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{x²}+{x}=0.99 is true for some x, but {x²}+{x}=1 isn´t, in Q

Source: Romanian District Olympiad 2004, Grade IX, Problem 3

October 7, 2018
fractional partalgebra

Problem Statement

a) Show that there are infinitely many rational numbers x>0 x>0 such that {x2}+{x}=0.99. \left\{ x^2 \right\} +\{ x \} =0.99. b) Show that there are no rational numbers x>0 x>0 such that {x2}+{x}=1. \left\{ x^2 \right\} +\{ x \} =1.
{} \{\} denotes the usual fractional part.
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