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 2 of 8: Every pair has the same parity

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In plain words

Pairwise averages cannot be half-integers.

x_{ij}=\frac{a_i+a_j}{2}\in\mathbb Z

Detailed analysis

For every pair of cards, the arithmetic mean equals a geometric mean. It is a positive integral root of an integer, so Step 1 makes it an integer. Thus $a_i+a_j$ is even for every pair, and all cards have the same parity.