For positive real numbers
$x$
and
$y$
, define their
*
special mean
*
to be the average of their arithmetic and geometric means. Find the total number of pairs of integers
$(x,y)$
, with
$x \le y$
, from the set of numbers
$\{1,2,3...,2016\}$
, such that the special mean of
$x$
and
$y$
is a perfect square.

Other problems: Check your Calibre

The answer is 506.

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The special mean of $x$ and $y$ is given as

$\text{S.M.} = \dfrac{\text{A.M. } + \text{ G.M.}}{2} = \dfrac{\frac{x+y}{2} + \sqrt{xy}}{2} = {\left( \dfrac{\sqrt x + \sqrt y}{2} \right)}^2$

Which is a perfect square integer for only perfect squares $x$ and $y$ with same parity.

Odd parity:$x, y \in \{ 1^2,3^2,5^2, \cdots, 43^2 \}$Even parity:$x, y \in \{ 2^2,4^2,6^2, \cdots, 44^2 \}$As $x \le y$ , we have $\dbinom{22}{2} + 22$ pairs for $(x, y)$ in each case.

This the total number of pairs is $2 \left[ \dbinom{22}{2} + 22 \right] = 506$ .