38th IMO 1997 shortlist

Let x₁ ≥ x₂ ≥ x₃ ... ≥ x_n+1 = 0. Show that √(x₁ + x₂ + ... + x_n) ≤ (√x₁ - √x₂) + (√2) (√x₂ - √x₃) + ... + (√n) (√x_n - √x_n+1).

Solution

Put y_k = √x_k - √x_k+1. Then (y_k + y_k+1 + ... + y_n)² = x_k. Hence lhs² = x₁ + x₂ + ... + x_n = ∑ ky_k² + 2 ∑_1≤h<k≤n hy_hy_k.

rhs = ∑ (√k) y_k, so rhs² = ∑ky_k² + 2 ∑_1≤h<k≤n √(hk) y_ky_l. The result follows since h ≤ √(hk) for h < k.

38th IMO shortlist 1997

© John Scholes
jscholes@kalva.demon.co.uk
2 Dec 2003
Last corrected/updated 2 Dec 2003