ÖMO 2002 Problem 4

Let $a_0, a_1, ..., a_{2002}$ be real numbers. a) Show that the smallest of the values $a_k (1-a_{2002-k})$ ( $0 \le k \le 2002$ ) the following applies: it is smaller or equal to $1/4$ . b) Does this statement always apply to the smallest of the values $a_k (1-a_{2003-k})$ ( $1 \le k \le 2002$ ) ? c) Show for positive real numbers $a_0, a_1, ..., a_{2002}$ : the smallest of the values $a_k (1-a_{2003-k})$ ( $1 \le k \le 2002$ ) is less than or equal to $1/4$ .