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2017-06-05 Advanced

         

Given any polygon with perimeter 1, can a circle with radius 14 \frac14 always enclose the polygon?

Do there exist 3 distinct positive integers (x,y,z) (x, y, z ) such that x+y x+y, y+zy+z, z+xz+x, and x+y+z x + y + z are all perfect squares?

A circular disk of radius RR consists of two uniform halves of equal area with masses 23M\frac{2}{3}M and 13M\frac{1}{3}M. 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 23M\frac23 M 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=AgBπR.\omega = \sqrt{\frac{Ag}{B \pi R}}.

If AA and BB are coprime positive integers, determine A+BA + B.

There exists a unique, positive-valued, non-constant, continuous and differentiable function y=f(x)y = f(x) such that

  • over any specified interval, the area between f(x)f(x) and the xx-axis is equal to the arc length of the curve, and
  • f(0)=1.f(0) = 1.

If ln2ln5f(x)dx=ab\displaystyle \int_{\ln2}^{\ln5} f(x) \, dx = \dfrac{a}{b}, where aa and bb are coprime positive integers, then find a+ba + b.

Find the number of permutations ff on {1,2,3,,32}\{1, 2, 3, \ldots, 32\} such that if mm divides nn, then f(m)f(m) divides f(n)f(n).

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