# 2021 JMPSC Accuracy Problems/Problem 13

## Contents

## Problem

Let and be nonnegative integers such that Find the sum of all possible values of

## Solution

Notice that since and are both integers, and are also both integers. We can then use casework to determine all possible values of :

Case 1: .

The solutions for and are the roots of , which are not real.

Case 2: .

The solutions for and are the roots of , which are not real.

Case 3: .

The solutions for and are the roots of , which are and .

Case 4:

The solutions for and are the roots of , which are and .

Therefore, the answer is .

~kante314

~Revised and Edited by Mathdreams

~Some Edits by BakedPotato66

## Solution 2

Note we are dealing with Pythagorean triples, so , and we have is a member of the set too. We see has work, but has nothing work. If , we have work. The answer is ~Geometry285

## See also

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.