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Romania NMO 2023 Grade 8 P3

Source: Romania National Olympiad 2023

April 14, 2023
floor functionalgebra

Problem Statement

We say that a natural number nn is interesting if it can be written in the form
n=⌊1a⌋+⌊1b⌋+⌊1c⌋, n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor, where a,b,ca,b,c are positive real numbers such that a+b+c=1.a + b + c = 1.
Determine all interesting numbers. ( ⌊x⌋\lfloor x \rfloor denotes the greatest integer not greater than xx.)
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