# 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.

## Contents

## Properties

- for all real .
- Hermite's Identity:

## Examples

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.

## Alternate Definition

Another common definition of the floor function is

where is the fractional part of .

## Problems

### Introductory Problems

- Let denote the largest integer not exceeding . For example, , and . How many positive integers satisfy the equation .

(2017 PCIMC)

### Intermediate Problems

- Find the integer satisfying . Here denotes the greatest integer less than or equal to .

(1999-2000 Hong Kong IMO Prelim)

- What is the units (i.e., rightmost) digit of

(1986 Putnam Exam, A-2)

- How many of the first 1000 positive integers can be expressed in the form

,

where is a real number, and denotes the greatest integer less than or equal to ?

### Olympiad Problems

- If is a positive real number, and is a positive integer, prove that

where denotes the greatest integer less than or equal to .

(1981 USAMO, #5) (Discussion 1) (Discussion 2)

- 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

(1986 IMO, #6)