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{a}+{1/a} =1 implies {a^n} +{1/a^n} =1

Source: Romanian District Olympiad 2011, Grade IX, Problem 4

October 8, 2018
algebrafractional part

Problem Statement

Let be a nonzero real number a, a, and a natural number n. n. Prove the implication: {a}+{1a}=1  ⟹  {an}+{1an}=1, \{ a \} +\left\{\frac{1}{a}\right\} =1 \implies \{ a^n \} +\left\{\frac{1}{a^n}\right\} =1 , where {} \{\} is the fractional part.
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