Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Prove it

Given natural numbers $x_{1}, \ldots, x_{n}$ . Prove that the number $\left(1+x_{1}^{2}\right) \ldots\left(1+x_{n}^{2}\right)$ can be represented as the sum of squares of two integers.

Solution

The statement immediately follows from the fact that the product of two sums of two squares is a sum of two squares (see problem $\underline{61078})$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.