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Epression is an integer if and only if {a}+{b}+{c} is

Source: Pan African MO 2006 Q3

May 1, 2013
floor functionalgebra unsolvedalgebra

Problem Statement

For a real number xx let ⌊x⌋\lfloor x\rfloor be the greatest integer less than or equal to xx and let {x}=x−⌊x⌋\{x\} = x - \lfloor x\rfloor. If a,b,ca, b, c are distinct real numbers, prove that a3(a−b)(a−c)+b3(b−a)(b−c)+c3(c−a)(c−b)\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)} is an integer if and only if {a}+{b}+{c}\{a\} + \{b\} + \{c\} is an integer.
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