messages listlengths 2 2 |
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[
{
"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": "<think>\nOkay, s... |
[
{
"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 drawn balls is red?\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 $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 4$, $f(2) = 11$, and $f(3) = 22$. 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 $P(x)$ 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: Let $f(x) = x^2 - 4x + 6$. If $a$ and $b$ are distinct real numbers such that $f(a) = f(b)$, find the value of $a+b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of a + b... |
[
{
"content": "Question: A committee of 5 people is to be chosen from a group of 6 men and 4 women. What is the probability that the committee consists of exactly 3 men and 2 women, given that at least one woman must be on the committee?\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 $P(x)$.\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: 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 drawn balls is red?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<thin... |
[
{
"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: 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: 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: 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, let's see. I need to find the polynomial P(x - 1) given that P(x² + 1) eq... |
[
{
"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: How many ways can the letters of the word 'GEOMETRY' be arranged such that the two 'E's are not 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 the letter... |
[
{
"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 $P(x^2+1) = x^4 + 4x^2 + 5$. 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: Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial such that $f(1) = 4$, $f(2) = 11$, and $f(3) = 22$. Find $f(4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of f(4) ... |
[
{
"content": "Question: Find the number of positive integers $x$ less than or equal to 2023 such that $x^2 + 2x + 1$ is a perfect square. Hint: $x^2 + 2x + 1 = (x+1)^2$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I nee... |
[
{
"content": "Question: In a triangle $ABC$, $AB=10$, $BC=12$, and $AC=14$. 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 t... |
[
{
"content": "Question: Let $N = 2^{10} \\cdot 3^{10} \\cdot 5^{10}$. What is the number of positive divisors of $N^2$ that are perfect cubes?\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 pos... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 2024 such that $n^2 + 2n + 1$ is divisible by $n+1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out how to solve ... |
[
{
"content": "Question: Find the sum of all real numbers $x$ such that $(x^2 - 7x + 11)^{x^2 - 9x + 20} = 1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve the equation $(x^2 - 7x + 11)^{x^2 - 9x + 20} = 1... |
[
{
"content": "Question: A committee of 5 people is to be chosen from a group of 6 men and 5 women. What is the probability that the committee has at least 3 men and at least 1 woman, given that there is at least 1 man and at least 1 woman on the committee?\nPlease reason step by step, and put your final answer ... |
[
{
"content": "Question: Let $N$ be the number of positive integers less than or equal to 2023 that are not divisible by 3 and not divisible by 5. 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... |
[
{
"content": "Question: In $\\triangle ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is perpendicular 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"
},
{
"co... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 100 such that $n^2 + 2n + 1$ is divisible by 4.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out how to solve this ... |
[
{
"content": "Question: In a certain game, a player starts with $100. Each turn, the player flips a fair coin. If it's heads, the player doubles their money. If it's tails, the player loses $10. What is the expected amount of money the player has after 3 turns?\nPlease reason step by step, and put your final an... |
[
{
"content": "Question: Let $P(x) = x^3 - 3x^2 + 5x - 7$. If $r_1, r_2, r_3$ are the roots of $P(x)$, find the value of $(r_1+1)(r_2+1)(r_3+1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of $(... |
[
{
"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: How many positive integers $n$ less than or equal to 1000 have the property that $n^2 + 2n + 1$ is a perfect square?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is asking how ... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 12, and the sum of the digits of $N$ 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 ... |
[
{
"content": "Question: Let $f(x) = x^2 - 6x + c$ for some constant $c$. If the roots of $f(x)=0$ are $r_1$ and $r_2$, and the roots of $f(f(x))=0$ are $x_1, x_2, x_3, x_4$, find the sum $x_1+x_2+x_3+x_4$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
[
{
"content": "Question: An isosceles trapezoid has parallel sides of length 10 and 22. The non-parallel sides each have length 10. Find the area of the trapezoid.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fi... |
[
{
"content": "Question: Find the remainder when $2^{2023}$ is divided by 13.\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 2^2023 is divided by 13. Hmm, that seems like a pretty big expon... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Let $M$ be the midpoint of $BC$. Find the length of $DM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Question: 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: 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... |
[
{
"content": "Question: How many ways are there to arrange the letters of the word 'GEOMETRY' such that no two vowels 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 ar... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by 20, $N$ is divisible by 24, and $N$ is a perfect square. Find the sum of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think... |
[
{
"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 $N = 2^{2023} \\cdot 5^{2020}$. 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 divis... |
[
{
"content": "Question: Let $P(x) = x^3 - ax^2 + bx - c$ be a polynomial with integer coefficients. We are given that $P(1)=0$, $P(2)=0$, and $P(3)=6$. Find the value of $c$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so ... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $N$ is divisible by $12$ and the sum of its digits is $12$. Find the sum of the squares of the digits of $N$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\... |
[
{
"content": "Question: Let $N$ be the smallest positive integer such that $2023N$ is a perfect square. What is the sum of the digits of $N$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the smallest posit... |
[
{
"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 $P(x)$ 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: How many ways are there 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": "<think>\nOkay, so I need to figure out how many ways there... |
[
{
"content": "Question: Let $f(x) = x^2 - 4x + 1$ and $g(x) = x^2 - 6x + 7$. Find the sum of all values of $x$ such that $f(x) = g(x-1)$.\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 values ... |
[
{
"content": "Question: In a certain game, a player starts with 1 point. In each turn, the player rolls a fair six-sided die. If the die shows an odd number, the player's score is doubled. If the die shows an even number, the player's score is decreased by 1. What is the expected score after 3 turns?\nPlease re... |
[
{
"content": "Question: In a triangle $ABC$, let $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $E$ be the foot of the altitude from $B$ to $AC$. Let $H$ be the orthocenter of triangle $ABC$. Find the length of segment $DH$.\nPlease reason step by step, and put your... |
[
{
"content": "Question: A standard 8x8 chessboard has its squares colored alternately black and white. How many ways are there to place two identical rooks on two different squares of the chessboard such that they do not attack each other?\nPlease reason step by step, and put your final answer within \\boxed{}.... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $x+y+z=1$. Find the minimum value of $\\frac{1}{x} + \\frac{4}{y} + \\frac{9}{z}$.\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: Find the number of positive integers $n$ such that $n^2 + 2023$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of positive integers $ n $ such that $ n... |
[
{
"content": "Question: Find the number of positive integer solutions $(x, y)$ to the equation $2x + 3y = 100$.\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 integer solutions (x, y) ... |
[
{
"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 $f(x) = x^2 - 2x + 2$. Define a sequence by $a_0 = 0$ and $a_{n+1} = f(a_n)$ for $n \\ge 0$. Find the value of $a_{2023}$ modulo 100.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, let's try to fig... |
[
{
"content": "Question: In $\\triangle ABC$, $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE$ is the median to $AC$. Let $AD$ and $BE$ intersect at point $P$. Find the ratio $AP/PD$.\nPlease reason step by ... |
[
{
"content": "Question: Find the number of positive integers $n$ less than or equal to 1000 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": "<think>\nOkay, so I need to find how many positive integer... |
[
{
"content": "Question: Let $ABCD$ be a square with side length 6. Let $M$ be the midpoint of $AB$, and $N$ be a point on $CD$ such that $CN = 2DN$. Let $P$ be the intersection of $AN$ and $DM$. Find the area of triangle $APM$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role... |
[
{
"content": "Question: A sequence is defined by $a_1 = 3$ and $a_{n+1} = \\frac{1}{2} \\left( a_n + \\frac{9}{a_n} \\right)$ for $n \\ge 1$. Find the value of $a_5$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need t... |
[
{
"content": "Question: If $x$ and $y$ are real numbers such that $x^2 + y^2 = 1$ and $x^3 + y^3 = 1$, find the value of $x^4 + y^4$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of $ x^4 + y^4 $... |
[
{
"content": "Question: Let $ABCD$ be a square with side length 6. Let $E$ be a point on $BC$ such that $BE=2$, and $F$ be a point on $CD$ such that $CF=3$. 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: Let $N = 2^{2023} \\cdot 5^{2020}$. How many positive integers are divisors of $N^2$ but not divisors of $N^3$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many positive ... |
[
{
"content": "Question: Suppose $P(x)$ is 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 polynomia... |
[
{
"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 $P(x)$.\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: A regular hexagon has side length 6. A circle is inscribed in the hexagon. Another regular hexagon is inscribed in the circle. What is the area of the smaller hexagon?\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 $1 \\le n \\le 1000$ and $n^2+n+1$ is divisible by 7.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many positive integers n be... |
[
{
"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... |
[
{
"content": "Question: Find the number of positive integer solutions $(x, y)$ to the equation $2x + 3y = 100$ such that $x$ and $y$ are both prime numbers.\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: 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: A fair six-sided die is rolled four times. What is the probability that the sum of the four rolls is 10?\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 rolli... |
[
{
"content": "Question: Find the sum of all real numbers $x$ such that $(x^2 - 5x + 5)^{x^2 - 9x + 20} = 1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all real numbers x such that (x² - 5x + 5) raised t... |
[
{
"content": "Question: A standard deck of 52 cards is shuffled. What is the probability that the top three cards are all of the same suit?\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... |
[
{
"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: In triangle $ABC$, point $D$ is on $BC$ such that $AD$ bisects $\\angle BAC$. If $AB=10$, $AC=15$, and $BC=20$, 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... |
[
{
"content": "Question: Find the smallest positive integer $n$ such that $2023^n \\equiv 1 \\pmod{100}$.\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 2023^n is con... |
[
{
"content": "Question: Let $x$ be a positive real number such that $x^2 + \\frac{1}{x^2} = 7$. Find the value of $x^3 + \\frac{1}{x^3}$.\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 $ x^3 + \... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $AC=15$. Let $D$ be a point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$. Find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOka... |
[
{
"content": "Question: Let $x, y, z$ be positive real numbers such that $x+y+z=1$. Find the minimum value of $\\frac{x^2}{y} + \\frac{y^2}{z} + \\frac{z^2}{x}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to fin... |
[
{
"content": "Question: Let $A, B, C, D$ be points in the plane such that $ABCD$ is a trapezoid with $AB \\parallel CD$. Let $AB = 10$ and $CD = 20$. Let $P$ be a point on $AD$ such that $AP/PD = 1/2$. Let $Q$ be a point on $BC$ such that $BQ/QC = 1/2$. Find the length of segment $PQ$.\nPlease reason step by st... |
[
{
"content": "Question: Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = 2023$ and $P(1) = 2024$. What is the smallest possible value of $|P(2)|$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need... |
[
{
"content": "Question: How many positive integers less than 1000 have the property that the sum of their digits is equal to 10?\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... |
[
{
"content": "Question: Let $N = 2^{2023} + 2^{2020}$. Find the sum of the digits of $N$ in base 4.\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 = 2^2023 + 2^2020 in base 4. Hmm,... |
[
{
"content": "Question: How many sequences of length 7 consisting of the digits {0, 1, 2, 3, 4} are there such that no two adjacent digits are the same, and the first digit is 0?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay,... |
[
{
"content": "Question: Let $N$ be the number of positive integers $x < 1000$ such that $x^2 + 7x$ 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 how many positive integers x ... |
[
{
"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: In a certain game, a player rolls a standard six-sided die. If the number rolled is $1$ or $2$, the player earns $3$ points. If the number rolled is $3$, $4$, $5$, or $6$, the player earns $1$ point. What is the expected number of points a player earns in $5$ rolls?\nPlease reason step b... |
[
{
"content": "Question: A rectangle has side lengths $a$ and $b$. If its area is $72$ and its perimeter is $36$, what is the length of its diagonal?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the length ... |
[
{
"content": "Question: Consider a sequence of integers $a_1, a_2, a_3, \\dots$ 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 $a_{2023}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<th... |
[
{
"content": "Question: Let $S(n)$ denote the sum of the digits of a positive integer $n$. Find the smallest positive integer $n$ such that $S(n) = 17$ and $n$ is divisible by 17.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay... |
[
{
"content": "Question: A committee of 5 people is to be chosen from a group of 6 men and 4 women. What is the probability that the committee consists of exactly 3 men and 2 women, given that at least one man and at least one woman must be on the committee?\nPlease reason step by step, and put your final answer... |
[
{
"content": "Question: A fair six-sided die is rolled $n$ times. What is the smallest positive integer $n$ such that the probability of rolling at least one 6 is greater than $0.9$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"content": "Question: Let $f(x)$ be a function such that $f(x) + 2f\\left(\\frac{1}{x}\\right) = 3x$ for all $x \\neq 0$. Find the value of $f(2)$.\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: A bag contains 5 red balls and 3 blue balls. If 4 balls are drawn at random without replacement, what is the probability that exactly 2 red balls are drawn?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOka... |
[
{
"content": "Question: Let $f$ be a function such that $f(x) + f(y) = f(x+y)$ for all real numbers $x, y$. If $f(1) = 3$, find the value of $f(-2/3)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the valu... |
[
{
"content": "Question: A rectangle has vertices at $(0,0)$, $(6,0)$, $(6,4)$, and $(0,4)$. A line with equation $y = mx$ divides the rectangle into two regions of equal area. Find the value of $m$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"conte... |
[
{
"content": "Question: In triangle $ABC$, $AB=13$, $BC=14$, and $AC=15$. Let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. Let $E$ be a point on $AC$ such that $BE$ bisects $\\angle ABC$. If $AD$ and $BE$ intersect at $I$, find the ratio of the area of triangle $ABI$ to the area of triangle $ABC... |
[
{
"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": "<think>\nOkay, so I need to find the radius of a... |
[
{
"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... |
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