Question 1
Consider pairs of reduced fractions $m/n\lt 1$ and $p/q\lt 1$, where $m,n,p,q$ are positive integers, such that if $m/n=\tan\alpha$ and $p/q=\tan\beta$, then $\alpha+\beta=45^\circ$.
(a) Given $m,n$, determine $p,q$. When does a solution exist?
(b) Given $n,q$, determine $m,p$. When does a solution exist?
(c) Given $m,q$, determine $n,p$. When is there one solution? When are there two?
(a) Given $m,n$, determine $p,q$. When does a solution exist?
(b) Given $n,q$, determine $m,p$. When does a solution exist?
(c) Given $m,q$, determine $n,p$. When is there one solution? When are there two?