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 4 of 8: Choose a prime factor of the maximum

Previous step Next step

In plain words

A nonconstant normalized deck has an odd maximum with an odd prime divisor.

M=\max_i a_i,\qquad M\ge3,\qquad p\mid M

Detailed analysis

Assume the cards are not all equal. The positive odd maximum $M$ is at least $3$ , so it has an odd prime divisor $p$ .