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# Problems of the Week

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

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?

A circular disk of radius $$R$$ consists of two uniform halves of equal area with masses $$\frac{2}{3}M$$ and $$\frac{1}{3}M$$. 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 $$\frac23 M$$ 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:

$\omega = \sqrt{\frac{Ag}{B \pi R}}.$

If $$A$$ and $$B$$ are coprime positive integers, determine $$A + B$$.

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

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

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

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

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