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 3 of 8: Normalize the gcd

Previous step Next step

In plain words

Scaling does not change the hypothesis or the conclusion.

d=\gcd(a_1,\ldots,a_n),\qquad b_i=\frac{a_i}{d}

Detailed analysis

Dividing every card by $d$ preserves the property because both arithmetic and geometric means scale by $d$ . We may assume $\gcd(a_1,\ldots,a_n)=1$ . Step 2 then implies that all cards are odd.