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Floors and Fracs

Source: PRMO Leg 2 P30

August 25, 2019
number theoryfloor functionPRMO

Problem Statement

For any real number xx, let ⌊x⌋\lfloor x \rfloor denote the integer part of xx; {x}\{ x \} be the fractional part of xx ({x}\{x\} == x−x- ⌊x⌋\lfloor x \rfloor). Let AA denote the set of all real numbers xx satisfying {x}=x+⌊x⌋+⌊x+(1/2)⌋20\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20} If SS is the sume of all numbers in AA, find ⌊S⌋\lfloor S \rfloor
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