Here are some problems by Harm Derksen, who made problems for the Dutch Mathematical Olympiad and the International Mathematical Olympiad.

Problem

Here are some problems by Harm Derksen, who made problems for the Dutch
Mathematical Olympiad and the International Mathematical Olympiad.  All
can be solved with rather elementary methods, on the level of first year
undergraduate study, but that doesn't mean they are easy.

0. Divide the first figure in three parts, and fit them together in
   another way, such that the result will be the second figure.
   [ Editor: the two figures were an image (divideproblem.gif) on
   Derksen's web page and are not in the collected material. ]

1. If f in R[x] is a polynomial with real coefficients such that f maps
   the interval (0,infinity) into itself, then there exists a positive
   integer n such that (1+x)^n f has only nonnegative coefficients.

2. 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.)

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

4. Suppose that e is a real number satisfying 0 <= e < 1.  Assume that
   a sequence of integers a_1, a_2, a_3, ... satisfies

       (a_1 + a_2 + ... + a_k)/k <= (a_1 + a_2 + ... + a_l + e)/l

   for all positive integers k, l.  Prove that the sequence a_1, a_2,
   a_3, ... is periodic.

5. (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).

6. (IMO 1990) Construct a convex polygon with side lengths 1^2, 2^2,
   3^2, ..., 1990^2 such that all angles are equal.

7. (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.

-- Harm Derksen <hderksen@math.unibas.ch>, "Mathematical Recreation"
   web page, April 1996