Problem 5

A deck of

n>1

cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which

n

does it follow that the numbers on the cards are all equal?

Step 1 of 8: A rational integral power forces integrality

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In plain words

The only rational roots of positive integers are integers.

x^m=N,\ x\in\mathbb Q,\ N\in\mathbb Z_{>0}\Longrightarrow x\in\mathbb Z

Detailed analysis

Write $x=u/v$ in lowest terms. From $u^m=Nv^m$ and $(u,v)=1$ , we get $v^m\mid u^m$ , hence $v=1$ and $x$ is an integer.