2024 SSMO Team Round Problems/Problem 10

Problem

The side-lengths of a convex cyclic quadrilateral are integers and . Find all possible values of the perimeter of .

Solution

Let From Ptolemy's Theorem, we have So, the equation is equivalent to \begin{align*} (ad+bc)^2 &= (ac+bd)^2-72\implies\\ 72&=a^2c^2+b^2d^2-a^2d^2-b^2c^2\implies\\ 72&=(a^2-b^2)(c^2-d^2). \end{align*} WLOG, assume that Since the difference of two perfect squares cannot be we have Clearly, has no solutions, as For we have and This means Finally, for we have and This gives It is easy to verify that all three of these possibilities work, giving an answer of

~SMO_Team