42nd IMO 2001 shortlist

Show that x₁/(1 + x₁²) + x₂/(1 + x₁² + x₂²) + ... + x_n/(1 + x₂² + ... + x_n²) < √n for all real numbers x_i.

Solution

Using Cauchy ∑ a_ib_i ≤ √(∑ a_i²) √(&sum b_i²) with a_i = 1, b_i = x_i/(1+x₁²+x₂²+...+x_i²) we have lhs ≤ rhs √(∑ b_i²).

Now use b₁² ≤ x₁²/(1+x₁²) = 1 - 1/(1+x₁²)

42nd IMO shortlist 2001

© John Scholes
jscholes@kalva.demon.co.uk
20 Oct 2003
Last corrected/updated 20 Oct 03