messages listlengths 2 2 |
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[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n$ divides $20!$ and $n+1$ divides $21!$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of positive integers $ n $ such th... |
[
{
"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 $P(x+1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the polynomial P(x+1) given that P(x² + 1) equals x⁴ + 4x² + 5. ... |
[
{
"content": "Question: Find the smallest positive integer $n$ such that $n$ has exactly 12 positive divisors, and the sum of its divisors is 360.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the smallest positive ... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 1$. Suppose $r_1, r_2, r_3, r_4$ are distinct real numbers such that $f(f(r_1)) = f(f(r_2)) = f(f(r_3)) = f(f(r_4)) = 5$. 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: Let $P(x)$ be a polynomial of degree 3 such that $P(1)=1, P(2)=4, P(3)=9,$ and $P(4)=15$. Find $P(5)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find P(5) for a cubic polynomial P(x) that sati... |
[
{
"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": "Okay,... |
[
{
"content": "Question: Let $ABCD$ be a square with side length 6. Point $E$ is on $CD$ such that $CE=2$. Point $F$ is on $BC$ such that $BF=3$. Let $P$ be the intersection of $AE$ and $DF$. Find the area of triangle $APF$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n^2 + 2023$ is a perfect square. Find the value of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of positive integers... |
[
{
"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 21. 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": "O... |
[
{
"content": "Question: Let $f(x) = x^2 + ax + b$. If $f(x+1) - f(x) = 2x+3$ for all real numbers $x$, find the value of $f(5)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let me try to figure out this problem. So, we have a quadr... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, $P(1)=2$, and $P(2)=4$. If $P(3)=k$, find the minimum possible value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with real coefficients such that $P(x^2+1) = (P(x))^2+1$ for all real $x$. Given that $P(0) = 0$, find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ 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 $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that its graph passes through the points $(1, -1)$ and $(3, -1)$, and its maximum value is $7$. Find the value of $a+b+c$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: Find the smallest positive integer $n$ such that $2023n$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the smallest positive integer n such that 2023n is a perfect square... |
[
{
"content": "Question: How many ways are there to arrange the letters of the word \"MATHEMATICS\" such that no two A's are adjacent and no two T's are adjacent?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to figure out h... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $1 \\le x \\le 2023$ and $x^2 + x + 1$ is divisible by $13$. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positive... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 = 100$ and $x > y$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of ordered pairs of integers (x, y) suc... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I n... |
[
{
"content": "Question: A sequence is defined by $a_1=1$ and $a_{n+1} = a_n + n^2$ for $n \\ge 1$. Find the value of $a_{10}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the value of a₁₀ for this sequence. The se... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_{n+2} = a_{n+1} + a_n$ for $n \\ge 1$. Find the sum of the first 10 terms whose value is a multiple of 3.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so ... |
[
{
"content": "Question: A positive integer $n$ is called \"lucky\" if the sum of its digits is a divisor of $n$. For example, $12$ is lucky because $1+2=3$ and $3$ divides $12$. Find the smallest lucky number greater than $100$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"rol... |
[
{
"content": "Question: In a triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $1 \\le n \\le 2023$ and $n^2 + 2n + 2$ is divisible by $5$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positive integers ... |
[
{
"content": "Question: In a certain game, a player starts with 1 dollar. In each round, the player flips a fair coin. If it's heads, the player doubles their money. If it's tails, the player loses 1 dollar. The game ends when the player has 0 dollars or 4 dollars. What is the probability that the player ends t... |
[
{
"content": "Question: Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $AC=8$. Let $M$ be the midpoint of $BC$. Find the length of the altitude from $A$ to $BC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the l... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $100 \\le n \\le 999$ and the sum of the digits of $n$ is $10$. Find the number of elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to f... |
[
{
"content": "Question: A circle is inscribed in a rhombus. The diagonals of the rhombus are 10 and 24. What is the radius of the inscribed circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the radius of a circl... |
[
{
"content": "Question: A sequence is defined by $a_1 = 3$ and $a_{n+1} = \\frac{a_n^2 + 5}{2a_n}$ 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": "Okay, let's see. I need to find the value of a₅ in th... |
[
{
"content": "Question: Find the number of positive integer solutions $(a, b, c)$ to the equation $a+b+c=10$ such that $a, b, c$ are distinct.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of positive int... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 10, and the sum of the digits of $N$ is 20. Find 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": "Okay... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n \\le 2023$ such that $n^2 + n + 1$ is a perfect square. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positive integers n ≤ 20... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find $P(x+1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find P(x + 1) where P is a polynomial such that P(x² + 1) = x⁴ + 4x² ... |
[
{
"content": "Question: Given a polynomial $P(x) = x^3 - 9x^2 + kx - 27$, where $k$ is a real number. If the roots of $P(x)$ form a geometric progression, find the value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $M$ be the midpoint of $BC$. A point $D$ is chosen on $AM$ such that $AD = 2DM$. Find the area of triangle $BDC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "O... |
[
{
"content": "Question: Two circles, $C_1$ and $C_2$, intersect at points $A$ and $B$. The radius of $C_1$ is 5 and the radius of $C_2$ is 12. The distance between the centers of $C_1$ and $C_2$ is 13. Let $P$ be a point on $C_1$ and $Q$ be a point on $C_2$. What is the maximum possible length of the segment $P... |
[
{
"content": "Question: How many ways are there to arrange the letters of the word \"MATHEMATICS\" such that no two A's are adjacent?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to figure out how many ways there are to ar... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 = 2025$ and $x > y$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of ordered pairs of integers (x, y) su... |
[
{
"content": "Question: A circle is inscribed in a rhombus with diagonals of length 12 and 16. Find the radius of the circle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the radius of a circle inscribed in a rhomb... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 2$. For how many integers $n$ is $f(f(n))$ a prime number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to figure out how many integers n make f(f(n)) a prime number, where f(... |
[
{
"content": "Question: Let $ABCD$ be a square with side length $10$. Let $M$ be the midpoint of $AB$. A point $P$ is chosen on $CD$ such that the area of triangle $AMP$ is $30$. Find the length of $CP$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Three balls are drawn randomly without replacement. What is the probability that exactly two of the drawn balls are red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Oka... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 1000 such that $n^2 + 7n + 10$ is divisible by $n+3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positive integers n... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n$ divides $n^2 + 2023$ and $n < 2023$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of positive integers n such that n divides... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x,y)$ such that $x^2+y^2 \\le 25$ and $x \\equiv y \\pmod{3}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let me try to figure out how to solve this problem. T... |
[
{
"content": "Question: Two circles, $C_1$ and $C_2$, have radii $r_1 = 5$ and $r_2 = 12$ respectively. The distance between their centers is $13$. A line tangent to both circles intersects the line connecting their centers at a point $P$. Find the distance from $P$ to the center of $C_1$.\nPlease reason step b... |
[
{
"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": "Okay, let's see. I need to find the number of po... |
[
{
"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 sum of all $n$ such that $a_n$ is a multiple of 7 and $n \\le 100$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"content": "Question: Regular hexagon $ABCDEF$ has side length 6. Point $P$ is in the interior of the hexagon such that the distance from $P$ to $AB$ is 2 and the distance from $P$ to $BC$ is 3. Find the square of the distance from $P$ to $CD$.\nPlease reason step by step, and put your final answer within \\b... |
[
{
"content": "Question: Let $f(n)$ be the number of positive integers $k$ such that $k \\le n$ and $k^2+1$ is divisible by $n$. Find the sum of all $n$ for which $f(n)=2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to fi... |
[
{
"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 $N$ 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 squares of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay... |
[
{
"content": "Question: Two circles, $C_1$ and $C_2$, have radii 3 and 5 respectively. The distance between their centers is 10. A line $L$ is tangent to both circles such that both circles are on the same side of $L$. The points of tangency are $A$ on $C_1$ and $B$ on $C_2$. Find the length of the segment $AB$... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12 and the sum of its digits is 12. Find 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": "Okay, so I ... |
[
{
"content": "Question: Suppose $P(x)$ is 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 values of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I ... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, $N$ is a multiple of 15, and $N$ is a multiple of 20, and the sum of the digits of $N$ is 15. Find the value of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x < 2024$ such that $x^2 + x + 1$ is divisible by $7$. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positive integers x less th... |
[
{
"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 21. 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": "O... |
[
{
"content": "Question: A positive integer $n$ has exactly 8 divisors. If the sum of the divisors of $n$ is 280, what is the value of $n$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find a positive integer n that has ... |
[
{
"content": "Question: A circle is inscribed in a rhombus with side length 6 and an angle of $60^\\circ$. What is the area of the circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the area of a circle that's in... |
[
{
"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's money is doubled. 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 th... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 20n - 2200$ 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": "Okay, so I need to find all positi... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2023$ is divisible by $n+1$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positive integers n ≤ ... |
[
{
"content": "Question: Two circles, $C_1$ and $C_2$, have radii 1 and 2 respectively, and their centers are 5 units apart. A line tangent to both circles intersects the line connecting their centers at a point $P$. Find the distance from $P$ to the center of $C_1$.\nPlease reason step by step, and put your fin... |
[
{
"content": "Question: In triangle $ABC$, $AB = 5$, $BC = 7$, and $AC = 8$. Let $M$ be the midpoint of $BC$. Find the length of the altitude from $A$ to $AM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the lengt... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 20n - 100$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of positive integers n such that n² + 20n - ... |
[
{
"content": "Question: In a certain game, a player starts with $1 and flips a fair coin. If the coin lands heads, the player gains $1. If it lands tails, the player loses $0.50. The game ends when the player either has $0 or $3. What is the probability that the player ends the game with $3?\nPlease reason step... |
[
{
"content": "Question: How many ordered pairs of positive integers $(a, b)$ are there such that $a^2 + b^2 = 2025$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of ordered pairs of positive integers (a,... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n$ divides $20!$ and $\\text{gcd}(n, 10!) = 5!$. Find the remainder when $N$ is divided by $1000$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I n... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $M$ be the midpoint of $BC$. A point $P$ is chosen on $AM$ such that $AP = 2PM$. Find the area of triangle $BPC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "O... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $100 \\le n \\le 999$ and the sum of the digits of $n$ is 10. Find the number of elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to fin... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 10, and $N$ has exactly 10 divisors. What is the sum of the digits of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to fin... |
[
{
"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 digit of $a_{2023}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the last digit of... |
[
{
"content": "Question: Let $f(n)$ be the number of positive integers $k$ such that $1 \\le k \\le n$ and $\\gcd(k, n) = 1$. If $f(n) = 24$ and $n$ has exactly two distinct prime factors, find the sum of all possible values of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"... |
[
{
"content": "Question: A regular octagon has side length 2. A smaller regular octagon is formed by connecting the midpoints of the sides of the larger octagon. What is the area of the smaller octagon?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"co... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with real coefficients $a, b, c$. Suppose that $f(x)$ has roots $r_1$ and $r_2$, and $f(x+1)$ has roots $r_1-1$ and $r_2-1$. If $f(0)=6$ and $f(1)=12$, find the value of $f(2)$.\nPlease reason step by step, and put your final answer wi... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2$ has exactly three positive divisors. Find the sum of all elements in $S$ that are less than $100$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, ... |
[
{
"content": "Question: Consider a sequence defined by $a_1 = 1$, $a_2 = 2$, and $a_{n+2} = 2a_{n+1} + a_n$ for $n \\ge 1$. Find the remainder when $a_{10}$ is divided by 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need t... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$ and $a_{n+1} = \\frac{a_n}{1+na_n}$ for $n \\ge 1$. Find the value of $a_{2023}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the value of $ a_{2023} $ in... |
[
{
"content": "Question: Two circles, $\\omega_1$ and $\\omega_2$, intersect at points $A$ and $B$. A line through $A$ intersects $\\omega_1$ at $C$ and $\\omega_2$ at $D$. Let $P$ be a point on $\\omega_1$ such that $PB$ is tangent to $\\omega_2$ at $B$. If $AC=3$, $AD=5$, and $PB=6$, find the length of $PC$.\n... |
[
{
"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 positive integer value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $1 \\le n \\le 1000$ and $\\lfloor \\frac{n}{3} \\rfloor$ is a multiple of $5$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many posit... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=8$. Let $D$ be the point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $AC=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE$ is the median to side $AC$. Let $F$ be the intersection of $AD$ and $BE$. Find the area of triangle $ABF$.\nPlease... |
[
{
"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 $2023$ have the property that the sum of the digits of $n$ is equal to the sum of the digits of $2n$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I nee... |
[
{
"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 $3$ dollars. What is the probability that t... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x < 1000$ such that $x^2 + 1$ is divisible by $101$. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of positive integers x less... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2023 such that the sum of the digits of $n$ is even.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to figure out how many positive integ... |
[
{
"content": "Question: Two positive integers $a$ and $b$ are chosen such that $a^2 + b^2 = 2025$. What is the greatest possible value of $a+b$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find two positive integers a ... |
[
{
"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 prime numbers $p$ for which $p \\in S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12 and $N$ has exactly 12 distinct positive divisors. Find the sum of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so... |
[
{
"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 $BE$ is an altitude to $AC$. Find the length of $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"ro... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n$ divides $(n-1)! + 1$. Find the sum of all such $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find all positive integers n such tha... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE \\perp AC$. Find the area of triangle $ADE$.\nPlease reason step by step, and put your final answer within \\boxed{... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x + 4$. 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": "Okay, so I need to figure out how many distinct integer values of n m... |
[
{
"content": "Question: Two circles with radii 9 and 16 are externally tangent to each other. A common external tangent to the two circles is drawn. Find the area of the quadrilateral formed by the two centers of the circles and the two points of tangency on the common external tangent.\nPlease reason step by s... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=10$ and $P(1)=13$. If $P(x)$ has two distinct integer roots $r_1$ and $r_2$, find the value of $|r_1-r_2|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"con... |
[
{
"content": "Question: Let $P(x) = x^3 - ax^2 + bx - c$ be a polynomial with integer coefficients. If the roots of $P(x)$ are $\\sin^2\\left(\\frac{\\pi}{7}\\right)$, $\\sin^2\\left(\\frac{2\\pi}{7}\\right)$, and $\\sin^2\\left(\\frac{3\\pi}{7}\\right)$, find the value of $a+b+c$.\nPlease reason step by step, ... |
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