# Floor function

(Redirected from Greatest integer 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 $x$ is usually denoted by $\lfloor x \rfloor$ or $[x]$. 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

• $\lfloor a+b\rfloor\ge \lfloor a\rfloor+\lfloor b \rfloor$ for all real $(a,b)$.
• Hermite's Identity: $$\lfloor na\rfloor = \left\lfloor a\right\rfloor+\left\lfloor a+\frac{1}{n}\right\rfloor+\ldots+\left\lfloor a+\frac{n-1}{n}\right\rfloor$$

## Examples

• $\lfloor 3.14 \rfloor = 3$
• $\lfloor 5 \rfloor = 5$
• $\lfloor -3.2 \rfloor = -4$

A useful way to use the floor function is to write $\lfloor x \rfloor=\lfloor y+k \rfloor$, 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 $$\lfloor x \rfloor = x-\{x\}$$

where $\{x\}$ is the fractional part of $x$.

## Problems

### Introductory Problems

• Let $[x]$ denote the largest integer not exceeding $x$. For example, $[2.1]=2$, $=4$ and $[5.7]=5$. How many positive integers $n$ satisfy the equation $\left[\frac{n}{5}\right]=\frac{n}{6}$.

(2017 PCIMC)

### Intermediate Problems

• Find the integer $n$ satisfying $\left[\frac{n}{1!}\right]+\left[\frac{n}{2!}\right]+...+\left[\frac{n}{10!}\right]=1999$. Here $[x]$ denotes the greatest integer less than or equal to $x$.

(1999-2000 Hong Kong IMO Prelim)

• What is the units (i.e., rightmost) digit of $$\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor$$ (1986 Putnam Exam, A-2)

• If $x$ is a positive real number, and $n$ is a positive integer, prove that $$[nx] \geq \frac{[x]}{1} + \frac{[2x]}{2} + \frac{[3x]}{3} + ... + \frac{[nx]}{n},$$ where $[t]$ denotes the greatest integer less than or equal to $t$.

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

• Let $[x]$ denote the integer part of $x$, i.e., the greatest integer not exceeding $x$. If $n$ is a positive integer, express as a simple function of $n$ the sum $$\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+...+\left[\frac{n+2^k}{2^{k+1}}\right]+\ldots$$

(1986 IMO, #6)