Macedonian Mathematical Olympiad

We consider the magical square:

$A_1$	$A_2$	$A_3$
$B_1$	$B_2$	$B_3$
$C_1$	$C_2$	$C_3$

$$\begin{array}{rl}{A}_1+A_2+A_3& =B_1+B_2+B_3=C_1+C_2+C_3=A_1+B_1+C_1\\ & =A_2+B_2+C_2=A_3+B_3+C_3=A_1+B_2+C_3=C_1+B_2+A_3=S,\end{array}$$ or, equivalently $$\begin{array}{rl}4S& =(B_1+B_2+B_3)+(A_2+B_2+C_2)+(A_1+B_2+C_3)+(C_1+B_2+A_3)\\ & =(A_1+A_2+A_3)+(B_1+B_2+B_3)+(C_1+C_2+C_3)+3B_2=3S+3B_2.\end{array}$$ We get $S=3B_2$. In what follows we will denote the central element $B_2$ by $x$.

Picture 3. Next we consider the colored square in Picture 4. Because $2a+c=3x$ and $2b+d=3x$ we get that the rectangle is filled in the following way:



Analogously to the way the colored square was filled in Picture 3, we get that $c=a$, $b=d$. But then

, from where $a=b=c=d=x$ i.e. all elements of the rectangle have to be equal. Let $n>3$, $m>3$. Then, because of the previous discussion, the rectangle of width $3$ and length $m$ has to be filled with one number (Picture 5). Picture 5. For the same reasons, the same holds for the colored rectangle and every rectangle obtained by vertical translation. Finally, if $n=m=3$, then the rectangle can be filled with $9$ different numbers. If $n>3$ or $m>3$, then the rectangle can be filled only with a single number.