APMO 1994

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that (i) For all $x, y \in \mathbb{R}$,

$$f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y)$$

(ii) For all $x \in[0,1), f(0) \geq f(x)$, (iii) $-f(-1)=f(1)=1$.

Find all such functions $f$.

Given a nondegenerate triangle $A B C$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|O H|<3 R$.

Let $n$ be an integer of the form $a^{2}+b^{2}$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $a b$. Determine all such $n$.

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

You are given three lists $A, B$, and $C$. List $A$ contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1 . Lists $B$ and $C$ contain the same numbers translated into base 2 and 5 respectively:

$A$	$B$	$C$
10	1010	20
100	1100100	400
1000	1111101000	13000
$\vdots$	$\vdots$	$\vdots$

Prove that for every integer $n>1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.