Mathematical Reflections 2015, Issue 1 - Problem J325

Problem:
For positive real numbers $a$ and $b$, define their \emph{perfect mean} to be half of the sum of their
arithmetic and geometric means. Find how many unordered pairs of integers $(a,b)$ from the set $\{1,2,\ldots,2015\}$ have their perfect mean a perfect square.

Proposed by Ivan Borsenco.

Solution:

We have $$\dfrac{\frac{a+b}{2}+\sqrt{ab}}{2}=n^2 \qquad n \in \mathbb{N},$$ i.e. $$\sqrt{a}+\sqrt{b}=2n, \qquad n \in \mathbb{N}.$$ As $a,b \in \{1,2,\ldots,2015\}$, then $a$ and $b$ must be perfect squares and $n \leq 44$. Assume that $a \leq b$. If $a$ is an odd perfect square, then $b$ is an odd perfect square, and making a case by case analysis for $a \in \{1,3^2,\ldots,43^2\}$ we get $22+21+\ldots+2+1=253$ unordered pairs. Similarly, if $a$ is an even perfect square, then $b$ is an even perfect square and we get $22+21+\ldots+2+1=253$ unordered pairs. So, there are $253+253=506$ unordered pairs which satisfy the given conditions.