Team selection tests for JBMO 2018

Review the given condition as $$pq(2r+50)=pr(7q+55)=qr(8p+12)=A.$$ This implies that $A$ is a multiple of $p$, $q$ and $r$ so the value $K=\frac{A}{pqr}$ is an integer. Dividing through, we have $$K=8+\frac{12}{p}=7+\frac{55}{q}=2+\frac{50}{r}.$$ Hence, $p\mid 12$, $q\mid 55$, $r\mid 50$. So we have 3 cases as follow - $p=2$, $q=11$, $r=5$. - $p=3$, $q=11$, $r=5$. - $p=3$, $q=5$, $r=2$. We can check that only $(p,q,r)=(3,11,5)$ works so $K=12$ and the value of $A$ is $1980$. $\square$