MathDB

A nice inequality with a+b+c=sqrt2

Source: BMO 2012

June 28, 2013

inequalities3-variable inequalitysquare root inequalityBalkanlagrangeBMOimo 65th gold champion

Problem Statement

Let

a,b,c\ge 0

and

a+b+c=\sqrt2

. Show that

\frac1{\sqrt{1+a^2}}+\frac1{\sqrt{1+b^2}}+\frac1{\sqrt{1+c^2}} \ge 2+\frac1{\sqrt3}

In general if

a_1, a_2, \cdots , a_n \ge 0

and

\sum_{i=1}^n a_i=\sqrt2

we have

\sum_{i=1}^n \frac1{\sqrt{1+a_i^2}} \ge (n-1)+\frac1{\sqrt3}