ÖMO 1991

Consider a convex solid $K$ in space and two parallel planes $\epsilon _1$ and $\epsilon _2$ on the distance $1$ tangent to $K$ . A plane $\epsilon$ between $\epsilon _1$ and $\epsilon _2$ is on the distance $d_1$ from $\epsilon _1$ . Find all $d_1$ such that the part of $K$ between $\epsilon _1$ and $\epsilon$ always has a volume not exceeding half the volume of $K$ .

Find all functions $f: \mathbb{Z} - \{ 0 \} \rightarrow \mathbb{Q}$ satisfying:

$f \left( \frac{x+y}{3} \right)=\frac {f(x)+f(y)}{2},$ whenever $x,y,\frac{x+y}{3} \in \mathbb{Z} - \{ 0 \}.$

$(a)$ Prove that $91$ divides $n^{37}-n$ for all integers $n$ . $(b)$ Find the largest $k$ that divides $n^{37}-n$ for all integers $n$ .

The sequence $(a_n)$ is given by $a_1=1,a_2=0$ and:

$a_{2k+1}=a_k+a_{k+1}, a_{2k+2}=2a_{k+1}$ for $k \in \mathbb{N}.$

Find $a_m$ for $m=2^{19}+91.$

For all positive integers $n$ prove the inequality:

$\left( \frac{1+(n+1)^{n+1}}{n+2} \right)^{n-1}>\left( \frac{1+n^n}{n+1} \right)^n.$

Find the number of ten-digit natural numbers (which do not start with zero) containing no block $1991$ .