1. Find all polynomials f(x) such that f(2x) = f '(x) f ''(x).

2. ABCD is a square side 1. P and Q lie on the side AB and R lies on the side CD. What are the possible values for the circumradius of PQR?

3. Find all pairs (m, n) of integers such that n^{2} - 3mn + m - n = 0.

4. Which of the following statements are true?
(A) X implies Y, or Y implies X, where X is the statement, the lines L_{1}, L_{2}, L_{3} lie in a plane, and Y is the statement, each pair of the lines L_{1}, L_{2}, L_{3} intersect.
(B) Every sufficiently large integer n satisfies n = a^{4} + b^{4} for some integers a, b.
(C) There are real numbers a_{1}, a_{2}, ... , a_{n} such that a_{1} cos x + a_{2} cos 2x + ... + a_{n} cos nx > 0 for all real x.

5. Find the largest cube which can be placed inside a regular tetrahedron with side 1 so that one of its faces lies on the base of the tetrahedron.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.