messages listlengths 2 2 |
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[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + n + 1$ is a prime number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive i... |
[
{
"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: How many positive integers $n$ less than or equal to 1000 are there such that $n$ is a multiple of 3 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 figure out ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, $P(1)=2$, and $P(2)=4$. 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: 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: A point $P$ is chosen randomly in the interior of a square with side length 1. What is the probability that the distance from $P$ to the nearest side of the square is less than or equal to $1/4$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 1000 are there such that $n$ is 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 positiv... |
[
{
"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: Find the number of positive integers $n$ such that $n^2 + 2023$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of positive integers n such that n² + 20... |
[
{
"content": "Question: Let $P(x)$ be a polynomial 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 $N$ 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": "<th... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the sum of all 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: How many positive integers less than 1000 have an odd number of positive divisors?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integers less than 1000 have ... |
[
{
"content": "Question: In a certain game, a player rolls a standard six-sided die. If the die shows a 1 or a 2, the player earns 3 points. If the die shows a 3, 4, 5, or 6, the player earns 1 point. What is the expected number of rolls needed to earn exactly 10 points?\nPlease reason step by step, and put your... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $x+y+z=1$. Find the minimum value of $\\frac{x^2}{y} + \\frac{y^2}{z} + \\frac{z^2}{x}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: In 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: 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: Let $f(x) = x^2 + ax + b$ be a quadratic polynomial. If $f(x+1) - f(x) = 2x+3$ for all real $x$, and $f(0) = 5$, find the value of $f(5)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fi... |
[
{
"content": "Question: Let $N$ 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 the numb... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, and the sum of the digits of $N$ is 12. Find the sum of the digits of $N+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, l... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, and the sum of the digits of $N$ is 12. Find the sum of the digits of $N+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: 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 $a_1=1$, $a_2=2$, 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 to ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 5$ and $P(5) = 13$. What is the remainder when $P(x)$ is divided by $(x-1)(x-5)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 1$ is divisible by $n+1$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many posit... |
[
{
"content": "Question: Two distinct numbers are chosen randomly from the set $\\{1, 2, 3, \\dots, 10\\}$. What is the probability that their product is an even number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need... |
[
{
"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 circle is inscribed in a rhombus with side length 6. If the ratio of the lengths of the diagonals of the rhombus is $1:\\sqrt{3}$, find the area of the circle.\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 $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE$ is the median to $AC$. Let $F$ be the intersection of $AD$ and $BE$. Find the area of triangle $ABF$.\nPlease reas... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $1 \\le n \\le 1000$ and $n$ is not divisible by 2, 3, or 5. Find the sum of the elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 7, and the sum of the digits of $N$ is 15. 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": "<t... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ for which $n^2 + 20n - 2023$ 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 a... |
[
{
"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 $P(x) = x^3 - 3x^2 + 2x - 1$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $(r_1+1)(r_2+1)(r_3+1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of (r... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $BC$. Find the length of $DM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $M$ be the midpoint of $BC$. A point $D$ is chosen on $AC$ such that $AD=2DC$. Find the length of $MD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOk... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2) = x^4 + 5x^2 + 6$. Find the sum of the roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the roots of the polynom... |
[
{
"content": "Question: 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 $1 \\le n \\le 2023$ and $a_n$ is a multiple of 3.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"co... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, and the sum of the digits of $N$ is 12. Find the sum of the digits of $N+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: In a certain school, 60% of the students are girls. 70% of the girls have brown eyes, and 40% of the boys have brown eyes. What percentage of the students in the school do not have brown eyes?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"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: How many positive integers less than 1000 are divisible by 3 or 5 but not by 7?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integers less than 1000 are divi... |
[
{
"content": "Question: Two distinct numbers are chosen randomly and independently from the set $\\{1, 2, 3, \\dots, 10\\}$. What is the probability that their sum is even?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"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 13. 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: Find the number of positive integers $n$ such that $n$ divides $2^{n-1} + 1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all positive integers n such that n divides 2^{n-1} + 1. L... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 20n + 11$ is a perfect square. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all... |
[
{
"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: A circle is inscribed in a rhombus with side length 10. If the ratio of the lengths of the diagonals of the rhombus is $3:4$, what is the radius of the inscribed circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $M$ be the midpoint of $BC$. A point $D$ is 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": "<think>\nOkay, so ... |
[
{
"content": "Question: Let $N = 2^{10} \\cdot 3^{10} \\cdot 5^{10}$. How many positive divisors of $N$ 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... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 2023 are there such that $n^2 + 2n + 13$ 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 i... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, and $N$ has exactly 12 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": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of $\\frac{a^2}{b} + \\frac{b^2}{c} + \\frac{c^2}{a}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. 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 point $P$ is chosen uniformly at random from the interior of a square with vertices at $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$. What is the probability that the distance from $P$ to the nearest side of the square is less than $1/4$?\nPlease reason step by step, and put your final answer... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 12, and the sum of the digits of $N$ is 12. Find the sum of the digits of $N+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, s... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 \\le 25$ and $x+y$ is an even number.\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 p... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $BC$. Find the length of $DM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: In a triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $BC$. Find the length of $DM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thin... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $x+y+z=1$. What is 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 ... |
[
{
"content": "Question: In a certain game, a player starts with $100. If the player flips a fair coin and it lands on heads, their score doubles. If it lands on tails, their score decreases by $10. What is the expected score after two flips?\nPlease reason step by step, and put your final answer within \\boxed{... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2023 such that $n^2 + 2n + 1$ is divisible by $n+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of positi... |
[
{
"content": "Question: For a positive integer $n$, let $S(n)$ be the sum of its digits. Find the smallest positive integer $n$ such that $S(n) = 20$ and $S(n+1) = 2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with real coefficients such that $P(x^2+1) = (P(x))^2+1$ for all real $x$. If $P(0) = 0$, find the value of $P(2)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2024$ is a perfect square. Find the sum of all elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all pos... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $M$ be the midpoint of $AD$. Find the length of $BM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 1$. Suppose $a$ and $b$ are distinct real numbers such that $f(a) = f(b)$. Find the value of $a+b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of ... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. 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: In a certain game, a player rolls a standard six-sided die. If the die shows a 1, 2, or 3, the player moves forward 1 step. If the die shows a 4 or 5, the player moves forward 2 steps. If the die shows a 6, the player moves forward 3 steps. What is the expected number of rolls it takes f... |
[
{
"content": "Question: Six friends, Alice, Bob, Carol, David, Eve, and Frank, are to be seated around a circular table. Alice and Bob insist on sitting next to each other. Carol and David refuse to sit next to each other. How many distinct seating arrangements are possible?\nPlease reason step by step, and put... |
[
{
"content": "Question: Let $N = 2^{2023}$. 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, so I need to figure out how many pos... |
[
{
"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 14. 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: A triangle has side lengths $a, b, c$. If $a^2+b^2+c^2 = 200$ and the area of the triangle is $15\\sqrt{3}$, 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... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2024$ is a perfect square. Find the sum of the 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 $P(x^2+1) = x^4 + 5x^2 + 6$. 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, let me try to figure out this problem. So, we have ... |
[
{
"content": "Question: How many ordered pairs of integers $(x, y)$ satisfy $x^2 + y^2 = 25$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many ordered pairs of integers (x, y) satisfy the equati... |
[
{
"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 $n \\ge 3$. 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": "Question: A sequence is defined by $a_1=1$, $a_2=2$, and $a_n = \\frac{a_{n-1}^2 + 1}{a_{n-2}}$ for $n \\ge 3$. Find the value of $a_6$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 2023$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of positive integers n such that n² + 20... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2+6n+5$ is divisible by $n+3$. 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 $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ is a divisor of $N^2$ and $k \ne N$?\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 ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 10, and the sum of the digits of $N$ is 10. What is the sum of the digits of $N+100$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOk... |
[
{
"content": "Question: Triangle $ABC$ has side lengths $AB=7$, $BC=8$, and $CA=9$. Let $M$ be the midpoint of $BC$. Find the length of the median $AM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the len... |
[
{
"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": "<think>\nOkay, so I need to figure out how many positiv... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $M$ be the midpoint of $BC$. Find the length of $DM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},... |
[
{
"content": "Question: Let $P(x) = x^3 - ax^2 + bx - c$ be a polynomial with real coefficients. If the roots of $P(x)$ are $r_1, r_2, r_3$ and $r_1^2 + r_2^2 + r_3^2 = 14$, $r_1r_2 + r_2r_3 + r_3r_1 = 11$, and $r_1r_2r_3 = 6$, find the value of $a^2$.\nPlease reason step by step, and put your final answer with... |
[
{
"content": "Question: Let $P(x) = x^3 - 3x^2 + 2x - 1$. If $r_1, r_2, r_3$ are the roots of $P(x)=0$, find the value of $(r_1+r_2)^2 + (r_2+r_3)^2 + (r_3+r_1)^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to ... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 19n + 48$ 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... |
[
{
"content": "Question: How many positive integers less than 1000 are multiples of 3 or 5 but not multiples of 15?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integers less than 10... |
[
{
"content": "Question: 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. What is the sum of all elements in $S$ that are less than 1000?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE$ is the angle bisector of $\\angle ABC$. If $AD$ and $BE$ intersect at $I$, find the ratio of the area of triangle ... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $O$ be the circumcenter of triangle $ABD$. Find the length of $OD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 8$. 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 1000 are there such that $n^2 + n + 1$ is divisible by 7?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive i... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 1000 are such that $n$ is a multiple of 3 or $n$ is a perfect square, but not both?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to fig... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 1000 have the property that $n^2 + 1$ is divisible by 5?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive in... |
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