2017 USAJMO Problems/Problem 4

Problem

Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2) + 12$ is prime that properly divides the positive number $a^2 + b^2 + c^2 + abc - 2017$ ?

Solution

Yes. Take $a=b=c=1$ then $11 \mid -2013 = 1^2 + 1^2 + 1^2 + 1\times1\times1 - 2017$ .

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

2017 USAJMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6
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2017 USAJMO Problems/Problem 4

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