Prove a Floor Inequality

During a math class, a proposed problem on a handout is as follows:

Prove that $\lfloor x\rfloor +\dfrac{\lfloor 2x\rfloor}{2}+\dfrac{\lfloor 3x\rfloor}{3}+\cdots +\dfrac{\lfloor nx\rfloor}{n}\le \lfloor nx\rfloor$ for all positive reals $$x$$ and integer $$n$$.

Can someone help me prove it? I tried to use a periodic argument, proving it true for $$x\in [0,1)$$ then plugging in $$x=x'+k$$ for positive integer $$k$$ to prove the inequality for all other $$x$$. However it didn't work, because my inequality substitutions were too strict and I ended up with $$(n-1)x\le \lfloor nx\rfloor$$ which is not true.

Thanks!

Note by Daniel Liu
4 years, 3 months ago

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- 4 years, 3 months ago

I don't know if I am right or wrong ... So please correct me ... Thanks in advance ... here is my Solution : As we know, floor(x) = x - {x} , where {x} is fractional part. if we replace all the floor values in the above eqn will reduce to

LHS : nx - [ {x} + ({2x}/2) + ({3x}/3) + ({4x}/4) + ........... + ({nx}/n) ]

What is interesting to see is that {x} = {2x}/2 = {3x}/3 = ..... = {nx}/n (You can check it with any value of x :) )

Hence LHS will become : nx - n{x}

Solving RHS we got : floor(nx) = nx - {nx} As the fractional part of {nx} will always lie between 0 and 1, i.e. 0 <= {nx} < 1

And it is easy to see that the product of n and {x} in LHS is always greater than or equal to (in case of 0 as fractional value) {nx} in RHS.

Probably this will prove it ... What say guys ???

- 4 years, 3 months ago

that should prove it man :)...... i don't find any serious anomaly..... good job

- 4 years, 3 months ago

{x} = {2x}/2 = {3x}/3 = ..... = {nx}/n is false (see my other comments).

- 4 years, 3 months ago

yes and it is just the special case where inequality turns to equality...... it gives maximum value of LHS

- 4 years, 3 months ago

$$n\{x\}\neq \{nx\}$$. Try, e.g., $$x=0.6$$, $$n=2$$. Then $$2\cdot \{0.6\}=1.2\neq\{2\cdot 0.6\}=0.2$$.

- 4 years, 3 months ago

When did I say n{x} = {nx} ??? Please check it once more :)

- 4 years, 3 months ago

"What is interesting to see is that {x} = {2x}/2 = {3x}/3 = ..... = {nx}/n"

I could have equally said that $$\{x\}\neq \frac{\{nx\}}{n}$$, just multiply both sides of your equality by $$n$$ (we can, since $$n\neq 0$$). Both of the equalities are equivalent.

- 4 years, 3 months ago

Yups got my mistake... thank you... :)

- 4 years, 3 months ago

@Daniel Liu @Finn Hulse : Guys can you please check the above solution ???

- 4 years, 3 months ago

I still don't think it proves it. @Finn Hulse

Your solution depends on the fact that $\{x\}+\dfrac{\{2x\}}{2}+\cdots+\dfrac{\{nx\}}{n}\ge \{nx\}$ but you didn't prove that.

- 4 years, 3 months ago

Meh I'm tired. :P

- 4 years, 3 months ago

It's good as far as I can tell! :D

- 4 years, 3 months ago

I think using induction approach should also work

- 4 years, 3 months ago

Apparently this is USAMO 1981 #5.

- 4 years, 3 months ago

Yep.

- 4 years, 3 months ago

I was just gonna say...

- 4 years, 3 months ago