messages listlengths 2 2 |
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[
{
"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? Express your answer as a common fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"r... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2) = x^4 + 5x^2 + 6$. Find the sum of all coefficients of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see. I need to find the sum of all coefficients o... |
[
{
"content": "Question: Consider a sequence 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 the digits 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 su... |
[
{
"content": "Question: Given that $x, y, z$ are positive real numbers such that $xy=24$, $yz=36$, and $xz=6$. Find the value of $xyz$.\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 xyz given th... |
[
{
"content": "Question: An equilateral triangle is inscribed in a circle of radius 6. A smaller circle is inscribed in the equilateral triangle. What is the area of the region inside the equilateral triangle but outside the smaller circle?\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 100 such that $n^2 + 2n + 1$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find how many positive integers n ≤ 1... |
[
{
"content": "Question: Let $f(x) = x^2 - 6x + c$. If the roots of $f(x)$ are $r_1$ and $r_2$, and $r_1^2 + r_2^2 = 20$, find the value of $c$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the value of $ c $ in the... |
[
{
"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": "Okay, so I need to find the value of P(3) where P is a polynomial such that P... |
[
{
"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": "Okay, so I need to find the sum of the roots of the polynomial P(x),... |
[
{
"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": "Okay, so I need to find the sum of the roots of the polynomial P(x),... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $n$ divides $2024^2$ and $n$ is a multiple of $22$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see. The problem is asking for the number of pos... |
[
{
"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": "Oka... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 1$. Suppose $r_1$ and $r_2$ are the roots of the equation $f(x) = 0$. Find the value of $(r_1^2 + 1)(r_2^2 + 1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the valu... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $1 \neq n$ and $n^2+1$ 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": "Okay, so I need to find all integers ... |
[
{
"content": "Question: An arithmetic progression has first term $a_1 = 1$ and common difference $d$. A geometric progression has first term $b_1 = 1$ and common ratio $r$. If $a_2 = b_2$ and $a_4 = b_4$, and $d \\ne 0$, find the value of $r$.\nPlease reason step by step, and put your final answer within \\boxe... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $1 \\le n \\le 2023$ and $n^2 + 2n + 2$ is a multiple of $5$. 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 positiv... |
[
{
"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, let me try to figure out this problem. So, we have a polynomial P(x) such that P(x... |
[
{
"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": "Okay, let me try to figure out this problem. So, we have a polynomial P(x) su... |
[
{
"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": "Okay, so I need to find the ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 7x + 10$ is a multiple of $x+2$. 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 fin... |
[
{
"content": "Question: A sequence of integers $a_0, a_1, a_2, \\dots$ is defined by $a_0 = 1$ and $a_{n+1} = a_n^2 + a_n$ for $n \\ge 0$. Find the number of integers $n$ such that $n \\le 2024$ and $a_n$ is a multiple of 3.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": ... |
[
{
"content": "Question: Let $f(x) = x^3 - 3x^2 + 2x - 1$. If $r_1, r_2, r_3$ are the roots of $f(x)=0$, 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": "Okay, so I need to find the value of $(r_1 + 1... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n^2 + 2023$ is a perfect square. Find the sum of the 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 positive int... |
[
{
"content": "Question: A circle is inscribed in a rhombus. The diagonals of the rhombus are 10 and 24. Find the radius of the inscribed circle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the radius of a circle i... |
[
{
"content": "Question: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with real coefficients. If $P(1) = 10$, $P(2) = 20$, and $P(3) = 30$, find the value of $P(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see. I need ... |
[
{
"content": "Question: How many positive integers $n$ are there such that $n$ divides $n^2+n+1$ and $n < 1000$?\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 less than 1000 satisfy that... |
[
{
"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": "Okay, so I need to find the value of a₆... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2+1$ is divisible by $n+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of positive integers n such that n² + 1 is divisible... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 2$. Find the sum of the roots of $P(x) = 0$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the sum of the roots of the polynomi... |
[
{
"content": "Question: In a certain game, a player starts with a score of 0. In each turn, the player rolls a standard six-sided die. If the die shows an odd number, the player's score is doubled. If the die shows an even number, 1 is added to the player's score. What is the expected score after two turns?\nPl... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to $1000$ are there such that $n$ is divisible by $3$ and $n$ is not divisible by $5$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to figure out ho... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 2x + 2$ is a multiple of 5. 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 positiv... |
[
{
"content": "Question: In $\\triangle ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $M$ be the midpoint of $BC$. Find the length of the altitude from $M$ to $AC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the length... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $1 \\le n \\le 2024$ and $n^2 + 2n + 2$ 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 positive integers... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find $P(x)$.\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) such that P(x² + 1) equals x⁴ + 4x² + 5. Hmm, ... |
[
{
"content": "Question: How many positive integers $n$ are there such that $1 \\le n \\le 1000$ and $n^2 + 5n + 6$ is divisible by $n+2$?\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 integer... |
[
{
"content": "Question: Determine the number of positive integers $n$ less than or equal to 100 such that $n^2 + 5n + 6$ is divisible by $n+2$.\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 i... |
[
{
"content": "Question: In $\\triangle ABC$, $AB=5$, $BC=7$, and $AC=8$. Let $M$ be the midpoint of $BC$. Find the length of $AM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the length of AM in triangle ABC where... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 12, $N$ is divisible by 15, and $N$ is divisible by 18. 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... |
[
{
"content": "Question: Find the number of ways to arrange the letters in the word \"MISSISSIPPI\" such that no two 'S's are adjacent.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to figure out how many ways there are to a... |
[
{
"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 their centers is 13. Find the length of the common chord $AB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role"... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 2x + 2$ 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 positi... |
[
{
"content": "Question: In a certain game, a player starts with $100. Each round, the player flips a fair coin. If it lands on heads, the player gains 10% of their current money. If it lands on tails, the player loses 10% of their current money. After 4 rounds, what is the expected value of the player's money?\... |
[
{
"content": "Question: How many positive integers $n$ are there such that $n$ is a divisor of $10^{10}$ and $n^2$ is a multiple of $10^5$?\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: A positive integer $N$ has exactly 6 divisors. If the sum of the divisors of $N$ is 28, 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 a positive integer N that has exac... |
[
{
"content": "Question: How many positive integers less than 1000 are multiples of 3 or 5 but not 15?\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 integers less than 1000 are multiples of 3 ... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=6$, and $CA=7$. 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: 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": "Okay, so I need to find the number of positive integers n such that n² + 2023 is a p... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 1$. Suppose $r_1, r_2, r_3, r_4$ are the roots of the equation $f(f(x)) = 0$. Find the sum $r_1^2 + r_2^2 + r_3^2 + r_4^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to fin... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 2$ and $P(2) = 5$. What is the remainder when $P(2023)$ is divided by $2021$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to f... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $1 \\le x \\le 2023$ and $x^2 + 2x$ is a multiple of $5$. 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 num... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $E$ be the point on $AC$ such that $DE$ is parallel to $AB$. Find the length of $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: Let $P(x)$ 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 +... |
[
{
"content": "Question: Consider a sequence defined by $a_1 = 3$ and $a_{n+1} = a_n^2 - 2a_n + 2$ for $n \\ge 1$. Find the last two digits of $a_{10}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the last two digi... |
[
{
"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: Define a sequence by $a_1=1$, $a_2=3$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 3$. Find the remainder when $a_{2023}$ is divided by 5.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see. I need to find ... |
[
{
"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 squares of the digits of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "... |
[
{
"content": "Question: Let $N$ be the number of positive integers $n$ such that $1 \\le n \\le 2023$ and $n^2 + n + 1$ is divisible by $13$. 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 how ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x < 1000$ such that $x^2 + 2x + 1$ is divisible by $x+1$. 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... |
[
{
"content": "Question: In triangle $ABC$, $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. 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 th... |
[
{
"content": "Question: How many positive integers less than 1000 are not divisible by 3, 5, or 7?\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 integers less than 1000 are not divisible by 3... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + k$. If the roots of $f(x)$ are $\\alpha$ and $\\beta$, and $\\alpha^2 + \\beta^2 = 10$, find the value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see. I need to find the... |
[
{
"content": "Question: Let $N = 2^{2023}$. Find the number of ordered pairs of positive integers $(a, b)$ such that $ab = N$ and $\\gcd(a, b) = 2^{10}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the number of o... |
[
{
"content": "Question: Let $S$ be the set of all integers $n$ such that $1 \\le n \\le 1000$ and $\\gcd(n, 1000) = 1$. 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 the sum of a... |
[
{
"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 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": "Okay, so I n... |
[
{
"content": "Question: A sequence is defined by $a_1 = 1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2}$ for $n ≥ 3$. Find the remainder when $a_{2023}$ is divided by $4$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find t... |
[
{
"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": "Okay, so I ... |
[
{
"content": "Question: Triangle $ABC$ has side lengths $AB=7$, $BC=8$, and $CA=9$. Let $M$ be the midpoint of $BC$. A point $D$ is on $AC$ such that $BD$ bisects $\\angle ABC$. Let $AM$ and $BD$ intersect at point $P$. Find the ratio $AP/PM$.\nPlease reason step by step, and put your final answer within \\boxe... |
[
{
"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": "Okay, so I need to find the value of P(3) where P is a polynomial such that P... |
[
{
"content": "Question: In a certain game, there are $n$ cards, numbered 1 to $n$. A player picks two distinct cards at random. If the sum of the numbers on the cards is 10, the player wins. If the probability of winning is $\\frac{1}{15}$, what is the value of $n$?\nPlease reason step by step, and put your fin... |
[
{
"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 $M$ to $AC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the length of t... |
[
{
"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. Find the number of positive divisors of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so... |
[
{
"content": "Question: In triangle $ABC$, $AB=5$, $BC=7$, and $AC=8$. Let $D$ be a point on $AC$ such that $BD$ bisects $\\angle ABC$. Let $E$ be a point on $BC$ such that $AE \\perp BD$. Find the length of $DE$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. Find the value of $P(3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the value of P(3) where P is a polynomial such that P... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x < 1000$ such that $x^2 + 1$ is divisible by $10$. 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 than ... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 7x + 1$ is divisible by 11. 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 po... |
[
{
"content": "Question: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with integer coefficients. Given that $P(0)=6$, $P(1)=10$, and $P(2)=18$, 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 f... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x,y)$ such that $x^2 + y^2 \\le 25$ and $x \\ge 0$, $y \\ge 0$.\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 int... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ less than or equal to 2024 such that $x^2 + 7x + 10$ is divisible by $x+3$. 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... |
[
{
"content": "Question: Find the number of ordered pairs of integers $(x, y)$ such that $x^2 + y^2 \\le 25$ and $x \\ge 0$.\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... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $2^n + 3^n = 5^n$.\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 n such that 2^n + 3^n equals 5^n. Let me st... |
[
{
"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": "Okay, so I need t... |
[
{
"content": "Question: How many positive integers $n$ less than or equal to 1000 are there such that $n$ is a multiple of 3 or $n$ is a multiple of 5, but not a multiple of 7?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, let's see.... |
[
{
"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 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 how m... |
[
{
"content": "Question: Find the number of ways to arrange the letters in the word \"MISSISSIPPI\" such that no two 'S's are adjacent.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to figure out how many ways there are to a... |
[
{
"content": "Question: Let $ABCD$ be a square with side length 10. Point $E$ is on $BC$ such that $BE=4$. Point $F$ is on $CD$ such that $CF=3$. Let $M$ be the intersection of $AE$ and $BF$. Find the area of triangle $AMB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": ... |
[
{
"content": "Question: A rectangle $ABCD$ has $AB=6$ and $BC=8$. Point $E$ is on $BC$ such that $BE=3$. Point $F$ is on $CD$ such that $CF=2$. Let $P$ be the intersection of $AE$ and $BF$. Find the area of triangle $APF$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "u... |
[
{
"content": "Question: In a certain game, a player starts at the origin $(0,0)$ on the coordinate plane. In each step, the player can move one unit right to $(x+1, y)$ or one unit up to $(x, y+1)$. How many distinct paths are there from $(0,0)$ to $(6,4)$ that do not pass through the point $(3,2)$?\nPlease rea... |
[
{
"content": "Question: Find the number of positive integers $n$ such that $n^2 + 20n - 22$ 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 - 2... |
[
{
"content": "Question: Consider a sequence of integers $a_1, a_2, \\dots, a_n$ where $a_1=1$, $a_2=2$, and $a_k = a_{k-1} + a_{k-2}$ for $k \\ge 3$. How many terms in the sequence $a_1, a_2, \\dots, a_{2023}$ are divisible by 4?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"ro... |
[
{
"content": "Question: A sequence of numbers $a_0, a_1, a_2, \\dots$ is defined by $a_0 = 1, a_1 = 2$, and $a_n = a_{n-1} + a_{n-2}$ for $n \\ge 2$. Find the value of $\\sum_{n=0}^{\\infty} \\frac{a_n}{3^n}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 5x^2 + 6$. Find the sum of the roots of $P(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the sum of the roots of the polynomial P(x... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2024 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": "Okay, so I need to find how many positive integers n ≤ 202... |
[
{
"content": "Question: A sequence is defined by $a_1 = 3$ and $a_{n+1} = a_n^2 - 2a_n + 2$ for $n \\ge 1$. Find the last two digits of $a_{10}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to find the last two digits of ... |
[
{
"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": "Okay, so I need to find the value of P(3) where P is a polynomial such that P... |
[
{
"content": "Question: A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need ... |
[
{
"content": "Question: Regular hexagon $ABCDEF$ has side length 6. Let $M$ be the midpoint of $DE$. Let $P$ be a point on $AM$ such that $BP$ is perpendicular to $AM$. Find the length of $BP$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": ... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 1000 such that $n$ is not divisible by 3, not divisible by 5, and not divisible by 7.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to f... |
[
{
"content": "Question: For how many pairs of positive integers $(a,b)$ with $a \\le b$ is $a^2+b^2+2ab$ a perfect square less than or equal to 100?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Okay, so I need to figure out how many pairs... |
[
{
"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": "Okay, so I need to find the sum of the coefficients of the ... |
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