Problems

Michelle and Mondo play the following game, with Michelle going first. They start with a regular polygon, and take it in turns to move. A move is to pick two non-adjacent points in one polygon, connect them, and split that polygon into two new polygons. A player wins if their opponent cannot move - which happens if there are only triangles left. See the diagram below for an example game with a pentagon. Prove that Michelle has the winning strategy if they start with a decagon (\(10\)-sided polygon).

Let \(n\) be a positive integer. Show that \(1+3+3^2+...+3^{n-1}+3^n=\frac{3^{n+1}-1}{2}\).

Show that all integers greater than or equal to \(8\) can be written as a sum of some \(3\)s and \(5\)s. e.g. \(11=3+3+5\). Note that there’s no way to write \(7\) in such a way.

Let \(\phi(n)\) be Euler’s function. Namely \(\phi(n)\) counts how many integers from \(1\) to \(n\) inclusive are coprime with \(n\). For two natural numbers \(m\), \(n\) such that \(\gcd(m,n)=1\), prove that \(\phi(mn) = \phi(m)\phi(n)\).