MathDB

6

Part of 2006 South africa National Olympiad

Problems(1)

This average of floor functions fluctuates

Source: South African MO 2006 Q6

5/27/2012
Consider the function ff defined by f(n)=1n(⌊n1⌋+⌊n2⌋+⋯+⌊nn⌋)f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right ) for all positive integers nn. (Here ⌊x⌋\lfloor x\rfloor denotes the greatest integer less than or equal to xx.) Prove that
(a) f(n+1)>f(n)f(n+1)>f(n) for infinitely many nn.
(b) f(n+1)<f(n)f(n+1)<f(n) for infinitely many nn.
functionfloor functionalgebra unsolvedalgebra