RO-NMO 2016 Problem 102

Let be a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ satisfying the conditions: $$\left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. ,$$ for all real numbers $x,y,t$ with $t\in [0,1] .$ Prove that:a) $f(b)+f(c)\le f(a)+f(d) ,$ for any real numbers $a,b,c,d$ such that $a\le b\le c\le d$ and $d-c=b-a.$ b) for any natural number $n\ge 3$ and any $n$ real numbers $x_1,x_2,\ldots ,x_n,$ the following inequality holds. $$f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right)$$