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{x} +{1/x} =1 for rationals

Source: 1990 Greece MO Grade XII p4

September 6, 2024
algebrafloor functionInteger Partfractional partinteger and fractional part

Problem Statement

Froa nay real xx, we denote [x][x], the integer part of xx and with {x}\{x\} the fractional part of xx, such that x=[x]+{x}x=[x]+\{x\}. a) Find at least one real xx such that{x}+{1x}=1\{x\}+\left\{\frac{1}{x}\right\}=1 b) Find all rationals xx such that {x}+{1x}=1\{x\}+\left\{\frac{1}{x}\right\}=1
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