British Maths Olympiad 2012 Round 1 Problem 4

My solution to an Olympiad problem.

By Vamshi Jandhyala in mathematics

September 12, 2019

Find all positive integers $n$ such that $12n-119$ and $75n-539$ are both perfect squares.



$$ \begin{align} 12n-119 &&=&& x^{2} \label{y2012r1p4e1}\\
75n-539 &&=&& y^{2} \label{y2012r1p4e2} \end{align} $$

From $\ref{y2012r1p4e1}$ and $\ref{y2012r1p4e2}$, we have

$$ \begin{align} 4y^{2}-25x^{2} = (2y-5x)(2y+5x) = 3 \cdot 273=9 \cdot 91 = 39 \cdot 21 = 13 \cdot 63 \label{y2012r1p4e3} \end{align} $$

Solving the sets of simultaneous equations resulting from $\ref{y2012r1p4e3}$ and restricting ourselves to integer solutions, we get the following solutions \((x,y)=(27,69)\) and \((x,y)=(5,19)\).

Using $\ref{y2012r1p4e1}$ and $x=5$ we get $n=12$. For $x=27$, $n$ is not an integer.