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How many x are there such that x,[x],{x} are in harmonic progression

Source: MTRP 2015 Class 11-Multiple Choice Question: Problem 7 :-

September 21, 2020
algebraGreatest Integer Function

Problem Statement

How many xx are there such that x,[x],{x}x,[x],\{x\} are in harmonic progression (i.e, the reciprocals are in arithmetic progression)? (Here [x][x] is the largest integer less than equal to xx and {x}=xāˆ’[x]\{x\}=x-[ x] )
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