51st Ukrainian National Mathematical Olympiad, 3rd Round

Suppose that five years ago family had $n$ sons and $m$ daughters, total age of sons was $N$, total age of daughters was $M$. If $k$ years ago one daughter was born, then the total age of sons now is $N+5n$, for daughters it is $M+5m+k$. Since $N=M+2$, we get that $$(N+5n)+2=M+5m+k\Rightarrow M+5n+4=M+5m+k\Rightarrow k=5(n-m)+4$$ and $0\le k\le 5$, then $k=4$ and $n-m=0$. So two years ago the total age of sons was $N+3n$ and daughters was $M+3m+2$ were equal: $$N+3n-(M+3m+2)=(N-M)+3(n-m)-2=2-2=0.$$ If $k$ years ago one son was born. The total age of sons now is $N+5n+k$, daughters - $M+5m$. We have $$(N+5n+k)+2=M+5m\Rightarrow M+5n+k+4=M+5m\Rightarrow k=5(m-n)-4.$$ Since $0\le k\le 5$, then $k=1$ and $m-n=1$. Two years ago the difference was: $$M+3m-(N+3n)=(M-N)+3(m-n)=-2+3=1.$$