2nd Swedish 1962 Problem 1 Find all polynomials f(x) such that f(2x) = f '(x) f ''(x).
4x 3 /9 Solution Suppose deg f = n. Then deg f ' = n-1 and deg f " = n-2, so n = n-1+n-2. Hence n = 3. So put f(x) = ax 3 + bx 2 + cx + d. Then we have 8ax 3 + 4bx 2 + 2cx + d = (3ax 2 + 2bx + c)(6ax + 2b) = 18a 2 x 3 + 18abx 2 + (6ac+4b 2 )x + 2bc. Comparing x 3 , a = 4/9. Then comparing x 2 , b = 0. Comparing x, c = 0. Comparing constant term, d = 0. 2nd Swedish 1962 © John Scholes jscholes@kalva.demon.co.uk 30 December 2003 Last corrected/updated 30 Dec 03