# 15th APMO 2003 – Problem 1

**Question**

The polynomial a_{8}x^{8} +a_{7}x^{7} + … + a_{0} has a_{8} = 1, a_{7} = -4, a_{6} = 7 and all its roots positive and real. Find the possible values for a_{0}.

**Answer**

Let the roots be x_{i}. We have Sum x_{i}^{2} = 4^{2} – 2·7 = 2. By Cauchy-Schwarz inequality, we have (x_{1}·1 + … + x_{8}·1) ≤ (x_{1}^{2} + … + x_{8}^{2})^{1/2}(1^{2} + … + 1^{2})^{1/2} with equality iff all x_{i} are equal. Hence all x_{i} are equal. So they must all be 1/2.