Competition problem — Number theory

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Number theory Difficulty 4.8 AIME Prove it United States

Problem:

Let $k$ be a rational number greater than $1$ (correction by Fengning Ding). Prove that there exist positive integers $a, b, c$ satisfying the equations
$\begin{aligned} a^{2}+b^{2} & =c^{2} \\ \frac{a+c}{b} & =k . \end{aligned}$

Solution

Solution:

Let $k = x / y$ , where $x$ and $y$ are positive integers. We find that $x > y$ . It suffices to note that
$a = x^{2} - y^{2}, \quad b = 2 x y, \quad c = x^{2} + y^{2}$
are positive integers satisfying both of the given equations.

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