2017-06-05 Advanced

         

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

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Áron Bán-Szabó by Áron Bán-Szabó

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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Mat Met by Mat Met

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 \(A + B\).

Steven Chase by Steven Chase

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 \(a + b\).

Brian Charlesworth by Brian Charlesworth

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

Khang Nguyen Thanh by Khang Nguyen Thanh
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