FI-MO 1997

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

Circles with radii $R$ and $r$ ( $R > r$ ) are externally tangent. Another common tangent of the circles in drawn. This tangent and the circles bound a region inside which a circle as large as possible is drawn. What is the radius of this circle?

$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others. $5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?

Count the sum of the four-digit positive integers containing only odd digits in their decimal representation.

For an integer $n\geq 3$ , place $n$ points on the plane in such a way that all the distances between the points are at most one and exactly $n$ of the pairs of points have the distance one.