KURSCHAK 1950

Several people visited a library yesterday. Each one visited the library just once (in the course of yesterday). Amongst any three of them, there were two who met in the library. Prove that there were two moments $T$ and $T$ ' yesterday such that everyone who visited the library yesterday was in the library at $T$ or $T'$ (or both).

Three circles $C_1$ , $C_2$ , $C_3$ in the plane touch each other (in three different points). Connect the common point of $C_1$ and $C_2$ with the other two common points by straight lines. Show that these lines meet $C_3$ in diametrically opposite points.

$(x_1, y_1,z_1)$ and $(x_2, y_2, z_2)$ are triples of real numbers such that for every pair of integers $(m,n)$ at least one of $x_{1m} + y_{1n} + z_1$ , $x_{2m} + y_{2n} + z_2$ is an even integer. Prove that one of the triples consists of three integers.