OMM 1998

A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$ . For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$ . Find infinitely many pairs of consecutive numbers each of which is lucky.

Rays $l$ and $m$ forming an angle of $a$ are drawn from the same point. Let $P$ be a fixed point on $l$ . For each circle $C$ tangent to $l$ at $P$ and intersecting $m$ at $Q$ and $R$ , let $T$ be the intersection point of the bisector of angle $QPR$ with $C$ . Describe the locus of $T$ and justify your answer.

Find all integers that can be written in the form $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{9}{a_9}$ where $a_1,a_2, ...,a_9$ are nonzero digits, not necessarily different.

The tangents at points $B$ and $C$ on a given circle meet at point $A$ . Let $Q$ be a point on segment $AC$ and let $BQ$ meet the circle again at $P$ . The line through $Q$ parallel to $AB$ intersects $BC$ at $J$ . Prove that $PJ$ is parallel to $AC$ if and only if $BC^2 = AC\cdot QC$ .

A plane in space is equidistant from a set of points if its distances from the points in the set are equal. What is the largest possible number of equidistant planes from five points, no four of which are coplanar?