15th APMO 2003

The polynomial a₈x⁸ +a₇x⁷ + ... + a₀ has a₈ = 1, a₇ = -4, a₆ = 7 and all its roots positive and real. Find the possible values for a₀.

Answer

1/2⁸

Solution

Thanks to Jonathan Ramachandran

Let the roots be x_i. We have Sum x_i² = 4² - 2·7 = 2. By Cauchy we have (x₁·1 + ... + x₈·1) ≤ (x₁² + ... + x₈²)^1/2(1² + ... + 1²)^1/2 with equality iff all x_i are equal. Hence all x_i are equal. So they must all be 1/2.

15th APMO 2003

© John Scholes
jscholes@kalva.demon.co.uk
6 Jul 2003
Last corrected/updated 6 Jul 03