OMM 1989

In a triangle $ABC$ the area is $18$ , the length $AB$ is $5$ , and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$ .

Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$ , but $a^5$ does not divide $b^6$

Prove that there is no $1989$ -digit natural number at least three of whose digits are equal to $5$ and such that the product of its digits equals their sum.

Find the smallest possible natural number $n = \overline{a_m ...a_2a_1a_0}$ (in decimal system) such that the number $r = \overline{a_1a_0a_m ..._20}$ equals $2n$ .

Let $C_1$ and $C_2$ be two tangent unit circles inside a circle $C$ of radius $2$ . Circle $C_3$ inside $C$ is tangent to the circles $C,C_1,C_2$ , and circle $C_4$ inside $C$ is tangent to $C,C_1,C_3$ . Prove that the centers of $C,C_1,C_3$ and $C_4$ are vertices of a rectangle.

Determine the number of paths from $A$ to $B$ on the picture that go along gridlines only, do not pass through any point twice, and never go upwards?