Olympiad problems

1989 All Soviet Union Mathematical Olympiad, 504

$ABC$ is a triangle. Points $D, E, F$ are chosen on $BC, CA, AB$ such that $B$ is equidistant from $D$ and $F$, and $C$ is equidistant from $D$ and $E$. Show that the circumcenter of $AEF$ lies on the bisector of $EDF$.

1940 Moscow Mathematical Olympiad, 060

Tags: geometry , equidistant , circle construction , geometric construction

Construct a circle equidistant from four points on a plane. How many solutions are there?

1938 Moscow Mathematical Olympiad, 038

Tags: geometry , combinatorial geometry , equidistant , 3d geometry , plane

In space $4$ points are given. How many planes equidistant from these points are there? Consider separately (a) the generic case (the points given do not lie on a single plane) and (b) the degenerate cases.

1992 All Soviet Union Mathematical Olympiad, 563

Tags: circles , reflection , symmetry , equidistant , geometry

$A$ and $B$ lie on a circle. $P$ lies on the minor arc $AB$. $Q$ and $R$ (distinct from $P$) also lie on the circle, so that $P$ and $Q$ are equidistant from $A$, and $P$ and $R$ are equidistant from $B$. Show that the intersection of $AR$ and $BQ$ is the reflection of $P$ in $AB$.

1947 Moscow Mathematical Olympiad, 132

Tags: geometry , locus , equidistant , distance

Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.