49. Can any rectangle be cut into two unequal but similar rectangles?
49. Suppose that rectangle $A B C D$ is cut into two unequal similar rectangles $A B M K$ and $K M C D$ (Fig. 71). From their similarity, we get: $|A K|:|K M|=|K M|:|M C|$. Let: $|A D|=a,|A B|=b,|A K|=x$. Now the proportion will take the form: $x: b=b:(a-x)$. From this, $x^{2}-a x+b^{2}=0$. Taking into account that $0 < x < a$, we have: $a^2 - 4 b^{2} > 0$, i.e., $a > 2 b$.