12th USAMO 1983

Show that the five roots of the quintic a₅x⁵ + a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀ = 0 are not all real if 2a₄² < 5a₅a₃.

Solution

Let the roots be r_i. If the condition holds, then 2 ∑ r_i < 5 ∑ r_ir_j. Expanding, 2 ∑ r_i² + 4 ∑ r_ir_j < 5 ∑ r_ir_j, or 2 ∑ r_i² < ∑ r_ir_j. But if r_i and r_j are real we have we have 2r_ir_j ≤ r_i² + r_j². So if all the roots are real, adding the 10 similar equations gives 2 ∑ r_ir_j ≤ 4 ∑ r_i². Contradiction. Hence not all the roots are real.

12th USAMO 1983

© John Scholes
jscholes@kalva.demon.co.uk
24 Aug 2002