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.
- for all real .
- Hermite's Identity:
A useful way to use the floor function is to write , where y is an integer and k is the leftover stuff after the decimal point. This can greatly simplify many problems.
Another common definition of the floor function is
where is the fractional part of .
- [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