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 6 of 8: The critical mean is not divisible by p

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In plain words

The mean lies strictly above the largest p-free card.

x=\frac{M+a_k}{2},\qquad a_k<x<M,\qquad p\nmid x

Detailed analysis

Both $M$ and $a_k$ are odd, so $x$ is an integer by Step 2, and $x>a_k$ . If $p\mid x$ , then $p\mid2x=M+a_k$ ; since $p\mid M$ and $p$ is odd, this would give $p\mid a_k$ , a contradiction.