Difference between revisions of "2021 JMPSC Accuracy Problems/Problem 13"
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#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]] | #[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]] | ||
{{JMPSC Notice}} | {{JMPSC Notice}} | ||
Revision as of 16:24, 11 July 2021
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
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. 
