#8 Measure your Calibre

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

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