Competition problem — Number theory

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Number theory Difficulty 5.1 AIME, harder Prove it Romania

Let $a, b, c, d$ be positive integers, and let $p = a + b + c + d$ . Prove that if $p$ is a prime, then $p$ is not a divisor of $ab - cd$ .

Solution

Consider the relation $(a+c)(b+c) = ab + ac + bc + c^2 = (a+b+c+d)c + ab - cd = pc + ab - cd$ .
If $p$ divides $ab - cd$ , then $p$ divides $a+c$ or $b+c$ .
On the other hand, $0 < a+c < p$ , and $0 < b+c < p$ , a contradiction.

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