51st Ukrainian National Mathematical Olympiad, 3rd Round

a) We construct a counterexample. Consider two squares $ABMN$ and $NMCD$ with side length $\frac{1}{4}$. Then, perimeter of the rectangle $ABCD$ is $2\cdot (\frac{1}{2}+\frac{1}{4})=\frac{3}{2}$ — non-integer, while both squares have integer perimeter.

b) Consider the side of the external square, and all squares that have one side which belongs to the side of the external square (see figure 1).



Let $a$ be the side of the external square, and the sides of small squares are $a_1,a_2,...,a_n$. Then $a=a_1+a_2+...+a_n$, and each side $a_i$, after multiplying by $4$, is integer. Therefore, perimeter $P$ of our square $ABCD$ is $P=4a=4a_1+4a_2+...+4a_n$ — integer.