Question bank APMO 1994

Find all functions $f:\mathbb R\to\mathbb R$ such that:
(i) $f(x)+f(y)+1\ge f(x+y)\ge f(x)+f(y)$ for all real $x,y$;
(ii) $f(0)\ge f(x)$ for all $x\in[0,1)$;
(iii) $-f(-1)=f(1)=1$.

Given a nondegenerate triangle $ABC$ with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH|\lt 3R$.

Let $n=a^2+b^2$, where $a,b$ are relatively prime integers, and suppose every prime $p\le\sqrt n$ divides $ab$. Determine all such integers $n$.

Is there an infinite set of points in the plane such that no three are collinear and the distance between any two is rational?

List $A$ contains the numbers $10^k$ in base $10$, for integers $k\ge1$. Lists $B,C$ contain the same numbers written in bases $2,5$, respectively. The first entries are: $10\leftrightarrow1010_2\leftrightarrow20_5$; $100\leftrightarrow1100100_2\leftrightarrow400_5$; $1000\leftrightarrow1111101000_2\leftrightarrow13000_5$.
Prove that for every integer $n\gt 1$, exactly one entry in exactly one of the lists $B,C$ has exactly $n$ digits.