RO-TST 1990

Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$ .

Prove that there are infinitely many n's for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$ , and prove that there are infinitely many $n$ 's for which there is no such partition.

The sequence $(x_n)_{n \geq 1}$ is defined by: $x_1=1$ $x_{n+1}=\frac{x_n}{n}+\frac{n}{x_n}$ Prove that $(x_n)$ increases and $[x_n^2]=n$ .

For a set $S$ of $n$ points, let $d_1 > d_2 >... > d_k > ...$ be the distances between the points. A function $f_k: S \to N$ is called a coloring function if, for any pair $M,N$ of points in $S$ with $MN \le d_k$ , it takes the value $f_k(M)+ f_k(N)$ at some point. Prove that for each $m \in N$ there are positive integers $n,k$ and a set $S$ of $n$ points such that every coloring function $f_k$ of $S$ satisfies $| f_k(S)| \le m$

The distance between any two of six given points in the plane is at least $1$ . Prove that the distance between some two points is at least $\sqrt{\frac{5+\sqrt5}{2}}$

Let $p,q$ be positive prime numbers and suppose $q>5$ . Prove that if $q \mid 2^{p}+3^{p}$ , then $q>p$ .

Laurentiu Panaitopol

In a group of $n$ persons, (i) each person is acquainted to exactly $k$ others, (ii) any two acquainted persons have exactly $l$ common acquaintances, (iii) any two non-acquainted persons have exactly $m$ common acquaintances. Prove that $m(n-k -1) = k(k -l -1)$ .