Home > Analysis, Olympiad > Traian Lalescu contest 2008 Problem 3

## Traian Lalescu contest 2008 Problem 3

Suppose $U\subset \Bbb{R}^2$ is a subset which contains the unit disk $D$, and consider a function $f \in C^1(D)$ such that $\displaystyle \left|\frac{\partial f}{\partial x}(P)\right|\leq 1, \left|\frac{\partial f}{\partial y}(P)\right|\leq 1,\ \forall P \in D$.

Consider a finite set of points $M_1,M_2,...,M_n \in D$ with the center of gravity $O$. Prove that for all $P \in D$ we have the inequality $\displaystyle \left|f(P)-\sum_{i=1}^n f(M_i)\right| \leq 2$.

Traian Lalescu student contest 2008