PE-MO 2015

If $C$ is a set of $n$ points in the plane that has the following property: For each point $P$ of $C$ , there are four points of $C$ , each one distinct from $P$ , which are the vertices of a square. Find the smallest possible value of $n$ .

Let $ABCDEF$ be a convex hexagon. The diagonal $AC$ is cut by $BF$ and $BD$ at points $P$ and $Q$ , respectively. The diagonal $CE$ is cut by $DB$ and $DF$ at points $R$ and $S$ , respectively. The diagonal $EA$ is cut by $FD$ and $FB$ at points $T$ and $U$ , respectively. It is known that each of the seven triangles $APB, PBQ, QBC, CRD, DRS, DSE$ and $AUF$ has area $1$ . Find the area of the hexagon $ABCDEF$ .

Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$ , such that $$\lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$ Prove that at least $n - 1$ of these numbers are perfect squares.

Clarification: Given a real number $x$ , $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$ . For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$ .