OBM 1997

Given $R, r > 0$ . Two circles are drawn radius $R$ , $r$ which meet in two points. The line joining the two points is a distance $D$ from the center of one circle and a distance $d$ from the center of the other. What is the smallest possible value for $D+d$ ?

Let $A$ be a set of $n$ non-negative integers. We say it has property $\mathcal P$ if the set $\{x + y \mid x, y \in A\}$ has $\binom{n}{2}$ elements. We call the largest element of $A$ minus the smallest element, the diameter of $A$ . Let $f(n)$ be the smallest diameter of any set $A$ with property $\mathcal P$ . Show that $n^2 \leq 4 f(n) < 4 n^3$ .

a) Show that there are no functions $f, g: \mathbb R \to \mathbb R$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x \in \mathbb R$ .

b) Let $S$ be the set of all real numbers greater than 1. Show that there are functions $f, g : S \to S$ satsfying the condition above.

Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$ , where $F_n$ is the Fibonacci sequence ( $F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$ )

Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$

Let $f(x)= x^2-C$ where $C$ is a rational constant. Show that exists only finitely many rationals $x$ such that $\{x,f(x),f(f(x)),\ldots\}$ is finite

$f$ is a plane map onto itself such that points at distance 1 are always taken at point at distance 1. Show that $f$ preserves distances.