messages listlengths 2 2 |
|---|
[
{
"content": "Question: A triangle has side lengths 13, 14, and 15. A circle is inscribed in the triangle. A second circle is tangent to two sides of the triangle and to the inscribed circle. Find the radius of this second circle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"r... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 6$. For how many distinct integer values of $n$ is $f(f(n))$ a prime number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many distinct integer valu... |
[
{
"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 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: 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: Triangle $ABC$ has side lengths $AB=7$, $BC=8$, and $CA=9$. Let $M$ be the midpoint of $BC$. Let $D$ be a point on $AC$ such that $AD = 3DC$. Find the length of $MD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<t... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 12, the sum of the digits of $N$ is 12, and the number of divisors of $N$ is 12. Find the sum of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2023$ is a perfect square. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all pos... |
[
{
"content": "Question: Let $P(x)$ 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 $x$ less than or equal to 2023 such that $x^2 + 2x + 2$ is divisible by 5. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 1$ is a perfect cube?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive ... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ 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 $n$ divides $n^2+2n+24$ and $n$ 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 ... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 20n - 2020$ 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: 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, let's see. I need to find the sum of the coe... |
[
{
"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: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 = 25$ and $x^3 + y^3$ 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 the number of order... |
[
{
"content": "Question: A circle is inscribed in a rhombus. The diagonals of the rhombus are 10 and 24. Find the radius of the inscribed 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... |
[
{
"content": "Question: A committee of 5 people is to be chosen from a group of 6 men and 4 women. What is the probability that the committee has more women than men?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: Let $N$ be the 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: 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 $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},... |
[
{
"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: In triangle $ABC$, $AB=7$, $BC=8$, and $CA=9$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE$ bisects $\\angle ABC$. If $AD$ and $BE$ intersect at $I$, find the ratio of the area of triangle $ABI$ to the area of triangle $ABC$.\... |
[
{
"content": "Question: 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>\nOka... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $E$ be the point on $AC$ such that $DE$ is parallel to $AB$. Find the length of $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a multiple of 7, and the sum of the digits of $n$ is 10. What is the smallest integer in $S$?\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 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 $E$ be a point on $AC$ such that $BE$ is the angle bisector of $\\angle ABC$. If $AD$ and $BE$ intersect at $I$, find the ratio of the area of triangle ... |
[
{
"content": "Question: A sequence of integers $a_1, a_2, a_3, \\dots$ is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. How many terms in the first 1000 terms of this sequence are even?\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: 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 $S$ be the set of all positive integers $n$ such that $n$ has exactly three positive divisors. Find the sum of all elements in $S$ that are less than or equal to 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content":... |
[
{
"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: In triangle $ABC$, $AB=13$, $BC=14$, and $AC=15$. 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 $x$ such that $x \\le 2023$ and $x^2 + 2x + 2$ is divisible by $5$. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many positi... |
[
{
"content": "Question: Let $P(x)$ 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 $P(1)=1$, $P(2)=2$, and $P(3)=4$. If $P(4)=k$, find the sum of all possible integer values of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: Let $N = 1! + 2! + 3! + \\dots + 2023!$. Find the remainder when $N$ is divided by 100.\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 $N$ be the number of positive integers $x$ such that $x^2 + 2024$ is a perfect square. Find the sum of all such $x$.\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 p... |
[
{
"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 area of triangle $ABF$.\nPlease reason... |
[
{
"content": "Question: How many positive integers less than 1000 are multiples of 3 or 5 but not multiples 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 integers less than 100... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=3$. Prove that $\\frac{a^2}{b+c} + \\frac{b^2}{c+a} + \\frac{c^2}{a+b} \\ge \\frac{3}{2}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, $P(1)=2$, and $P(2)=4$. Find the smallest possible value of $|P(3)|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. ... |
[
{
"content": "Question: Let $P(x)$ 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 $\\angle A$. 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 $AB... |
[
{
"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 if the player's money reaches 0 or 8 dollars. What is the probability that the pla... |
[
{
"content": "Question: A bag contains 5 red balls and 7 blue balls. Two balls are drawn randomly without replacement. What is the probability that the two balls drawn are of different colors?\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 $E$ 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, so I need to figure out how many positive integers up to 2023 ... |
[
{
"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 distinct prime factors of the sum of all such $n$ where $n < 100$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of $\\frac{a^2}{b} + \\frac{b^2}{c} + \\frac{c^2}{a}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: How many ordered pairs of positive integers $(a, b)$ are there such that $a \\le b$ and $a^2 + b^2 = 2025$?\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 ... |
[
{
"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 such $n$ where $n < 100$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = 1$, $P(1) = 2$, and $P(2) = 4$. If $P(3) = k$, what is the smallest possible integer value of $k$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"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 the sum of the first 2023 terms, $\\sum_{k=1}^{2023} a_k$, is divided by 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: How many positive integers $n$ less than 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 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 certain game, a player starts with $1 and flips a fair coin. If it lands heads, their money doubles. If it lands tails, they lose $1. What is the probability that the player will have exactly $4 after exactly 3 flips?\nPlease reason step by step, and put your final answer within \\b... |
[
{
"content": "Question: Let $P(x) = x^3 - 6x^2 + 11x - 6$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $r_1^2 + r_2^2 + r_3^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of $... |
[
{
"content": "Question: A sequence is defined by $a_1 = 3$ and $a_{n+1} = \\frac{1}{2}a_n + 1$ for $n \\ge 1$. Find the value of $a_5$.\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 a₅ in this ... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the last two digits of $a_{2023}$.\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: Find the number of ordered pairs of positive integers $(a, b)$ 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 find the number of ordered pai... |
[
{
"content": "Question: A circle is inscribed in a rhombus with side length 10 and one angle of $60^°$. 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... |
[
{
"content": "Question: Two distinct numbers are chosen randomly 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": "<think>\nOkay, so I nee... |
[
{
"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: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. 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 polyn... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 = 25$ and $x^3 + y^3$ 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 the number of order... |
[
{
"content": "Question: Suppose that $a, b, c$ are 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 n... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ divides $N$ and $k^2$ also 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 positi... |
[
{
"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 positive divisors. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 12$ is divisible by $n+4$?\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 posi... |
[
{
"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 value of $P(10)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\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 $AD$. What is the length of $BM$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"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: A triangle has side lengths $a, b, c$. If $a^2+b^2+c^2 = 200$ and the area of the triangle is $10√3$, find the perimeter of the triangle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: Let $N = 2^{2024}$. 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: A circle is inscribed in a rhombus with side length 6. The ratio of the area of the circle to the area of the rhombus is $\\frac{\\pi}{8}$. Find the acute angle of the rhombus in degrees.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. 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 $a, b, c$ 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 $ABC$, $AB=13$, $BC=14$, 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: 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 $ABC$, $AB=5$, $BC=7$, 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: Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 5$. Find the smallest possible value of $|P(0)|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = -2$, $P(1) = 3$, and $P(2) = 10$. What is the remainder when $P(2023)$ is divided by $2023^2 - 2023$?\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 $P(1) = 2$, $P(2) = 3$, 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: 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 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: How many positive integers $n$ less than or equal to 1000 are there such that $n^2 + 2n + 13$ 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 how many positive int... |
[
{
"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 $n^2 + n + 1$ is divisible by $n+2$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay... |
[
{
"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 $N = 2^{2024}$. 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, let me try to figure out this proble... |
[
{
"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 $AD$ such that $AE = 3ED$. Find the area of triangle $BEC$.\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 a perfect cube?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive ... |
[
{
"content": "Question: 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: How many positive integers $n$ less than or equal to 1000 have the property that the sum of the digits of $n$ is 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... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.