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

Algebra Difficulty 5.2 AIME, harder Prove it Ukraine

Solve the equation:
$(x + 1)^5 + (x + 1)^4(x - 1) + (x + 1)^3(x - 1)^2 + \\ + (x + 1)^2(x - 1)^3 + (x + 1)(x - 1)^4 + (x - 1)^5 = 0.$

Solution

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 $x = 0$ .

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