Arithmetic mean and Geometric mean

**alexmahone** · October 22nd 2008, 01:22 AM

The difference between the arithmetic mean and the geometric mean of two positive integers $m$ and $n$ is 1. Prove that $\frac {m}{2}$ and $\frac {n}{2}$ are perfect squares.

**SimonM** · October 22nd 2008, 01:50 AM

$\frac {m + n}{2} = 1 + \sqrt {mn}$

Clearly $mn$ is a perfect square, therefore $m = a^2 t, \, n = b^2 t$ for some square free t

Therefore

$\frac {t(a^2 + b^2)}{2} = 1 + abt \Rightarrow t(a - b)^2 = 2 \Rightarrow t = 2$ as required

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