Q2 Matemática  (Balkan MO Shortlist 2012)

Seja $a,b,c\ge 0$ e $a+b+c=\sqrt{2}$. Mostre que $$\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\ge 2+frac1\sqrt{3}$$ Clique para revelar o texto ocultoEm geral, se $a_1,a_2,\cdots ,a_n\ge 0$ e ${\sum }_{i=1}^n{a}_i=\sqrt{2}$ temos $$\sum _{i=1}^n\frac{1}{\sqrt{1+a_i^2}}\ge (n-1)+\frac{1}{\sqrt{3}}$$