41st Swedish 2001 Problem 5 Find all polynomials p(x) such that p'(x) 2 = c p(x) p''(x) for some constant c.
p(x) = a(x+b) n Solution Let p(x) have degree n. Comparing the coefficients of the highest power of x, we get c = n/(n-1). Using calculus, we integrate p'/p = cp"/p' to get ln p = c ln p' + k, so p' = a p (n-1)/n . Hence p must be h(x + b) n for some h, b, and it is easily checked that that works. Comment. Can anyone see how to do this without calculus? I am not sure what the question intended since it uses derivatives in the question. 41st Swedish 2001 © John Scholes jscholes@kalva.demon.co.uk 21 December 2003 Last corrected/updated 21 Dec 03