KURSCHAK 2013

Let $a,b$ be positive real numbers satisfying $2ab=a-b$ . Denote for any positive integer $k$ $x_k$ and $y_k$ to be the closest integer to $ak$ and $bk$ , respectively (if there are two closest integers, choose the larger one). Prove that any positive integer $n$ appears in the sequence $(x_k)_{k\ge 1}$ if and only if it appears at least three times in the sequence $(y_k)_{k\ge 1}$ .

Consider the closed polygonal discs $P_1$ , $P_2$ , $P_3$ with the property that for any three points $A\in P_1$ , $B\in P_2$ , $C\in P_3$ , we have $[\triangle ABC]\le 1$ . (Here $[X]$ denotes the area of polygon $X$ .) (a) Prove that $\min\{[P_1],[P_2],[P_3]\}<4$ . (b) Give an example of polygons $P_1,P_2,P_3$ with the above property such that $[P_1]>4$ and $[P_2]>4$ .

Is it true that for integer $n\ge 2$ , and given any non-negative reals $\ell_{ij}$ , $1\le i<j\le n$ , we can find a sequence $0\le a_1,a_2,\ldots,a_n$ such that for all $1\le i<j\le n$ to have $|a_i-a_j|\ge \ell_{ij}$ , yet still $\sum_{i=1}^n a_i\le \sum_{1\le i<j\le n}\ell_{ij}$ ?