50th Mathematical Olympiad in Ukraine, Third Round (January 23, 2010)

Answer: $x=0$.

Multiplying both sides by $2=((x+1)-(x-1))$ yields $(x+1{)}^6-(x-1{)}^6=0$ or equivalently $(x+1{)}^2=(x-1{)}^2$. Solving this equation, we obtain the unique solution $x=0$.