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
[hide]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
(1968 IMO, #6)