# Problem From the 2016 IMO Shortlist

### Solution 1

The required inequality is equivalent to

$\displaystyle \ln(a^2+1)+\ln(b^2+1)+\ln(c^2+1)\le 3\ln\left[\left(\frac{a+b+c}{3}\right)^2+1\right].$

Consider the function $f:\,(0,\infty)\to\mathbb{R},\,$ defined by $f(x)=\ln(x^2+1).\,$ We have $\displaystyle f'(x)=\frac{2x}{x^2+1}\,$ and $\displaystyle f''(x)=-\frac{2(x^2-1)}{(x^2+1)^2}.\,$ It follows that function $f\,$ is concave on $\displaystyle \left[\frac{1}{2},\infty\right).$