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Saudi Arabian Mathematical Competitions, 2012

Number theory Difficulty 4.7 AIME Prove it Saudi Arabia

Let $a$ , $b$ , $c$ be positive integers. Prove that if the numbers $\frac{a^2}{a+b}$ , $\frac{b^2}{b+c}$ , $\frac{c^2}{c+a}$ are integers and primes, then $a = b = c$ .

Solution

We will use the following result:

Lemma. If $x$ and $y$ are positive integers such that $\frac{x^2}{x+y}$ is an integer and prime, then $y \ge x$ .

Proof of Lemma. Assume that $\frac{x^2}{x+y} = p$ , where $p$ is a prime.
We have $py = x(x - p)$ , so $p \mid x(x - p)$ , i.e. $p \mid x$ . Let $x = up$ , for some positive integer $u$ . We get
$y = x(up - p) = x(u - 1)p \ge x,$
and we are done, since $u \ge 2$ .

Applying Lemma in our situation, it follows $b \ge a$ , $c \ge b$ , $a \ge c$ , meaning that $a = b = c$ .

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