2nd Swedish 1962


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 n² - 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 L₁, L₂, L₃ lie in a plane, and Y is the statement, each pair of the lines L₁, L₂, L₃ intersect. (B) Every sufficiently large integer n satisfies n = a⁴ + b⁴ for some integers a, b. (C) There are real numbers a₁, a₂, ... , a_n such that a₁ cos x + a₂ 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.