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saurabh5

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MATH_3000_Filtered_olmo_completions_filtered

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[{"content":"A 8-gon is colored so that in clockwise order, the vertices are colored as follows: ver(...TRUNCATED)

["OK, that which means that we have at most 8,200 minus 2,000 kinds of red Para-winkle goldfish avai(...TRUNCATED)

[{"content":"Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ su(...TRUNCATED)

["Identify the constraints and variables in the problem. The constraints are that \\( a, b, c, d, \\(...TRUNCATED)

[{"content":"Let $S$ be the set \\{1,2, \\ldots, 2012\\}. A perfectutation is a bijective function $(...TRUNCATED)

["This problem is difficult and requires a deep understanding of bijective functions and permutation(...TRUNCATED)

[{"content":"Problem 7.1. (15 points) Find the smallest ten-digit natural number, all digits of whic(...TRUNCATED)

["Understood. To solve this problem, we need to find a ten-digit number with unique digits that sati(...TRUNCATED)

[{"content":"Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is div(...TRUNCATED)

["Using the Chinese Remainder Theorem, we can solve the congruences separately.\nFor $79$:\nWe can r(...TRUNCATED)

[{"content":"Four small spheres each with radius $6$ are each internally tangent to a large sphere w(...TRUNCATED)

["First, let's understand the diagram given. There's a large sphere with radius 17, and 4 smaller sp(...TRUNCATED)

[{"content":"A function $f$ is defined for all real numbers and satisfies $f(3+x)=f(3-x)$ and $f(8+x(...TRUNCATED)

["First, let's think about the given properties of the function f(x). \n\n1. Symmetry about x=3: f(3(...TRUNCATED)

[{"content":"A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular s(...TRUNCATED)

["Okay, let's tackle this problem step by step. We have a 2x2 grid of four sections, each to be plan(...TRUNCATED)

[{"content":"Suppose that we are given 40 points equally spaced around the perimeter of a square, so(...TRUNCATED)

["Let's break this down. Each side of the square has 10 points, and since there are four sides, we h(...TRUNCATED)

[{"content":"Given real numbers \\( x \\) and \\( y \\) satisfying\n\\[ 2^x + 3^y = 4^x + 9^y, \\]\n(...TRUNCATED)

["Based on the equation \\( 2^x + 3^y = 4^x + 9^y \\), it appears that we can rewrite \\( 4^x \\) as(...TRUNCATED)