2017 USAJMO Problems/Problem 4
Are there any triples of positive integers such that is prime that properly divides the positive number ?
The answer is no. Substitute . This means that . Then It is given in the problem that this is positive. Now, suppose for the sake of contradiction that is a prime. Clearly . Then we have is an integer greater than or equal to . This also implies that . Since is prime, we must have Additionally, must be odd, so that is odd while are even. So, if we must have Now suppose WLOG that and . Then we must have , impossible since . Again, suppose that and . Then we must have and since in this case we must have , this is also impossible. Then the final case is when are positive odd numbers. Note that if for positive integers , then for positive integers where . Then we only need to prove the case where , since is odd. Then one of is true, implying that or . But if , then is minimized when , so that . This is a contradiction, so we are done.
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