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: A sequence is defined by $a_1 = 3$ and $a_{n+1} = a_n^2 - 2a_n + 2$ for $n \\ge 1$. Find the last two digits of $a_{2024}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the last two... |
[
{
"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 at least 3 men, given that it has at least 1 woman?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"conten... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n < 1000$ such that $n$ is divisible by 6 and the sum of its digits is 9. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of $\\frac{a^2}{b} + \\frac{b^2}{c} + \\frac{c^2}{a}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: A triangle has side lengths 13, 14, and 15. The angle bisector of the largest angle divides the opposite side into two segments. Find the length of the shorter of these two segments.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 20 and the sum of the digits of $N$ is 20. Find the value of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find... |
[
{
"content": "Question: How many ways are there to arrange the letters of the word 'MISSISSIPPI' such that no two 'S's are adjacent?\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 ways there a... |
[
{
"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 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 integers n ... |
[
{
"content": "Question: A rectangle $ABCD$ has vertices $A=(0,0)$, $B=(6,0)$, $C=(6,4)$, and $D=(0,4)$. A point $P$ is chosen on side $CD$. Let $M$ be the midpoint of $AB$. If the area of triangle $APM$ is 10, what is the length of $DP$?\nPlease reason step by step, and put your final answer within \\boxed{}.",... |
[
{
"content": "Question: For positive real numbers $a, b, c$ 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 find t... |
[
{
"content": "Question: Suppose $x$ and $y$ are real numbers such that $x^2 + y^2 = 1$ and $x^3 + y^3 = 1$. What is the value of $x^4 + y^4$?\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 $ x^4... |
[
{
"content": "Question: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with integer coefficients. If $P(1) = 2$, $P(2) = 5$, and $P(3) = 10$, find the value of $P(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's se... |
[
{
"content": "Question: Two circles, $\\omega_1$ and $\\omega_2$, intersect at points $A$ and $B$. Let $CD$ be a common tangent to $\\omega_1$ and $\\omega_2$ with $C$ on $\\omega_1$ and $D$ on $\\omega_2$. If $AC=3$, $AD=4$, and $BC=x$, find $BD$ in terms of $x$.\nPlease reason step by step, and put your final... |
[
{
"content": "Question: In triangle $ABC$, $AB=7$, $BC=8$, and $CA=9$. Let $M$ be the midpoint of $BC$. A point $D$ is chosen on $AC$ such that the area of triangle $ADM$ is half the area of triangle $ABM$. Find the length of $CD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"... |
[
{
"content": "Question: A fair six-sided die is rolled repeatedly. What is the probability that the first time a '1' appears, it is immediately followed by a '2'?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fi... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $AC=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle A$. 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": "Question: Point $P$ is inside rectangle $ABCD$ such that $PA=5$, $PB=7$, and $PC=6$. Find $PD$.\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 PD where point P is inside rectangle... |
[
{
"content": "Question: Let $x, y$ be integers such that $x^2 - 4y^2 = 13$. Find the sum of all possible values of $x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all integer solutions (x, y) to the equa... |
[
{
"content": "Question: In triangle $ABC$, $D$ is a point on $BC$ such that $AD$ bisects $\\angle BAC$. If $AB=10$, $AC=15$, and $BD=6$, find the length of $CD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: How many positive integers less than or equal to 2023 are divisible by 3 or 5 but not by 7?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integers up to 2023 ... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k \\le N$ and $k$ divides $N^2$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integer... |
[
{
"content": "Question: A committee of 5 people is to be chosen from a group of 6 men and 4 women. If the committee must have at least 2 men and at least 1 woman, how many different committees can be formed?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"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 positive integers $n$ less than or equal to 1000 such that $n^2+1$ is divisible by $n+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many positive integers ... |
[
{
"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 triangle $ABC$, $AB=10$, $BC=12$, and $AC=14$. 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 t... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE \\perp AC$. Find the area of triangle $ADE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": ... |
[
{
"content": "Question: What is the smallest positive integer $n$ such that $n$ has exactly 12 positive divisors, and the sum of its divisors is 372?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the smalle... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum possible value of the expression $3x + 4y$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the maximum valu... |
[
{
"content": "Question: An equilateral triangle is inscribed in a circle of radius $R$. A smaller circle is inscribed in the equilateral triangle. What is the ratio of the area of the equilateral triangle to the area of the smaller circle?\nPlease reason step by step, and put your final answer within \\boxed{}.... |
[
{
"content": "Question: A rectangle has vertices at $(0,0)$, $(6,0)$, $(6,4)$, and $(0,4)$. A line with equation $y = mx + c$ passes through the point $(1,1)$ and divides the rectangle into two regions of equal area. Find the value of $m$.\nPlease reason step by step, and put your final answer within \\boxed{}.... |
[
{
"content": "Question: A sequence is defined by $a_1 = 3$ and $a_{n+1} = a_n^2 - 2a_n + 2$ for $n \\ge 1$. Find the last two digits of $a_{2024}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the last two... |
[
{
"content": "Question: In a certain game, a player starts at the origin $(0,0)$ and takes steps. Each step consists of moving one unit right or one unit up. What is the number of distinct paths from $(0,0)$ to $(4,3)$ that do not pass through the point $(2,2)$?\nPlease reason step by step, and put your final a... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 100 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 find the number of positive ... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 7 and the sum of the digits of $N$ is 16. What is the sum of the squares of the digits of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<th... |
[
{
"content": "Question: How many ways are there to arrange the letters in the word \"MISSISSIPPI\" such that no two 'S's are adjacent?\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 ways there... |
[
{
"content": "Question: Find the smallest positive integer $n$ such that $n$ leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 5 when divided by 7.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"content": "Question: A positive integer $N$ has exactly three prime factors. If $N = 2^a 3^b 5^c$ where $a, b, c$ are positive integers, and $N$ has 36 divisors, find the minimum possible value of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: In triangle $ABC$, $AB=7$, $BC=8$, and $CA=9$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\triangle A$. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, ... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $N$ is a divisor of $k^k$ and $k < N$?\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: In a certain school, there are 200 students. Each student takes at least one of three subjects: Math, Physics, and Chemistry. 120 students take Math, 100 take Physics, and 80 take Chemistry. If 50 students take exactly two subjects, and 20 students take all three subjects, how many stude... |
[
{
"content": "Question: How many positive integers less than or equal to 1000 are not divisible by 2, 3, or 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 up to 1000 aren'... |
[
{
"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 $I$ be the incenter of triangle $ABC$. Find the ratio of the area of triangle $ADI$ to the area of triangle $ABC$.\nPlease reason step by step, and put your final answer wi... |
[
{
"content": "Question: Six points are equally spaced on a circle. How many distinct triangles can be formed by choosing three of these points as vertices such that no two sides of the triangle are equal in length?\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 the sum of the digits of $n$ is 10 and $n$ is a multiple of 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: How many positive integers $n < 1000$ are there such that $n$ is divisible by 3 and the sum of the digits of $n$ 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 ou... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\geq 3$. Find the remainder when $a_{2024}$ is divided by 8.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I nee... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that the sum of its digits is 2024 and all digits of $N$ are even. Find the number of digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me ... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2023n + 1$ 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 al... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $1 \\le n \\le 100$ and $n^2 + n + 1$ is divisible by 7. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 \\le 25$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of ordered pairs of integers (x, y) such... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$. Let $E$ be a point on $AC$ such that $BE$ is an angle bisector of $\\angle B$. Find the length of $DE$.\nPlease reason step by step, and put your final answer within \\bo... |
[
{
"content": "Question: Let $P(x)$ be a polynomial of degree 3 such that $P(1) = 1$, $P(2) = 4$, $P(3) = 9$, and $P(4) = 11$. Find $P(5)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find P(5) where P(x) is a ... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 7 and the sum of its digits is 19. What is the sum of the squares of the digits of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, and $N$ has exactly 14 divisors. Find the sum of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need ... |
[
{
"content": "Question: A positive integer $n$ has exactly 6 divisors. If the sum of its divisors is 124, what is the value of $n$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find a positive integer n that ha... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 15n + 27$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of positive integers n such that n² ... |
[
{
"content": "Question: 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 $CA=15$. A point $D$ is chosen on $BC$ such that the inradii of triangles $ABD$ and $ACD$ are equal. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<t... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $AC=8$. Let $M$ be the midpoint of $BC$. A point $P$ is on $AM$ such that $AP = 2PM$. Find the length of $BP$.\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 such that $P(x^2+1) = x^4 + 5x^2 + c$ for some constant $c$. If the sum of the roots of $P(x)$ is $6$, find the value of $c$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=3$. 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 $ABC$ be an equilateral triangle with side length 6. Let $D$ be a point on $BC$ such that $BD=2$ and $E$ be a point on $AC$ such that $CE=2$. Let $AD$ and $BE$ intersect at point $F$. Find the area of triangle $ABF$.\nPlease reason step by step, and put your final answer within \\box... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2024 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 the number of positiv... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $M$ be the midpoint of $BC$. A circle passes through $A$ and $M$ and is tangent to $AC$ at $A$. Find the radius of this circle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: An equilateral triangle is inscribed in a circle of radius 6. Three smaller circles are then inscribed, each tangent to two sides of the triangle and to the large circle. Find the sum of the areas of these three smaller circles.\nPlease reason step by step, and put your final answer with... |
[
{
"content": "Question: Let $P(x) = x^3 - 6x^2 + ax - 6$ be a polynomial with integer coefficients. If all roots of $P(x)$ are positive integers, find the sum of all possible values of $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<th... |
[
{
"content": "Question: Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 1000 such that $n^2 + 1$ is divisible by $13$.\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: Find the number of positive integers $x$ less than 1000 such that $x^2 + 7x + 10$ is divisible by 13.\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 x less ... |
[
{
"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 $\triangle ABC$. Let $E$ be a point on $AC$ such that $BE$ is the median to $AC$. If $AD$ and $BE$ intersect at $P$, find the ratio $\frac{AP}{PD}$.\nPlease reason step b... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x - 1$. For how many integers $n$ is $f(f(n)) = 2$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many integer values of n satisfy the equation f(f(n)) =... |
[
{
"content": "Question: In a certain game, a player starts with $N$ points. In each round, the player either gains 3 points with probability $1/3$ or loses 1 point with probability $2/3$. What is the probability that the player's score eventually reaches 0, given that they start with 1 point? (Assume the game e... |
[
{
"content": "Question: Let $f(n)$ be the sum of the digits of $n$. For example, $f(123) = 1+2+3=6$. Find the smallest positive integer $n$ such that $f(n) + f(n+1) = 100$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"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 $N$ be the smallest positive integer such that $N$ is a multiple of 7, and the sum of the digits of $N$ is 16. What is the sum of the last two digits of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE$ 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: 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 $BE \\perp AC$. Find the length of $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"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, so I need t... |
[
{
"content": "Question: Ten friends, $A_1, A_2, \\dots, A_{10}$, are seated around a circular table. Each friend has a unique hat. They pass their hats to the person on their right. Then, each person randomly chooses one of the two hats currently available to them (their original hat, or the hat passed to them)... |
[
{
"content": "Question: Let $S$ be the set of all possible values of $n$ for which the polynomial $P(x) = x^3 - (n-1)x^2 + (n+4)x - 2n$ has three distinct integer roots. Find the sum of the elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"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: A positive integer $n$ is called \"digit-sum perfect\" if the sum of its digits is equal to the sum of the digits of its prime factors (counted with multiplicity). For example, 12 is digit-sum perfect because $1+2=3$ and $12=2^2 \\cdot 3$, so $2+2+3=7 \\ne 3$. Find the sum of all digit-s... |
[
{
"content": "Question: Let $a, b, c$ be positive integers such that $a+b+c=20$. What is the maximum possible value of $a^2b^3c^4$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the maximum value of $ a^2b^... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that the sum of its digits is 2024 and all digits of $N$ are distinct. What is the value of the units digit of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: A positive integer $n$ has exactly 12 divisors. If the sum of the divisors of $n$ is 2520, find the value of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this problem. S... |
[
{
"content": "Question: Find the number of ordered pairs of positive integers $(a,b)$ such that $a^2 + b^2 = 2025$.\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 of positive inte... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $xy+yz+zx=1$. Find the minimum value of $\\frac{1}{x^2+1} + \\frac{1}{y^2+1} + \\frac{1}{z^2+1}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: How many positive integers less than 1000 have the property that the sum of their digits 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 inte... |
[
{
"content": "Question: Find the number of pairs of positive integers $(a,b)$ such that $a^2b^2 + a^2 + b^2 = 2024$.\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 pairs of positive integers (a... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 \\leq 100$ and $x \\equiv y \\pmod{3}$.\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 number of... |
[
{
"content": "Question: Let $f(x) = x^3 - 3x$. For how many distinct real values of $x$ is $f(f(x)) = 2$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many distinct real values of x satisfy the e... |
[
{
"content": "Question: Let $n$ be the smallest positive integer such that $n$ has exactly 12 positive divisors, and the sum of its divisors is $S(n) = 288$. Find the value of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOka... |
[
{
"content": "Question: Let $ABCD$ be a convex quadrilateral such that $AB=6$, $BC=7$, $CD=8$, and $DA=9$. If the diagonals $AC$ and $BD$ intersect at point $P$, and $\triangle APB$ has area 12, what is the area of $\triangle CPD$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"... |
[
{
"content": "Question: In triangle $ABC$, point $D$ is on $BC$ such that $AD$ bisects $\\angle BAC$. If $AB=6$, $AC=10$, and the area of triangle $ABD$ is 12, find the area of triangle $ABC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2+n+1$ is a prime number. Find the largest integer in $S$ that is less than or equal to 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, ... |
[
{
"content": "Question: In a certain game, a player starts with $100. If the player flips a fair coin and it lands on heads, they gain 10. If it lands on tails, they lose 5. What is the probability that the player's money reaches exactly $150 before it reaches $50?\nPlease reason step by step, and put your fina... |
[
{
"content": "Question: A standard deck of 52 cards is shuffled. What is the probability that the top 5 cards are all of the same suit (e.g., all clubs, or all diamonds, etc.)?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
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