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Minimum of Floor Functions

Source: Benelux MO 2014 Problem 1

July 17, 2014
functionfloor functionalgebra unsolvedalgebra

Problem Statement

Find the smallest possible value of the expression ⌊a+b+cd⌋+⌊b+c+da⌋+⌊c+d+ab⌋+⌊d+a+bc⌋\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor in which a, b, ca,~ b,~ c, and dd vary over the set of positive integers.
(Here ⌊x⌋\lfloor x\rfloor denotes the biggest integer which is smaller than or equal to xx.)
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