messages listlengths 2 2 |
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[
{
"content": "Question: An urn contains 5 red balls and 7 blue balls. Two balls are drawn randomly 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: 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: Points A, B, C are vertices of an equilateral triangle with side length 6. Let S be the set of all points P in the plane such that the sum of the squares of the distances from P to A, B, and C is 54. Find the area of the region formed by S.\nPlease reason step by step, and put your final... |
[
{
"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": "<think>\nOkay, so I need to find the polynomial P(x) such that P(x² + 1) equals x⁴ + 4x² +... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. A point $D$ is chosen on $BC$ such that the inradii of $\\triangle ABD$ and $\\triangle ADC$ are equal. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"c... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 7 and the sum of its digits is 15. What is the value of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the ... |
[
{
"content": "Question: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with real coefficients. If $P(1) = 4$, $P(2) = 9$, and $P(3) = 16$, what is the value of $P(5)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I nee... |
[
{
"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: A regular octagon is inscribed in a circle of radius 1. The vertices are $V_1, V_2, \\dots, V_8$ in counterclockwise order. Find the area of the quadrilateral $V_1V_3V_5V_7$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"conte... |
[
{
"content": "Question: Let $P(x) = x^3 - ax^2 + bx - c$ be a polynomial with integer coefficients. If $P(1) = 5$, $P(2) = 10$, and $P(3) = 17$, find the value of $P(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $N = 2^{10} \\cdot 3^8 \\cdot 5^6$. How many positive integer 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 integer d... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 2$. How many real solutions does the equation $f(f(f(x))) = 2$ have?\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 real solutions the equation f... |
[
{
"content": "Question: Let $N$ be a three-digit number such that when its digits are reversed, the new number is $M$. If $N - M = 297$ and the sum of the digits of $N$ is 15, find the number $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 9, $N$ has exactly 9 distinct positive divisors, and the sum of its divisors is a perfect square. Find $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"c... |
[
{
"content": "Question: Consider the polynomial $P(x) = x^3 - 7x^2 + ax - 15$. If the roots of $P(x)$ are in arithmetic progression, find the value of $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x) = x^3 - ax^2 + bx - c$. If the roots of $P(x)$ are $r, s, t$, and the roots of $P(x+1)$ are $r-1, s-1, t-1$, such that $r, s, t$ are positive integers, find the smallest possible value of $c$.\nPlease reason step by step, and put your final answ... |
[
{
"content": "Question: Let $f(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients $a$ and $b$. If the roots of $f(x)$ are $\\alpha$ and $\\beta$, and the roots of $f(f(x))$ are $r_1, r_2, r_3, r_4$, such that $r_1, r_2$ are real and $r_3, r_4$ are non-real, find the smallest possible value of $... |
[
{
"content": "Question: Find the smallest positive integer $n$ such that $n$ has exactly 12 positive divisors and the sum of its divisors is 372.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the smallest p... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = 2023$ and $P(1) = 2024$. What is the remainder when $P(2)$ is divided by $7$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ... |
[
{
"content": "Question: In a 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"
},
{
"co... |
[
{
"content": "Question: Let $N$ be a five-digit number such that its digits are $a, b, c, d, e$ in that order. If $N = 41(a+b+c+d+e)$, find the sum of all possible values of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $N = \frac{10^{2023} + 1}{10^{23} + 1}$. 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 find the sum of the digits of N, where N is (10^2023 + 1)/(10... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $AC=15$. Let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$. Let $E$ be a point on $AC$ such that $BE$ is an angle bisector of $\\angle B$. Find the length of $DE$.\nPlease reason step by step, and put your final answer within \\bo... |
[
{
"content": "Question: In a triangle $ABC$, $D$ is a point on $BC$ such that $AD$ bisects $\\angle BAC$. If $AB = 6$, $AC = 10$, and the area of triangle $ABD$ is 15, what is the area of triangle $ABC$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"... |
[
{
"content": "Question: Let $ABCD$ be a convex quadrilateral with $AB=3$, $BC=4$, $CD=5$, and $DA=6$. If $\\angle B = 90^\\circ$ and $\\angle D = 90^\\circ$, find the area of quadrilateral $ABCD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content... |
[
{
"content": "Question: In a circle with center $O$ and radius 6, chords $AB$ and $CD$ intersect at point $P$. If $AP=8$, $PB=3$, and $CP=2$, find the length of $PD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: An equilateral triangle is inscribed in a circle of radius $R$. A smaller circle is tangent to the equilateral triangle's three sides and the larger circle. Find the ratio of the area of the smaller circle to the area of the larger circle.\nPlease reason step by step, and put your final ... |
[
{
"content": "Question: Given that $\\sin x + \\cos x = \\frac{1}{2}$, find the value of $\\sin^3 x + \\cos^3 x$.\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 sin³x + cos³x given that sinx + c... |
[
{
"content": "Question: Suppose $P(x)$ is a polynomial with integer coefficients such that $P(1) = 5$ and $P(5) = 13$. What is the remainder when $P(2)$ is divided by $4$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I n... |
[
{
"content": "Question: A triangle has side lengths $a, b, c$. If $a^2+b^2+c^2 = 200$ and $ab+bc+ca = 100$, find the area of the triangle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the area of a triangl... |
[
{
"content": "Question: Let $a_1, a_2, \\dots, a_n$ be a sequence of positive real numbers. If $\\sum_{i=1}^n a_i = 10$ and $\\sum_{i=1}^n a_i^2 = 20$, find the maximum possible value of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<... |
[
{
"content": "Question: In triangle $ABC$, point $D$ is on $BC$ such that $AD$ bisects $\\angle BAC$. If $AB = 6$, $AC = 9$, and the area of triangle $ABD$ is 12, find the area of triangle $ABC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content"... |
[
{
"content": "Question: 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: Let $ABCD$ be a square with side length 6. Let $E$ be a point on $CD$ such that $CE=2$. Let $F$ be a point on $BC$ such that $BF=3$. Let $P$ be the intersection of $AE$ and $DF$. Find the area of triangle $APF$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"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 a certain game, a player starts with $N$ points. In each round, the player either gains 3 points or loses 2 points. After 10 rounds, the player has 15 points. How many different possible values could $N$ have, given that $N$ is a positive integer?\nPlease reason step by step, and put ... |
[
{
"content": "Question: How many ways are there to arrange the letters of the word 'MISSISSIPPI' such that no two 'S's are adjacent?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many ways there a... |
[
{
"content": "Question: Let $S$ be the set of all positive integers $n$ such that $n$ is a divisor of $10^{10}$ and $n$ is a multiple of $10^5$. Find the number of elements in $S$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOka... |
[
{
"content": "Question: A rectangle $ABCD$ has sides $AB=3$ and $BC=4$. A point $P$ is chosen on side $BC$ such that $BP=1$. A point $Q$ is chosen on side $CD$ such that $CQ=2$. The line segment $AP$ intersects the line segment $BQ$ at point $R$. Find the area of triangle $ARQ$.\nPlease reason step by step, and... |
[
{
"content": "Question: Let $P(x) = x^3 - ax^2 + bx - c$ be a polynomial with integer coefficients. We are given that $P(1)=0$, $P(2)=3$, and $P(3)=10$. Find the value of $P(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 12, the sum of the digits of $N$ is 12, and $N$ ends with the digit 2. Find the sum of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"con... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $x+y+z=1$. What is the minimum value of $\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find t... |
[
{
"content": "Question: 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: Find the smallest positive integer $n$ such that $n^2 + 7n + 1$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the smallest positive integer n such that n² + 7n +... |
[
{
"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 $a_{10}$.\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 $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 2$. 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 ... |
[
{
"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": "<think>\nOkay, so I need to find P(x+1) given that P(x² + 1) = x⁴ + 4x² + 5. Hmm, let me... |
[
{
"content": "Question: Let $f(n)$ be the number of ways to write $n$ as a sum of distinct positive integers. For example, $f(4)=2$ since $4=4$ and $4=1+3$. What is the value of $f(7)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. If two balls are drawn at random without replacement, what is the probability that they are both red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I ne... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $xy+yz+zx=1$. Find the minimum value of $\\frac{x^2}{y^2+z^2} + \\frac{y^2}{x^2+z^2} + \\frac{z^2}{x^2+y^2}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"content": "Question: A rectangle $ABCD$ has sides $AB = 3$ and $BC = 4$. Point $E$ is on $BC$ such that $BE = 1$. Point $F$ is on $CD$ such that $CF = 1$. Lines $AE$ and $BF$ intersect at point $P$. Find the area of triangle $APF$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
[
{
"content": "Question: Six people, A, B, C, D, E, and F, are to be seated around a circular table. If A and B must sit next to each other, and C and D must not sit next to each other, how many distinct seating arrangements are possible?\nPlease reason step by step, and put your final answer within \\boxed{}.",... |
[
{
"content": "Question: In $\\triangle ABC$, point $D$ is on $BC$ such that $AD \\perp BC$. If $AB = 13$, $AC = 15$, and $BC = 14$, find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the... |
[
{
"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 $CA$ such that $AD = 3 \times DC$. Find the length of $MD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<... |
[
{
"content": "Question: A regular octagon is inscribed in a circle of radius 10. The vertices are $V_1, V_2, \\dots, V_8$ in counterclockwise order. Let $A$ be the area of the quadrilateral $V_1V_3V_5V_7$. Find $A$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. If the first ball drawn is red, what is the probability that the second ball drawn is also red?\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": "<think>\nOkay, so I need to find the value of P(3) where P is a polynomial su... |
[
{
"content": "Question: A triangle $ABC$ has side lengths $AB=13$, $BC=14$, 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 $DE$ is parallel to $AB$. Find the length of $DE$.\nPlease reason step by step, and put your final answe... |
[
{
"content": "Question: Suppose $x$ and $y$ are positive real numbers such that $x^2 + y^2 = 1$ and $x^4 + y^4 = \frac{1}{2}$. Find the value of $x^6 + y^6$.\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: Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial with integer coefficients such that $P(1) = 10$, $P(2) = 20$, and $P(3) = 30$. Find the value of $P(0) + P(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<... |
[
{
"content": "Question: Two complex numbers $z_1$ and $z_2$ satisfy $|z_1| = 1$, $|z_2| = 2$, and $|z_1 + z_2| = 3$. Find the value of $|z_1 - z_2|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value ... |
[
{
"content": "Question: Let $P(x)$ be a polynomial such that $P(x^2+1) = x^4 + 4x^2 + 5$. 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 polyn... |
[
{
"content": "Question: A committee of 5 people is to be chosen from a group of 6 men and 4 women. If the committee must have at least 2 men and at least 1 woman, how many different committees can be formed?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: A bag contains 5 red balls and 3 blue balls. If 3 balls are drawn randomly without replacement, what is the probability that at least two of them are red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 1000 such that $n^2 + 2n$ 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: How many positive integers less than or equal to 2024 are not divisible by 3, 5, or 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 2024 aren'... |
[
{
"content": "Question: Let $f(x)$ be a polynomial such that $f(x^2+1) = x^4 + 4x^2 + 5$. Find $f(x)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the polynomial f(x) such that f(x² + 1) equals x⁴ + 4x² +... |
[
{
"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 $N = 2^{2023} \\cdot 3^{2024} \\cdot 5^{2025}$. How many positive integers are divisors of $N$ but not divisors of $M = 2^{2022} \\cdot 3^{2023} \\cdot 5^{2024}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<t... |
[
{
"content": "Question: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with real coefficients. If $f(1) = 2$, $f(2) = 5$, 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 ne... |
[
{
"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 20. What is the sum of the digits of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thi... |
[
{
"content": "Question: How many positive integers less than 1000 have exactly three 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 exactly thre... |
[
{
"content": "Question: Let $N$ be a positive integer such that $N$ leaves a remainder of 3 when divided by 5, a remainder of 5 when divided by 7, and a remainder of 7 when divided by 9. What is the smallest possible value of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"r... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, $P(1)=2$, and $P(2)=4$. What is the smallest possible integer value of $P(3)$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so... |
[
{
"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: Let $N = 2^{2023}$. How many positive integer divisors of $N^N$ 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 integer divisors of $ N^N $... |
[
{
"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 $b$. If its area is 36 and its perimeter is 30, find the value of $a^2 + b^2$.\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 $ a^2 +... |
[
{
"content": "Question: Let $f(x) = x^2 - 2x + 4$. For how many distinct integers $n$ is $f(n)$ 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 distinct integers n make f(n) =... |
[
{
"content": "Question: A bag contains 3 red marbles and 5 blue marbles. Two marbles are drawn randomly without replacement. What is the probability that at least one of the drawn marbles is red?\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, let's see. I need to find the v... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2023 such that $n^2 + 2n + 13$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is asking for th... |
[
{
"content": "Question: A regular hexagon is inscribed in a circle of radius $R$. A smaller circle is inscribed in the hexagon. What is the ratio of the area of the smaller circle to the area of the larger circle?\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": "<think>\nOkay, so I need to find the sum of the roots of the pol... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x$ such that $1 \\le x \\le 2023$ and $x^2 + 2x + 1$ is divisible by $10$. 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 num... |
[
{
"content": "Question: Let $f(x)$ be a polynomial such that $f(x^2) = x^4 + 5x^2 + 6$. Find the sum of the roots of $f(x)$.\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: 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 $x, y$ be positive real numbers such that $x+y=1$. Find the minimum value of $\\left(x + \\frac{1}{x}\\right)^2 + \\left(y + \\frac{1}{y}\\right)^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, ... |
[
{
"content": "Question: Point $P$ is inside equilateral triangle $ABC$. If the distances from $P$ to sides $AB$, $BC$, and $CA$ are 1, 2, and 3 respectively, find the area of triangle $ABC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<t... |
[
{
"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 $CA$ such that $AD=5$. Find the length of $MD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOk... |
[
{
"content": "Question: How many ways are there to arrange the letters of the word 'MISSISSIPPI' such that no two 'S's are adjacent?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many ways there a... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = 20$, $P(1) = 23$, and $P(2) = 28$. Find the minimum possible value of $|P(3)|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let... |
[
{
"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 $P(x) = x^3 - 3x^2 + ax - 4$ be a polynomial such that its roots $r_1, r_2, r_3$ satisfy $r_1^2 + r_2^2 + r_3^2 = 11$. Find the value of $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I nee... |
[
{
"content": "Question: How many positive integers less than 1000 are multiples of 3 but not multiples of 5?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive integers less than 1000 are... |
[
{
"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: Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. A point $D$ is on $BC$ such that $AD$ bisects angle $A$. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need... |
[
{
"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 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 $P(x^2+1) = x^4 + 5x^2 + 7$. 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 $N = 2^{2023}$. What is the remainder when $N$ 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 find the remainder when $ N = 2^{2023} $ is divided by 7. Hmm, let me thi... |
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