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 8 of 8: Contradiction and conclusion

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In plain words

The maximal counterexample cannot exist.

x=\left(\prod_{r=1}^{m}a_{i_r}\right)^{1/m}\le a_k\lt x

Detailed analysis

The geometric mean of numbers at most $a_k$ is at most $a_k$ , contradicting $x>a_k$ . Therefore all cards are equal, for every $n>1$ .