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12nd ibmo - mexico 1997/q1.

Source: Spanish Communities

April 22, 2006
floor functioncalculusintegrationrationumber theory unsolvednumber theory

Problem Statement

Let r≥1r\geq1 be areal number that holds with the property that for each pair of positive integer numbers mm and nn, with nn a multiple of mm, it is true that ⌊nr⌋\lfloor{nr}\rfloor is multiple of ⌊mr⌋\lfloor{mr}\rfloor. Show that rr has to be an integer number.
Note: If xx is a real number, ⌊x⌋\lfloor{x}\rfloor is the greatest integer lower than or equal to xx}.
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