OMM 1988

In how many ways can one arrange seven white and five black balls in a line in such a way that there are no two neighboring black balls?

If $a$ and $b$ are positive integers, prove that $11a+2b$ is a multiple of $19$ if and only if so is $18a+5b$ .

Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles.

In how many ways can one select eight integers $a_1,a_2, ... ,a_8$ , not necesarily distinct, such that $1 \le a_1 \le ... \le a_8 \le 8$ ?

If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$ .

Consider two fixed points $B,C$ on a circle $w$ . Find the locus of the incenters of all triangles $ABC$ when point $A$ describes $w$ .