- 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) - (AMM) A function f:
**Q**x**Q**-->**Q**has the following properties- f(a,b)=f(b,a) for all a,b in
**Q**, - f(f(a,b),c)=f(a,f(b,c))
for all a,b,c in
**Q**, - f(0,0)=0,
- f(a+c,b+c)=f(a,b)+c.

**Q**or f(a,b)=min(a,b) for all a,b in**Q**. - f(a,b)=f(b,a) for all a,b in
- Suppose that e is a real number satisfying 0<=e<1. Assume
that a sequance of integers
a
_{1},a_{2},a_{3},... satisfies(a for all positive integers k,l. Prove that the sequence a_{1}+a_{2}+...+a_{k})/k<= (a_{1}+a_{2}+...+a_{l}+e)/l_{1},a_{2},a_{3},... is periodic. - (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). - (IMO 1990)
Construct a convex polygon with side lengths 1
^{2}, 2^{2},3^{2},...,1990^{2}such that all angles are equal. - (IMO 1991)
Construct a bounded sequence of real numbers
a
_{1}, a_{2}, a_{3},.... such that |a_{i}-a_{j}||i-j|>=1 for all distinct positive integers i and j.

10/5/2000