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DeepSeek-R1-Distill-Qwen-1.5B_aime-24_expanded_prompt_1_eval

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"Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: \n\\[\\l(...TRUNCATED)

"Alright, so I have this problem where I need to find the absolute value of a logarithm expression i(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)| given three log(...TRUNCATED)

"Okay, so I have this problem with three equations involving logarithms and three variables x, y, an(...TRUNCATED)

"Okay, so I've got this problem here with three logarithmic equations involving x, y, and z. The goa(...TRUNCATED)

"Okay, so I've got this problem here with three equations involving logarithms and variables x, y, z(...TRUNCATED)

"Alright, so I've got this problem here where I need to find the value of |log₂(x⁴y³z²)|, give(...TRUNCATED)

"Okay, so I have this problem with three equations involving logarithms, and I need to find the valu(...TRUNCATED)

"Okay, so I've got this problem here with three logarithmic equations, and I need to find the value (...TRUNCATED)

"Okay, so I have this problem here where I need to find the value of |log₂(x⁴y³z²)| given thre(...TRUNCATED)

"Okay, so I've got this problem here where I need to find the value of |log₂(x⁴y³z²)| given so(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)|, and it's given(...TRUNCATED)

"Okay, so I've got this problem here with three logarithmic equations involving variables x, y, and (...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)| given three log(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)| given three log(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)|, given three lo(...TRUNCATED)

"Okay, so I have this problem here with three logarithmic equations involving variables x, y, and z.(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)|, and it's given(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)| given three log(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)|, given three lo(...TRUNCATED)

"Alright, so I have this problem where I need to find the value of \\( \\left|\\log_2(x^4y^3z^2)\\ri(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)| given three log(...TRUNCATED)

"Okay, so I have this problem with three equations involving logarithms, and I need to find the valu(...TRUNCATED)

"Okay, so I have this problem here with three logarithmic equations and I need to find the absolute (...TRUNCATED)

"Alright, so I have this problem here where I need to find the value of |log₂(x⁴y³z²)| given t(...TRUNCATED)

"Alright, so I have this problem here with three logarithmic equations involving x, y, and z. I need(...TRUNCATED)

"Okay, so I've got this problem here with three logarithmic equations involving x, y, and z. The goa(...TRUNCATED)

"Alright, so I have this problem here with three logarithmic equations, and I need to find the value(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)|, and it's given(...TRUNCATED)

"Alright, so I have this problem here with three logarithmic equations involving x, y, and z. The go(...TRUNCATED)

"Okay, so I have this problem with three logarithmic equations involving variables x, y, and z. The (...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)|, given three lo(...TRUNCATED)

"Okay, so I have this problem where I need to find the value of |log₂(x⁴y³z²)| given three log(...TRUNCATED)

"Let $O(0,0), A(\\tfrac{1}{2}, 0),$ and $B(0, \\tfrac{\\sqrt{3}}{2})$ be points in the coordinate pl(...TRUNCATED)

"Alright, let's tackle this problem step by step. It's a bit involved, so I need to take it slowly a(...TRUNCATED)

"Okay, so I have this problem where I need to find the square of the distance from the origin O to a(...TRUNCATED)

"Okay, so I have this problem with points O, A, and B. O is at (0,0), A is at (1/2, 0), and B is at (...TRUNCATED)

"Okay, so I have this problem here with points O, A, and B. O is at (0,0), A is at (1/2, 0), and B i(...TRUNCATED)

"Okay, so I have this geometry problem here, and I need to figure out the value of OC squared, where(...TRUNCATED)

"Alright, so I've got this problem here, and I need to figure out the answer step by step. Let me tr(...TRUNCATED)

"Okay, so I have this problem with points O, A, and B. O is at (0,0), A is at (1/2, 0), and B is at (...TRUNCATED)

"Okay, so I have this geometry problem here, and I need to find the square of the distance from the (...TRUNCATED)

"Okay, so I need to figure out this geometry problem. Let me start by understanding the problem stat(...TRUNCATED)

"Okay, so I have this problem here with points O, A, and B. Let me write down their coordinates to g(...TRUNCATED)

"Okay, so I have this geometry problem here. Let me try to understand it step by step. \n\nWe have t(...TRUNCATED)

"Okay, so I have this problem here with points O, A, and B. Let me write them down:\n\n- O is at (0,(...TRUNCATED)

"Okay, so I've got this geometry problem here, and I need to find the square of the distance from th(...TRUNCATED)

"Okay, so I have this problem where I need to find the square of the distance from point O to a spec(...TRUNCATED)

"Okay, so I've got this geometry problem here, and I need to figure out the square of the distance f(...TRUNCATED)

"Okay, so I've got this geometry problem here. Let me try to parse it step by step.\n\nWe have three(...TRUNCATED)

"Okay, let me try to figure out this problem step by step. So, we have three points: O at (0,0), A a(...TRUNCATED)

"Okay, so I need to solve this geometry problem. Let me read it again and try to understand what's b(...TRUNCATED)

"Okay, so I have this problem here with points O, A, and B. O is at (0,0), A is at (1/2, 0), and B i(...TRUNCATED)

"Alright, so I've got this problem here. Let me try to understand it step by step.\n\nWe have three (...TRUNCATED)

"Okay, so I have this problem where I need to find the square of the distance from point O to point (...TRUNCATED)

"Okay, let me try to solve this problem step by step. So, we have points O(0,0), A(1/2, 0), and B(0,(...TRUNCATED)

"Okay, so I have this problem here with points O at (0,0), A at (1/2, 0), and B at (0, sqrt(3)/2). T(...TRUNCATED)

"Alright, so I have this problem here, and I need to figure out the answer step by step. Let me read(...TRUNCATED)

"Alright, so I have this problem here with points O, A, and B given. O is at (0,0), A is at (1/2, 0)(...TRUNCATED)

"Okay, so I have this geometry problem here. Let me try to understand it step by step.\n\nWe have th(...TRUNCATED)

"Alright, so I have this geometry problem here, and I need to figure it out step by step. Let me try(...TRUNCATED)

"Okay, so I have this geometry problem here. Let me try to visualize it first. Points O, A, and B ar(...TRUNCATED)

"Alright, so I'm trying to solve this geometry problem. Let me read it again and make sure I underst(...TRUNCATED)

"Okay, so I have this problem with points O, A, and B. O is at (0,0), A is at (1/2, 0), and B is at (...TRUNCATED)

"Okay, so I have this problem where there are three points: O at (0,0), A at (1/2, 0), and B at (0, (...TRUNCATED)

"Okay, let me try to figure out this problem. So, we have three points: O at (0,0), A at (1/2, 0), a(...TRUNCATED)

"Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers(...TRUNCATED)

"Alright, so I have this probability problem to solve. Let me try to understand it step by step. \n\(...TRUNCATED)

"Okay, so Jen is entering this lottery by picking 4 distinct numbers from the set S, which is {1,2,3(...TRUNCATED)

"Okay, so Jen is playing this lottery where she picks 4 distinct numbers from the set S = {1, 2, 3, (...TRUNCATED)

"Okay, so Jen is playing a lottery where she picks 4 distinct numbers from the set S which contains (...TRUNCATED)

"Okay, so Jen is entering this lottery by picking 4 distinct numbers from the set S, which is {1,2,3(...TRUNCATED)

"Alright, so Jen is playing this lottery where she picks 4 distinct numbers from the set S = {1,2,3,(...TRUNCATED)

"Okay, so Jen is playing a lottery where she picks 4 distinct numbers from the set S = {1,2,3,...,10(...TRUNCATED)

"Okay, so I have this probability problem here about Jen entering a lottery. Let me try to understan(...TRUNCATED)

"Okay, so Jen is entering this lottery by picking 4 distinct numbers from the set S which is {1,2,3,(...TRUNCATED)

"Okay, so Jen is playing this lottery where she picks 4 distinct numbers from the set S = {1,2,3,...(...TRUNCATED)

"Okay, so I need to find the probability that Jen wins the grand prize given that she won a prize. T(...TRUNCATED)

"Okay, so Jen is entering this lottery by picking 4 distinct numbers from the set S, which is {1,2,3(...TRUNCATED)

"Okay, so Jen is entering this lottery where she picks 4 distinct numbers from the set S, which is {(...TRUNCATED)

"Okay, so Jen is entering this lottery where she picks 4 distinct numbers from the set S = {1,2,3,..(...TRUNCATED)

"Okay, so I've got this probability problem here about Jen entering a lottery. Let me try to break i(...TRUNCATED)

"Okay, so Jen is playing this lottery where she picks 4 distinct numbers from the set S = {1, 2, 3, (...TRUNCATED)

"Okay, so Jen is playing this lottery where she picks 4 distinct numbers from the set S = {1,2,3,...(...TRUNCATED)

"Okay, so Jen is entering this lottery where she picks 4 distinct numbers from the set S, which is {(...TRUNCATED)

"Okay, so Jen is entering this lottery where she picks 4 distinct numbers from the set S, which is {(...TRUNCATED)

"Alright, so Jen is playing this lottery where she picks 4 distinct numbers from the set S, which is(...TRUNCATED)

"Okay, so I need to figure out the probability that Jen wins the grand prize given that she won a pr(...TRUNCATED)

"Okay, so Jen is entering this lottery by picking 4 distinct numbers from the set S, which is {1,2,3(...TRUNCATED)

"Okay, so I have this probability problem here, and I need to figure it out step by step. Let me try(...TRUNCATED)

"Alright, so I have this probability problem here, and I need to figure out the probability that Jen(...TRUNCATED)

"Alright, so I have this probability problem here, and I need to figure out the probability that Jen(...TRUNCATED)

"Okay, so Jen is entering this lottery by picking 4 distinct numbers from the set S, which is {1,2,3(...TRUNCATED)

"Alright, so Jen is entering this lottery by picking 4 distinct numbers from the set S, which is {1,(...TRUNCATED)

"Okay, so Jen is playing this lottery where she picks 4 distinct numbers from the set S = {1, 2, 3, (...TRUNCATED)

"Alright, so I'm trying to figure out this probability problem. Let me read it again and make sure I(...TRUNCATED)

"Okay, so I have this probability problem here about Jen entering a lottery. Let me try to understan(...TRUNCATED)

"Okay, so Jen is entering this lottery by picking 4 distinct numbers from the set S which is {1,2,3,(...TRUNCATED)

"Okay, so I'm trying to figure out this probability problem. Jen is entering a lottery by picking 4 (...TRUNCATED)

"Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take tur(...TRUNCATED)

"Alright, so I have this problem where Alice and Bob are playing a game with a stack of n tokens. Th(...TRUNCATED)

"Okay, so Alice and Bob are playing this game with a stack of n tokens. Alice goes first, and on eac(...TRUNCATED)

"Okay, so I have this problem where Alice and Bob are playing a game with a stack of n tokens. They (...TRUNCATED)

"Okay, so Alice and Bob are playing this game with a stack of n tokens. They take turns, starting wi(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Alright, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Okay, so Alice and Bob are playing this game with a stack of n tokens. Alice goes first, and each t(...TRUNCATED)

"Okay, so Alice and Bob are playing this game with a stack of n tokens. They take turns removing eit(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing 1 or 4 tokens from a st(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Okay, so Alice and Bob are playing this game with a stack of n tokens. They take turns removing eit(...TRUNCATED)

"Okay, so I have this problem where Alice and Bob are playing a game with a stack of n tokens. They (...TRUNCATED)

"Okay, so I've got this problem where Alice and Bob are playing a game with a stack of n tokens. The(...TRUNCATED)

"Okay, so Alice and Bob are playing this game with a stack of n tokens. They take turns removing eit(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Alright, so I've got this problem where Alice and Bob are playing a game with a stack of n tokens. (...TRUNCATED)

"Okay, so I'm trying to solve this problem where Alice and Bob are playing a game with a stack of n (...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing tokens from a stack. Th(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing 1 or 4 tokens from a st(...TRUNCATED)

"Alright, so I have this problem where Alice and Bob are playing a game with a stack of n tokens. Th(...TRUNCATED)

"Alright, so I have this problem where Alice and Bob are playing a game with a stack of n tokens. Th(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing 1 or 4 tokens from a st(...TRUNCATED)

"Alright, so Alice and Bob are playing this game with a stack of n tokens. They take turns removing (...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Alright, so I'm trying to figure out this game problem where Alice and Bob take turns removing eith(...TRUNCATED)

"Okay, so Alice and Bob are playing this game with a stack of n tokens. They take turns removing eit(...TRUNCATED)

"Okay, so Alice and Bob are playing this game with a stack of n tokens. They take turns removing eit(...TRUNCATED)

"Okay, so Alice and Bob are playing this game where they take turns removing either 1 or 4 tokens fr(...TRUNCATED)

"Eight circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ a(...TRUNCATED)

"Alright, so I have this problem about circles tangent to sides of a triangle and some other circles(...TRUNCATED)

"Okay, so I have this problem about circles tangent to sides of a triangle and another problem invol(...TRUNCATED)

"Okay, so I have this problem about triangles and circles. Let me try to parse it step by step.\n\nW(...TRUNCATED)

"Okay, so I have this problem here about circles tangent to each other and two sides of a triangle. (...TRUNCATED)

"Okay, so I have this geometry problem here about circles tangent to sides of a triangle and each ot(...TRUNCATED)

"Alright, so I've got this problem here about triangle ABC. There are eight circles of radius 34 tha(...TRUNCATED)

"Okay, so I have this problem here about circles tangent to sides of a triangle and some arrangement(...TRUNCATED)

"Okay, so I have this geometry problem here. It involves a triangle ABC, with eight circles of radiu(...TRUNCATED)

"Okay, so I have this problem about triangle ABC. There are eight circles of radius 34 that are sequ(...TRUNCATED)

"Okay, so I have this problem here about circles tangent to a triangle and some other circles. Let m(...TRUNCATED)

"Okay, so I have this geometry problem here about a triangle ABC with three circles of radius 34 arr(...TRUNCATED)

"Okay, so I have this problem about triangle ABC with three circles of radius 34 arranged such that (...TRUNCATED)

"Okay, so I've got this geometry problem here, and I need to find the inradius of triangle ABC. The (...TRUNCATED)

"Okay, so I have this problem about circles tangent to sides of a triangle and finding the inradius.(...TRUNCATED)

"Okay, so I've got this geometry problem here, and I need to figure out the inradius of triangle ABC(...TRUNCATED)

"Okay, so I've got this geometry problem here. It says there are eight circles of radius 34 that are(...TRUNCATED)

"Okay, so I'm trying to solve this problem about circles tangent to the sides of a triangle and anot(...TRUNCATED)

"Okay, so I have this problem about triangle ABC with eight circles of radius 34, each tangent to ea(...TRUNCATED)

"Okay, so I have this problem here about triangle ABC and some circles tangent to its sides. Let me (...TRUNCATED)

"Alright, so I've got this problem here about circles tangent to each other and to the sides of a tr(...TRUNCATED)

"Okay, so I have this problem where there's a triangle ABC, and there are eight circles of radius 34(...TRUNCATED)

"Okay, so I have this geometry problem here about a triangle ABC. There are eight circles of radius (...TRUNCATED)

"Okay, so I have this problem about circles tangent to each other and to the sides of a triangle. Le(...TRUNCATED)

"Alright, so I have this problem about circles tangent to each other and to the sides of a triangle.(...TRUNCATED)

"Alright, so I have this problem here about circles tangent to a triangle and their radii. Let me tr(...TRUNCATED)

"Okay, so I have this geometry problem here about a triangle ABC with three circles of radius 34 tha(...TRUNCATED)

"Alright, so I have this geometry problem here about triangles, circles, and inradius. Let me try to(...TRUNCATED)

"Okay, so I have this geometry problem here. Let me try to visualize it first. There's a triangle AB(...TRUNCATED)

"Alright, so I have this problem here about triangles and circles. Let me try to parse it step by st(...TRUNCATED)

"Okay, so I have this problem about circles tangent to sides of a triangle and figuring out the inra(...TRUNCATED)

"Okay, so I have this problem about triangle ABC with eight circles of radius 34 each, arranged in a(...TRUNCATED)

"Okay, so I'm trying to solve this geometry problem. Let me read it again and make sure I understand(...TRUNCATED)

"Define $f(x)=|| x|-\\tfrac{1}{2}|$ and $g(x)=|| x|-\\tfrac{1}{4}|$. Find the number of intersection(...TRUNCATED)

"Alright, so I have this problem where I need to find the number of intersections of two graphs defi(...TRUNCATED)

"Okay, so I need to figure out how many intersections there are between the two graphs given by the (...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections of two graphs:\n\n1. (...TRUNCATED)

"Okay, so I have this problem here where I need to find the number of intersections of two graphs:\n(...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections of two graphs: \\( y (...TRUNCATED)

"Alright, so I've got this problem here where I need to find the number of intersections between two(...TRUNCATED)

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"Okay, so I have this problem where I need to find the number of intersections between the graphs of(...TRUNCATED)

"Okay, so I need to figure out the number of intersections between the graphs of \\( y = 4g(f(\\sin((...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections of two graphs defined(...TRUNCATED)

"Okay, so I need to find the number of intersections between the graphs of \\( y = 4 g(f(\\sin(2\\pi(...TRUNCATED)

"Okay, so I have this problem here where I need to find the number of intersections between two grap(...TRUNCATED)

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"Okay, so I have this problem where I need to find the number of intersections between two graphs de(...TRUNCATED)

"Okay, so I've got this problem here where I need to find the number of intersections of two graphs.(...TRUNCATED)

"Okay, so I've got this problem here where I need to find the number of intersections between two gr(...TRUNCATED)

"Okay, so I need to find the number of intersections of the graphs of \\( y = 4 g(f(\\sin(2\\pi x)))(...TRUNCATED)

"Okay, so I need to figure out the number of intersections between the graphs of \\( y = 4 g(f(\\sin(...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections between two graphs. T(...TRUNCATED)

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"Okay, so I have this problem where I need to find the number of intersections between two graphs:\n(...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections between the graphs of(...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections of two graphs defined(...TRUNCATED)

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"Alright, so I have this problem here where I need to find the number of intersections between two g(...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections of two graphs. The fu(...TRUNCATED)

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"Okay, so I need to find the number of intersections of the graphs given by y = 4g(f(sin(2πx))) and(...TRUNCATED)

"Alright, so I have this problem here where I need to find the number of intersections of two graphs(...TRUNCATED)

"Alright, so I have this problem here where I need to find the number of intersections between two g(...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections of two graphs: \n\ny (...TRUNCATED)

"Okay, so I have this problem where I need to find the number of intersections of two graphs defined(...TRUNCATED)

"Each vertex of a regular octagon is independently colored either red or blue with equal probability(...TRUNCATED)

"Alright, so I have this problem about a regular octagon where each vertex is colored either red or (...TRUNCATED)

"Okay, so I have this problem about coloring the vertices of a regular octagon either red or blue, e(...TRUNCATED)

"Okay, so I have this problem where each vertex of a regular octagon is colored either red or blue, (...TRUNCATED)

"Okay, so I have this problem about a regular octagon where each vertex is colored either red or blu(...TRUNCATED)

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"Okay, so I have this problem where each vertex of a regular octagon is colored either red or blue i(...TRUNCATED)

"Okay, so I have this problem where each vertex of a regular octagon is colored either red or blue, (...TRUNCATED)

"Okay, so I have this problem about a regular octagon where each vertex is colored either red or blu(...TRUNCATED)

"Okay, so I've got this problem about a regular octagon where each vertex is colored either red or b(...TRUNCATED)

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"Okay, so I'm trying to solve this probability problem about a regular octagon where each vertex is (...TRUNCATED)

"Okay, so I have this problem about a regular octagon where each vertex is colored either red or blu(...TRUNCATED)

"Okay, so I have this problem where each vertex of a regular octagon is colored either red or blue, (...TRUNCATED)

"Alright, so I've got this problem about coloring the vertices of a regular octagon either red or bl(...TRUNCATED)

"Okay, so I have this problem about a regular octagon where each vertex is colored either red or blu(...TRUNCATED)

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"Okay, so I have this problem about a regular octagon where each vertex is colored either red or blu(...TRUNCATED)

"Alright, so I have this problem about a regular octagon where each vertex is colored either red or (...TRUNCATED)

"Alright, so I have this problem about a regular octagon where each vertex is colored either red or (...TRUNCATED)

"Okay, so I have this problem about coloring the vertices of a regular octagon. Each vertex is color(...TRUNCATED)

"Alright, so I have this problem about a regular octagon where each vertex is colored either red or (...TRUNCATED)

"Okay, so I have this problem about a regular octagon where each vertex is colored either red or blu(...TRUNCATED)

"Alright, so I have this problem about coloring the vertices of a regular octagon either red or blue(...TRUNCATED)

"Okay, so I have this problem about a regular octagon where each vertex is colored either red or blu(...TRUNCATED)

"Okay, so I have this problem about a regular octagon where each vertex is colored either red or blu(...TRUNCATED)

"Okay, so I'm trying to solve this probability problem about a regular octagon where each vertex is (...TRUNCATED)

"Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a + b + c = 300$ and \\[a^(...TRUNCATED)

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"Okay, so I need to find the number of triples of nonnegative integers (a, b, c) such that a + b + c(...TRUNCATED)

"Okay, so I have this problem where I need to find the number of triples of nonnegative integers (a,(...TRUNCATED)

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