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(a-2)(b-2)(c-2)+12 is prime and divides ...

Source: JMO 2017 P4

April 20, 2017
JMO2017 USAJMOUSAJMO

Problem Statement

Are there any triples (a,b,c)(a,b,c) of positive integers such that (a−2)(b−2)(c−2)+12(a-2)(b-2)(c-2)+12 is a prime number that properly divides the positive number a2+b2+c2+abc−2017a^2+b^2+c^2+abc-2017?
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