinct real numbers such that $f(f(r_1)) = f(f(r_2)) = f(f(r_3)) = f(f(r_4)) = 2$. Find the sum $r_1 + r_2 + r_3 + r_4$.\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: Find the number of positive integers $n$ less than or equal to 2023 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 the number of positive i... |
[
{
"content": "Question: How many positive integers $n$ are there 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 figure out how many positive integers n satisfy the cond... |
[
{
"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 smallest such $N$.\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 $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$. What is the probability that the distance from $P$ to the nearest side of the square is less than $1/4$?\nPlease reason step by step, and put your final answer wi... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random 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": "<thi... |
[
{
"content": "Question: How many positive integers less than 2024 are relatively prime to 2024?\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 2024 are relatively p... |
[
{
"content": "Question: Two fair six-sided dice are rolled. What is the probability that the product of the two numbers rolled is a prime number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out the prob... |
[
{
"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: 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 $x, y, z$ be positive real numbers such that $x+y+z=1$. What is the minimum value of $\\frac{1}{xy} + \\frac{1}{yz} + \\frac{1}{zx}$?\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 $n$ 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 i... |
[
{
"content": "Question: A sequence is defined by $a_1=1$, $a_2=2$, and $a_n = \\frac{a_{n-1}^2 + 1}{a_{n-2}}$ for $n \\ge 3$. Find the value of $a_6$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find t... |
[
{
"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: 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: A circle is inscribed in a rhombus with diagonals of length $12$ and $16$. Find the area of the circle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the area of a circle that's insc... |
[
{
"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 $a_{2024}$ is divided by $7$.\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 positive integers $x$ such that $1 \\le x \\le 2024$ 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 the numb... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2023 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: 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: How many positive integers $n$ less than or equal to 1000 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: Find the number of positive integers $n$ less than or equal to 1000 such that $n^2 + 7n + 10$ is divisible by $n+2$.\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... |
[
{
"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 $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 2x + 2$ is divisible by $x+1$. 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... |
[
{
"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: A circle is inscribed in a rhombus with diagonals of length 12 and 16. What is the radius of the circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the radius of a circle inscribe... |
[
{
"content": "Question: An ant starts at a vertex of a regular tetrahedron. At each step, the ant moves from its current vertex to one of the three adjacent vertices with equal probability. What is the probability that the ant is back at its starting vertex after exactly 4 steps?\nPlease reason step by step, an... |
[
{
"content": "Question: A sequence is defined by $a_1 = 3$, $a_2 = 5$, 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 ne... |
[
{
"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 $AFB$ ... |
[
{
"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 less than 1000 are divisible by 3 but not 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 integers less than 1000 are divisible... |
[
{
"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 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 smallest such $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "... |
[
{
"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 $ABC$, $AB=13$, $BC=14$, 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 $P(x)$ be a polynomial with integer coefficients such that $P(1) = 2$ and $P(2) = 5$. 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, let's... |
[
{
"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: 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: 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 $n$ 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 the point on $BC$ such that $AD$ is the angle bisector of $\\angle A$. Let $E$ be the 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 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 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 $n^2 + n + 1$ 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 find how many positive integer... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 5$ and $P(5) = 13$. What is the remainder when $P(2023)$ is divided by 4?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see... |
[
{
"content": "Question: Let $ABC$ be an equilateral triangle with side length 6. Let $P$ be a point in the interior of the triangle. The distances from $P$ to the sides $AB$, $BC$, and $CA$ are $d_1, d_2, d_3$ respectively. Find the minimum possible value of $d_1^2 + d_2^2 + d_3^2$.\nPlease reason step by step,... |
[
{
"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 $r_1^2 + r_2^2 + r_3^2$.\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: 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 median to $AC$. Let $F$ be the intersection of $AD$ and $BE$. Find the ratio of the area of triangle $ABF$ to... |
[
{
"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: How many pairs of positive integers $(a,b)$ are there such that $a! + b! = 3^k$ for some integer $k \\ge 1$?\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 pairs of pos... |
[
{
"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 $P(x)$ 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 ... |
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