CMO 2015

Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$ . Show that

$$\left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).$$

Let $A, B, D, E, F, C$ be six points lie on a circle (in order) satisfy $AB=AC$ . Let $P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD$ . Let $K$ be a point lie on $ST$ satisfy $\angle QKS=\angle ECA$ .

Prove that $\frac{SK}{KT}=\frac{PQ}{QR}$

Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions:

i) $|A| = n$ , $|B| = m$ and $A$ is a subset of $B$ ii) For any distinct $x,y \in B$ , $x+y \in B$ iff $x,y \in A$ Determine the minimum value of $m$ .

Determine all integers $k$ such that there exists infinitely many positive integers $n$ not satisfying $$n+k |\binom{2n}{n}$$

Given $30$ students such that each student has at most $5$ friends and for every $5$ students there is a pair of students that are not friends, determine the maximum $k$ such that for all such possible configurations, there exists $k$ students who are all not friends.

Let $a_1,a_2,...$ be a sequence of non-negative integers such that for any $m,n$ $$\sum_{i=1}^{2m} a_{in} \leq m.$$ Show that there exist $k,d$ such that $$\sum_{i=1}^{2k} a_{id} = k-2014.$$