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Given any polygon with perimeter 1, can a circle with radius 41 always enclose the polygon?
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Do there exist 3 distinct positive integers (x,y,z) such that x+y, y+z, z+x, and x+y+z are all perfect squares?
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A circular disk of radius R consists of two uniform halves of equal area with masses 32M and 31M. The disk is free to rotate on an axle through its geometric center.
Initially, the disk is at rest as shown on the left such that the half of mass 32M is on top.
Suppose the disk is toppled by an extremely gentle nudge. At the instant the disk first reaches the orientation on the right, its angular speed can be expressed as follows:
ω=BπRAg.
If A and B are coprime positive integers, determine A+B.
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There exists a unique, positive-valued, non-constant, continuous and differentiable function y=f(x) such that
If ∫ln2ln5f(x)dx=ba, where a and b are coprime positive integers, then find a+b.
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Find the number of permutations f on {1,2,3,…,32} such that if m divides n, then f(m) divides f(n).
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