VMO 1971

Consider positive integers $m <n,p < q$ such that $(m, n) = 1, (p, q) = 1$ and satisfy the condition that if $\frac{m}{n}= tan\alpha$ and $\frac{p}{q} = tan\beta$ , then $\alpha + \beta = 45^o$ . i) Given $m, n$ , find $p, q$ . ii) Given $n, q$ , find $m, p$ . ii) Given $m, q$ , find $n, p$ .

$ABCDA'B'C'D'$ is a cube (with $ABCD$ and $A'B'C'D'$ faces, and $AA', BB', CC', DD'$ edges). $L$ is a line which intersects or is parallel to the lines $AA', BC$ and $DB'$ . $L$ meets the line $BC$ at $M$ (which may be the point at infinity). Let $m = |BM|$ . The plane $MAA'$ meets the line $B'C'$ at $E$ . Show that $|B'E| = m$ . The plane $MDB'$ meets the line $A'D'$ at $F$ . Show that $|D'F| = m$ . Hence or otherwise show how to construct the point $P$ at the intersection of $L$ and the plane $A'B'C'D'$ . Find the distance between $P$ and the line $A'B'$ and the distance between $P$ and the line $A'D'$ in terms of $m$ . Find a relation between these two distances that does not depend on $m$ . Find the locus of $M$ . Let $S$ be the envelope of the line $L$ as $M$ varies. Find the intersection of $S$ with the faces of the cube.