67th Romanian Mathematical Olympiad

(a) Let $A$ be a ring having property (P), let $m=|U(A)|$, and notice that $(-1{)}^m=1$. If $m$ is odd, then $-1=1$, so $1$ has order $2$ in the additive group $(A,+)$, and consequently $|A|$ is even. If $m$ is even, so must be $|A|$, since the former divides the latter.

(b) Let $m$ be a positive integer, let $n=2^{m+2}$, and let $A=\underset{m}{\underbrace{{\mathbb{Z}}_2\times \cdots \times {\mathbb{Z}}_2}}\times {\mathbb{Z}}_4$.

The multiplicative group $U(A)=\{(\hat{1},\dots ,\hat{1},\hat{1}),(\hat{1},\dots ,\hat{1},\hat{3})\}$ is easily seen to be isomorphic to the subgroup $\{(\hat{0},\dots ,\hat{0},\hat{0}),(\hat{0},\dots ,\hat{0},\hat{2})\}$ of the additive group $(A,+)$.