Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Prove it Saudi Arabia

Let $a$ , $b$ , $c$ , $d$ be positive integers such that $a + b + c + d = 2011$ . Prove that $2011$ is not a divisor of $a b - c d$ .

Solution

We have
$(a + c)(b + c) = a b + a c + b c + c^2 = (a + b + c + d) c + a b - c d = 2011 c + a b - c d$
Because $2011$ is a prime, if $2011 \mid a b - c d$ , then $2011 \mid a + c$ or $2011 \mid b + c$ . This is not possible since $0 < a + c < 2011$ , and $0 < b + c < 2011$ , a contradiction.

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