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{5{4{3{2{х}}}}}=1 nested fractional parts (II Soros Olympiad 1995-96 R3 9.8)

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June 4, 2024
fractional partalgebra

Problem Statement

Let a{a} be the fractional part of the number aa, that is, {a}=a−[a]\{a\} = a - [a], where[a] [a] is the integer part of a a. (For example, {1.7}=1.7−1=0.7\{1.7\} = 1.7 -1 = 0.7,{−2}=−2−(−3)=3−2\{-\sqrt2 \}= -\sqrt2 -(-3) = 3-\sqrt2.) a) How many solutions does the equation have {5{4{3{2{x}}}}}=1  ? \{5\{4\{3\{2\{x\}\}\}\}\}=1\,\, ? b) Find its greatest solution.
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