Floor and factorial functions

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Nikita

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I have two common(as I think) problems and no have idea how to solve them. I need to prove that:

1. [6x] + [x] >= [3x] + 2[2x]
2. 6n! * n! / (3n! * 2n! * 2n!) is integer

So, maybe anyone has any ideas?
 
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Nikita said:
I have two common(as I think) problems and no have idea how to solve them. I need to prove that:

1. 6[x] + [x] >= [3x] + 2[2x]
2. (6n)! * n! / (3n! * 2n! * 2n!) is integer

So, maybe anyone have some ideas?
Click to expand...

I am assuming you are indicating floor-function as [].

Since you have no idea regarding these problems - let us start with definitions:

What is the definition of floor function? What is the domain of floor function? Can you calculate floor of a negative number?

What is the definition of factorial function?
 
Subhotosh Khan said:
I am assuming you are indicating floor-function as [].

Since you have no idea regarding these problems - let us start with definitions:

What is the definition of floor function? What is the domain of floor function? Can you calculate floor of a negative number?

What is the definition of factorial function?
Click to expand...

definitions:
n! = 1*2*....*n
[x] = is the largest integer less than or equal to x

Domain of it is also very easy :)
 
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Nikita said:
definitions:
n! = 1*2*....*n
[x] = is the largest integer less than or equal to x

Domain of it also very easy :)
Click to expand...

What is true:

[x]+[x] ≥ [2x]

or

[x]+[x] ≤ [2x]
 
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And unfortunately I had mistakes in the task. Now it's ok
 
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Nikita said:
[2x] = [x] + [x+0.5] (can prove by seeing 2 cases: x - [x] <0.5 and x-[x] >= 0.5)
[x] + [x+0.5] >= [x] + [x] obviously
Click to expand...

How would you apply this knowledge to your first problem?
 
1. [6x] + [x] >= [3x] + 2[2x]

[6x] ≥ 2[3x]

[3x] ≥ [2x] + [x]

Continue....
 
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