CMOQR 2019

A function $f$ is called injective if when $f(n) = f(m)$ , then $n = m$ . Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$ . Prove $m = n$

Rosemonde is stacking spheres to make pyramids. She constructs two types of pyramids $S_n$ and $T_n$ . The pyramid $S_n$ has $n$ layers, where the top layer is a single sphere and the $i^{th}$ layer is an $i\times$ i square grid of spheres for each $2 \le i \le n$ . Similarly, the pyramid $T_n$ has $n$ layers where the top layer is a single sphere and the $i^{th}$ layer is $\frac{i(i+1)}{2}$ spheres arranged into an equilateral triangle for each $2 \le i \le n$ .

Let $f(x) = x^3 + 3x^2 - 1$ have roots $a,b,c$ . (a) Find the value of $a^3 + b^3 + c^3$ (b) Find all possible values of $a^2b + b^2c + c^2a$

Let $n$ be a positive integer. For a positive integer $m$ , we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$ , find the largest positive integer $m$ for which such a partition exists.