For each positive integer $n$, let $I_n$ denote the number of integers $p$ for which $50^n<7^p<50^{n+1}$. (a) Prove that, for each $n$, $I_n$ is either $2$ or $3$. (b) Prove that $I_n=3$ for infinitely many $n\in\mathbb N$, and find at least one such $n$.

Let be given a semi-sphere $\Sigma$ whose base-circle lies on plane $p$. A variable plane $Q$, parallel to a fixed plane non-perpendicular to $P$, cuts $\Sigma$ at a circle $C$. We denote by $C'$ the orthogonal projection of $C$ onto $P$. Find the position of $Q$ for which the cylinder with bases $C$ and $C'$ has the maximum volume.

Let us define a function $f:\mathbb N\to\mathbb N_0$ by $f(1)=0$ and, for all $n\in\mathbb N$, $$f(2n)=2f(n)+1,\qquad f(2n+1)=2f(n).$$Given a positive integer $p$, define a sequence $(u_n)$ by $u_0=p$ and $u_{k+1}=f(u_k)$ whenever $u_k\ne0$. (a) Prove that, for each $p\in\mathbb N$, there is a unique integer $v(p)$ such that $u_{v(p)}=0$. (b) Compute $v(1994)$. What is the smallest integer $p>0$ for which $v(p)=v(1994)$. (c) Given an integer $N$, determine the smallest integer $p$ such that $v(p)=N$.

Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. Proposed by United Kingdom.

Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$. (b) Calculate $f(n)$ for any natural number $n$.