# Problem Section

I like creating math puzzles. I've made puzzles for the Dutch Mathematical Olympiad as well as the International Mathematical Olympiad. Below is a selection of some of my math problems. All problems can be solved with rather elementary methods, say on the level of first year undergraduate math study. However, that doesn't mean that the problems are easy.
1. Suppose that 1<x<2. Prove that the sequence
[x],[2x],[3x],[4x],...
contains infinitely many powers of 2. ([y] is the largest integer <=y)
2. (AMM) A function f:Q x Q-->Q has the following properties
1. f(a,b)=f(b,a) for all a,b in Q,
2. f(f(a,b),c)=f(a,f(b,c)) for all a,b,c in Q,
3. f(0,0)=0,
4. f(a+c,b+c)=f(a,b)+c.
Proof that f(a,b)=max(a,b) for all a,b in Q or f(a,b)=min(a,b) for all a,b in Q.
3. Suppose that e is a real number satisfying 0<=e<1. Assume that a sequance of integers a1,a2,a3,... satisfies
(a1+a2+...+ak)/k<= (a1+a2+...+al+e)/l
for all positive integers k,l. Prove that the sequence a1,a2,a3,... is periodic.
4. (IMO 1989) Suppose that k is a positive integer and suppose there is a set S of n points in the Euclidean plane, such that for every point P of S there exists a circle with center P going through at least k points of S. Prove that
k< 1/2+sqrt(2n).
5. (IMO 1990) Construct a convex polygon with side lengths 12, 22,32,...,19902 such that all angles are equal.
6. (IMO 1991) Construct a bounded sequence of real numbers a1, a2, a3,.... such that |ai-aj||i-j|>=1 for all distinct positive integers i and j.

10/5/2000