Prove a Floor Inequality

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

Prove that x+2x2+3x3++nxnnx\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 xx and integer nn.

Can someone help me prove it? I tried to use a periodic argument, proving it true for x[0,1)x\in [0,1) then plugging in x=x+kx=x'+k for positive integer kk to prove the inequality for all other xx. However it didn't work, because my inequality substitutions were too strict and I ended up with (n1)xnx(n-1)x\le \lfloor nx\rfloor which is not true.

Thanks!

Note by Daniel Liu
5 years, 4 months ago

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http://mks.mff.cuni.cz/kalva/usa/usoln/usol815.html

mathh mathh - 5 years, 4 months ago

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Apparently this is USAMO 1981 #5.

David Lee - 5 years, 4 months ago

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Yep.

Daniel Liu - 5 years, 4 months ago

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I was just gonna say...

Kaan Dokmeci - 5 years, 4 months ago

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I think using induction approach should also work

Roger Lu - 5 years, 4 months ago

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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 ???

Vipin Kumar - 5 years, 4 months ago

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

Vipin Kumar - 5 years, 4 months ago

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

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

Daniel Liu - 5 years, 4 months ago

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@Daniel Liu Meh I'm tired. :P

Finn Hulse - 5 years, 4 months ago

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

Finn Hulse - 5 years, 4 months ago

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n{x}{nx}n\{x\}\neq \{nx\}. Try, e.g., x=0.6x=0.6, n=2n=2. Then 2{0.6}=1.2{20.6}=0.22\cdot \{0.6\}=1.2\neq\{2\cdot 0.6\}=0.2.

mathh mathh - 5 years, 4 months ago

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

Vipin Kumar - 5 years, 4 months ago

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

I could have equally said that {x}{nx}n\{x\}\neq \frac{\{nx\}}{n}, just multiply both sides of your equality by nn (we can, since n0n\neq 0). Both of the equalities are equivalent.

mathh mathh - 5 years, 4 months ago

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

Vipin Kumar - 5 years, 4 months ago

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

Abhinav Raichur - 5 years, 4 months ago

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

mathh mathh - 5 years, 4 months ago

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

Abhinav Raichur - 5 years, 4 months ago

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