49th Mathematical Olympiad in Ukraine

Let us add $a^2$ to both sides of our equality: $$a^2+ab+bc+ca=1+a^2\text{or}(a+b)(a+c)=1+a^2.$$ Analogously, $$(a+b)(b+c)=1+b^2\text{and}(c+b)(a+c)=1+c^2.$$ If we now multiply together all these equalities, we will obtain that $(1+a^2)(1+b^2)(1+c^2)=((a+b)(b+c)(a+c){)}^2$, as required.