# Floor function

The greatest integer function, also known as the **floor function**, gives the greatest integer less than or equal to its argument. The floor of is usually denoted by or . The action of this function is the same as "rounding down." On a positive argument, this function is the same as "dropping everything after the decimal point," but this is *not* true for negative values.

## Properties

- for all real .
- Hermite's Identity:

## Examples

- $\lfloor 5 \rfloor = ?

- $ (Error compiling LaTeX. ! Missing $ inserted.)\lfloor -3.2 \rfloor = -4\lfloor x \rfloor=\lfloor y+k \rfloor$, where y is an integer and k is the leftover stuff after the decimal point. This can greatly simplify many problems.

## Alternate Definition

Another common definition of the floor function is

where is the fractional part of .

## Olympiad Problems

- [1981 USAMO #5] If is a positive real number, and is a positive integer, prove that

where denotes the greatest integer less than or equal to .

- [1968 IMO #6] Let denote the integer part of , i.e., the greatest integer not exceeding . If is a positive integer, express as a simple function of the sum