Prove that [x] + $\frac{[2x]}{2} + ..... \frac{[nx]}{n}$ $ \le $ $[nx]$ $x \in R^+$
Where [.] is the greatest integer function. - USAMO 1981.
Usually, we can find the solutions of USAMO problems pretty easily but all solutions of this i found were wrong.
My attempt: I went on with induction and reached [kx +k] $\ge$ [kx] + $\frac{[kx+x]}{k+1} $ but turns out this wasn't true. So inequality must be very weak.
I am absolutely baffled by this problem, any solution would be really appreciated. Thanks!
I AM LOOKING FOR A NON CALCULUS solution