JBMO-SL 2002 Problem 6

Let $a_1,a_2,...,a_6$ be real numbers such that: $a_1 \not = 0, a_1a_6 + a_3 + a_4 = 2a_2a_5 \ \mathrm{and}\ a_1a_3 \ge a_2^2$ Prove that $a_4a_6\le a_5^2$ . When does equality holds?