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, ... , an such that a1 cos x + a2 cos 2x + ... + an 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.

Sweden home
© John Scholes
23 September 2003
Last corrected/updated 30 Dec 03