40th Hellenic Mathematical Olympiad

First we examine which values of the area of rectangles are acceptable: Since, $0<x,y\le 8$, the integer $40$ is written only as $40=5\cdot 8$. Since a $5\times 8$ rectangle can be put in the $8\times 8$ rectangle with $4$ ways horizontally and with $4$ ways vertically we have totally $8$ such rectangles. The numbers $39,38,37,34,33,31$ are not the product of two integers $x,y$ with $0<x,y\le 8$. The number $36$ can be written uniquely $36=6\cdot 6$. Since a $6\times 6$ rectangle can be put in the $8\times 8$ rectangle with $3^2$ ways, we have $9$ such rectangles. The number $35$ can be written uniquely $35=5\cdot 7$. Since a $5\times 7$ rectangle can be put in the $8\times 8$ rectangle with $2\cdot (4\cdot 2)=16$ ways, we have $9$ such rectangles. The number $32$ can be written uniquely $32=4\cdot 8$. A $4\times 8$ rectangle can be put in the $8\times 8$ rectangle with $2\cdot 5=10$ ways. Finally, we have $8+9+16+10=43$ rectangles with the required properties.