BALKAN 1993

Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that $$(a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6.$$ Find the maximum possible value of $f$ .

Cyprus

A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called monotone if $a_n\leq a_{n-1} \leq \cdots \leq a_0$ . Determine the number of monotone positive integers with at most 1993 digits.

Circles $\mathcal C_1$ and $\mathcal C_2$ with centers $O_1$ and $O_2$ , respectively, are externally tangent at point $\lambda$ . A circle $\mathcal C$ with center $O$ touches $\mathcal C_1$ at $A$ and $\mathcal C_2$ at $B$ so that the centers $O_1$ , $O_2$ lie inside $C$ . The common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $\lambda$ intersects the circle $\mathcal C$ at $K$ and $L$ . If $D$ is the midpoint of the segment $KL$ , show that $\angle O_1OO_2 = \angle ADB$ .

Greece

Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation $$\frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m$$ has a positive integer solution $(x, y) \neq (1, 1)$ if and only if $m = p$ .

Romania