59th Ukrainian National Mathematical Olympiad

Question

For which positive integers n 2, there exist n odd (not necessarily different) numbers a_1, a_2, , a_n such that a_1^2 + a_2^2 + + a_n^2 is a square of some positive integer?

Accepted Answer

Clearly, square of an integer number can give a remainder of

0

,

1

or

4

modulo

8

. Therefore, only for

n

of the form

8 k + r

, where

r \in {0, 1, 4}

, such numbers can exist. Let us show how they can be constructed.

n = 4 t, a_{1} = \dots = a_{n - 1} = 1, a_{n} = (2 t - 1) :

a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2} = (4 t - 1) \cdot 1^{2} + (2 t - 1)^{2} = (2 t)^{2} .

n = 8 t + 1, a_{1} = \dots = a_{n - 1} = 1, a_{n} = (2 t - 1) :

a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2} = (8 t) \cdot 1^{2} + (2 t - 1)^{2} = (2 t + 1)^{2} .