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 value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of P(3) where P is a polynomial su... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What is the probability that at least one of the balls drawn is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are of the same color?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn 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": "<think>\nOkay,... |
[
{
"content": "Question: In a certain school, 60% of the students are girls. 30% of the girls play soccer, and 50% of the boys play soccer. If a student is chosen randomly from the school and they play soccer, what is the probability that the student is a girl?\nPlease reason step by step, and put your final ans... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What is the probability that at least one of the balls drawn is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with $a, b, c$ real numbers. If $f(x)$ has roots $r_1$ and $r_2$ such that $r_1 + r_2 = 5$ and $r_1 r_2 = 6$, and $f(0) = 12$, find the value of $f(1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"ro... |
[
{
"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 bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn randomly without replacement. What is the probability that at least one of the marbles drawn is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$. 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 coefficie... |
[
{
"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, so I need to find the sum of all coefficient... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that its roots are $r_1$ and $r_2$. If $r_1 + r_2 = 5$ and $r_1 r_2 = 6$, and $f(0) = 12$, find the value of $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"content": "Question: Two standard dice are rolled. What is the probability that the sum of the numbers rolled is a prime number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the probability that when ro... |
[
{
"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 moves forward 3 spaces. If the die shows a 3, 4, or 5, the player moves forward 1 space. If the die shows a 6, the player moves backward 2 spaces. What is the expected number of spaces a p... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of P(3) where P is a polynomial su... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$. 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 coefficie... |
[
{
"content": "Question: In triangle $ABC$, point $D$ is on $BC$ such that $AD$ bisects $\\angle BAC$. If $AB=12$, $AC=18$, and $BD=8$, 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 find ... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn randomly without replacement. What is the probability that at least one of the balls drawn is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thin... |
[
{
"content": "Question: In a certain game, a player rolls a standard six-sided die. If the die shows an even number, the player scores that number of points. If the die shows an odd number, the player scores double that number of points. What is the expected value of the player's score on a single roll?\nPlease... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of P(3) where P is a polynomial su... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn without replacement. What is the probability that at least one of the drawn balls is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"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 receives 3 points. If the die shows a 3, 4, or 5, the player receives 1 point. If the die shows a 6, the player loses 2 points. What is the expected value of the points received in one rol... |
[
{
"content": "Question: Consider a sequence defined by $a_1 = 1$, $a_2 = 3$, and $a_n = 3a_{n-1} - 2a_{n-2}$ for $n \\ge 3$. Find the value of $a_8$.\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: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 4$, $f(2) = 9$, and $f(3) = 18$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the va... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 1$, $f(2) = 5$, and $f(3) = 13$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the va... |
[
{
"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 bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What is the probability that at least one of the drawn balls is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the value of P(3) where P is a polyn... |
[
{
"content": "Question: Two standard dice are rolled. What is the probability that the sum of the numbers rolled is a prime number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out the probability that w... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the polynomial P... |
[
{
"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 $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1)=7$, $f(2)=15$, and $f(3)=27$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value o... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(0) = 1$, $f(1) = 3$, and $f(2) = 7$. Find the value of $f(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the val... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial. If $f(1)=2$, $f(2)=5$, and $f(3)=10$, what is the value of $f(4)$?\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 val... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $xy=2$, $yz=3$, and $zx=5$. Find the value of $xyz$.\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 $ xyz $ given that $ ... |
[
{
"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: How many positive integers $n$ less than or equal to 2023 are relatively prime to 2023?\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 less than or ... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 2$, $f(2) = 5$, and $f(3) = 10$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the va... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with $a, b, c$ real numbers. If $f(x) \\ge 0$ for all real $x$, and $f(1) = 0$, $f(2) = 1$, find the value of $a$.\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 $f(1) = 3$, $f(2) = 11$, and $f(3) = 25$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the v... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a multiple of $7$ and $n < 1000$. What is the sum of the digits of the sum of all elements in $S$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thin... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of P(3) where P is a polynomial su... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of all roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the polyn... |
[
{
"content": "Question: What is the smallest positive integer $n$ such that $n^3 + 4n^2 - 11n - 30$ 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 find the smallest positive integer n such that... |
[
{
"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 $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 3$, $f(2) = 7$, and $f(3) = 13$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the va... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the sum of the coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of the coefficient... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle A$. Let $M$ 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: 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. If the distance between the centers of $C_1$ and $C_2$ is 13, find the length of the common chord $AB$.\nPlease reason step by step, and put your final answer within \\b... |
[
{
"content": "Question: What is the smallest positive integer $n$ such that $n!$ is divisible by $10^{10}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the smallest positive integer $ n $ such that $ n! $... |
[
{
"content": "Question: Find the number of positive integer solutions $(x, y, z)$ to the equation $x + y + z = 10$, where $x \\ge 1$, $y \\ge 2$, and $z \\ge 3$.\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-friendly\" if the sum of its digits divides $n$. For example, $21$ is digit-sum-friendly because $2+1=3$ and $3$ divides $21$. Find the smallest digit-sum-friendly integer $n > 100$ such that $n$ is a perfect square.\nPlease reason step by ste... |
[
{
"content": "Question: In a triangle $ABC$, the side lengths are $AB=13$, $BC=14$, 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": "Question: A sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = 2a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the remainder when $a_{10}$ is divided by 7.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see.... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 2$, $f(2) = 5$, and $f(3) = 10$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the va... |
[
{
"content": "Question: Let $P(x)$ be a polynomial of degree 3 such that $P(1)=1, P(2)=4, P(3)=9,$ and $P(4)=17$. Find the value of $P(5)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of P(5) wh... |
[
{
"content": "Question: What is the smallest positive integer $n$ 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 smallest positive integer n such that n² + 2023 ... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 1000 such 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 find how many positive integers n ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$. 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 coefficie... |
[
{
"content": "Question: Determine the number of positive integers $n$ less than or equal to 1000 such that $n$ is 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 m... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2+1$ is divisible by $n+1$. 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 positi... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $xy+yz+zx = 12$. Find the minimum value of $x^2+y^2+z^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the minimum value of $ x^2 + ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 2x$ is a multiple of 7. 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 number ... |
[
{
"content": "Question: Determine the number of positive integers $n$ less than or equal to 1000 such that $n$ is 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 m... |
[
{
"content": "Question: A circle is inscribed in a square. A smaller square is inscribed in the circle. What is the ratio of the area of the larger square to the area of the smaller square?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<th... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 2024$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of positive integers n such that n² + 20... |
[
{
"content": "Question: Let $a, b, c$ be positive real numbers. If $a+b+c=1$, what is the minimum value of $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the min... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of P(3) where P is a polynomial su... |
[
{
"content": "Question: A rectangle has side lengths $a$ and $b$. A circle is inscribed in the rectangle such that it is tangent to all four sides. What is the area of the region inside the rectangle but outside the circle, in terms of $a$ and $b$?\nPlease reason step by step, and put your final answer within \... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=8$. 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": "<think>\nOkay, so I need to find the le... |
[
{
"content": "Question: Two positive integers $a$ and $b$ are chosen such that $a < b$. The geometric mean of $a$ and $b$ is 12, and their arithmetic mean is 13. Find the value of $\\frac{b}{a}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $CA=6$. 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": "<think>\nOkay, so I need to find the le... |
[
{
"content": "Question: In triangle $ABC$, $AB=10$, $BC=12$, and $CA=14$. 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 the... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 3$, $f(2) = 7$, and $f(3) = 13$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the va... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=2023$, $P(1)=2024$, and $P(2)=2025$. Find the remainder when $P(2023)$ is divided by $2023$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: In triangle $ABC$, let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. If $AB=10$, $AC=15$, and $BD=6$, what is the length of $BC$?\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 of degree 3 such that $P(1)=1$, $P(2)=4$, $P(3)=9$, and $P(4)=12$. Find the value of $P(5)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find a cubic ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 2x + 1$ is a perfect square. 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 nu... |
[
{
"content": "Question: In a certain game, a player rolls a standard six-sided die. If the die shows an even number, the player gets that many points. If the die shows an odd number, the player loses that many points. What is the expected value of the player's score after one roll?\nPlease reason step by step, ... |
[
{
"content": "Question: Two circles, $C_1$ and $C_2$, intersect at points $A$ and $B$. A line through $A$ intersects $C_1$ at $P$ and $C_2$ at $Q$. Let $M$ and $N$ be the midpoints of $AP$ and $AQ$ respectively. If $PQ = 10$, what is the length of $MN$?\nPlease reason step by step, and put your final answer wit... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 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": "<think>\nOkay, so I need to find the numbe... |
[
{
"content": "Question: Let $P(x)$ be a polynomial of degree 3 such that $P(1)=1$, $P(2)=4$, $P(3)=9$, and $P(4)=16$. 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 the value of P(5) where P(x) ... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 2n - 48$ 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 $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$. 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 coefficie... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 2$, $f(2) = 5$, and $f(3) = 10$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the va... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 \\equiv 1 \\pmod{5}$. 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 number of p... |
[
{
"content": "Question: In a certain game, a player starts with 1 point. In each round, the player flips a fair coin. If it lands heads, their score doubles. If it lands tails, their score increases by 2. What is the expected score after 3 rounds?\nPlease reason step by step, and put your final answer within \\... |
[
{
"content": "Question: Let $N = 2^{2023}$. How many positive integers $k$ are there such that $k$ divides $N$ and $\\frac{N}{k}$ divides $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this probl... |
[
{
"content": "Question: Let $N$ be the number of positive integers less than or equal to 2024 that are not divisible by 3, 5, or 7. 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 the n... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 3$, $f(2) = 1$, and $f(3) = 6$. Find the value of $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the val... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 9 and $N$ has exactly 9 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 to ... |
[
{
"content": "Question: Let $N = 10^{100}$. What is the sum of the digits of the sum of the digits of $N^{10}$?\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 digits of the sum of the digits o... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$. 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 coefficie... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 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": "<think>\nOkay, so I need to find how many ... |
[
{
"content": "Question: What is the smallest positive integer $n$ such that $n+1$ is a multiple of 3, $n+2$ is a multiple of 5, and $n+3$ is a multiple of 7?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find th... |
[
{
"content": "Question: A circle is inscribed in a square. A smaller square is inscribed in the circle. What is the ratio of the area of the larger square to the area of the smaller square?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<th... |
[
{
"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: Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 2$. Find the sum of all roots of $P(x) = 0$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum of all roots of the... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is a multiple of 9 and $N$ has exactly 9 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": "<think>\nO... |
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