# 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 $x$ 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)