BG-MO 1996

Sequence $\{a_n\}$ it define $a_1=1$ and $$a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$$ for all $n\ge 1$ Prove that $\lfloor a_n^2\rfloor=n$ for all $n\ge 4.$

The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$ , while the diagonals $AC$ and $BD$ in the point $F$ . The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$ . Prove that $\angle EHF=90^\circ$ .

A square table of size $7\times 7$ with the four corner squares deleted is given. [list=a] [] What is the smallest number of squares which need to be colored black so that a $5-$ square entirely uncolored Greek cross (Figure 1) cannot be found on the table? [] Prove that it is possible to write integers in each square in a way that the sum of the integers in each Greek cross is negative while the sum of all integers in the square table is positive. [/list]

size(3.5cm); usepackage("amsmath");
MP("\text{Figure }1.", (1.5, 3.5), N);
DPA(box((0,1),(3,2))^^box((1,0),(2,3)), black);