British Maths Olympiad 2016 Round 1 Problem 3

My solution to an Olympiad problem.

By Vamshi Jandhyala in mathematics

September 11, 2019

Determine all pairs $(m, n)$ of positive integers which satisfy the equation

$$ \begin{align*} n^{2}-6n=m^{2}+m-10 \end{align*} $$

Solution

Completing squares on both sides we get

$$ \begin{align} (n-3)^{2}-9&&=&&(m+\frac{1}{2})^{2}-\frac{1}{4}-10 \label{y2016r1p34e1}\\
\implies (2m+1)^{2} - (2n-6)^{2} &&=&& 5 \label{y2016r1p34e2} \end{align} $$

From $\ref{y2016r1p34e1}$ and $\ref{y2016r1p34e2}$, we get the following sets of simultaneous equations,

$$ \begin{align} 2m-2n&&=&&-6 &&\lor&& 2m-2n &&=&&-2\\
2m+2n&&=&&10 &&\lor&& 2m+2n &&=&&6 \end{align} $$

Solving the above, we get $(m,n)=(1,4)$ or $(m,n)=(1,2)$.