Let $a_0,a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2}\geq a_n$ for all positive integers $n$. Show that $$\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}$$holds for all positive integers $n$.
The required inequality is equivalent to $\sum_{i =1}^{n}{(a_{n+1}-a_i)} \geq \sum_{i =1}^{n}{(a_{n+1-i}-a_0)}$.
So it's enough to show that, for each $i\in \{ 1,2,...,n\}$, $a_{n+1}-a_i\geq a_{n+1-i}-a_0$.
This is true since $a_{n+1}-a_i\geq a_{n}-a_{i-1}\geq a_{n-1} -a_{i-2}\geq ...\geq a_{n+1-i}-a_0$.
Here's something harder.
Poland 1970
Let $a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n}+a_{n+2}}{2}\geq a_{n+1}$ for all positive integers $n$.
Show that $$\forall_{n\in Z_+}\ \frac{a_1+a_3+...+a_{2n+1}}{n+1}\ge\frac{a_2+a_4+...+a_{2n}}{n}$$