Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Prove it United States

Problem:
A number is called cool if it is the sum of two nonnegative perfect squares. For example the numbers $17$ and $25$ are cool because $17=4^{2}+1^{2}$ and $25=5^{2}+0^{2}$ , but the number $15$ is not cool.
Show that if $k$ is cool, then $2k$ is cool.

Solution

Solution:
Observe that if $k = a^{2} + b^{2}$ , then $2k = (a-b)^{2} + (a+b)^{2}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.