Difference between revisions of "Floor function"
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=== Introductory Problems === | === Introductory Problems === | ||
* Let <math>[x]</math> denote the largest integer not exceeding <math>x</math>. For example, <math>[2.1]=2</math>, <math>[4]=4</math> and <math>[5.7]=5</math>. How many positive integers <math>n</math> satisfy the equation <math>\left[\frac{n}{5}\right]=\frac{n}{6}</math>. (2017 PCIMC) | * Let <math>[x]</math> denote the largest integer not exceeding <math>x</math>. For example, <math>[2.1]=2</math>, <math>[4]=4</math> and <math>[5.7]=5</math>. How many positive integers <math>n</math> satisfy the equation <math>\left[\frac{n}{5}\right]=\frac{n}{6}</math>. (2017 PCIMC) | ||
| + | |||
| + | ===Intermediate Problems === | ||
| + | * Find the integer <math>n</math> satisfying <math>\left[\frac{n}{1!}\right]+\left[\frac{n}{2!}\right]+...+\left[\frac{n}{10!}\right]=1999</math>. Here <math>[x]</math> denotes the greatest integer less than or equal to <math>x</math>. | ||
=== Olympiad Problems === | === Olympiad Problems === | ||
Revision as of 02:31, 29 August 2022
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 ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
. 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:
![\[\lfloor na\rfloor = \left\lfloor a\right\rfloor+\left\lfloor a+\frac{1}{n}\right\rfloor+\ldots+\left\lfloor a+\frac{n-1}{n}\right\rfloor\]](//latex.artofproblemsolving.com/8/4/d/84ddeb9332a5aedd6606f0b8711d40bd60fecc9e.png)
for all real 
.![\[\lfloor na\rfloor = \left\lfloor a\right\rfloor+\left\lfloor a+\frac{1}{n}\right\rfloor+\ldots+\left\lfloor a+\frac{n-1}{n}\right\rfloor\]](http://latex.artofproblemsolving.com/8/4/d/84ddeb9332a5aedd6606f0b8711d40bd60fecc9e.png)
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
![\[\lfloor x \rfloor = x-\{x\}\]](http://latex.artofproblemsolving.com/f/0/8/f08050f59ce7fe0a9ca89436b6fc4bc457641904.png)
where 
is the fractional part of .
Problems
Introductory Problems
- Let
![$[x]$](//latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denote the largest integer not exceeding 
. For example, ![$[2.1]=2$](//latex.artofproblemsolving.com/8/4/5/845a60595be566ae18cc6ab4335a3229209eec59.png)
, ![$[4]=4$](//latex.artofproblemsolving.com/3/b/c/3bcdeec719a406f28bd765d6cb61d4b6af47f7bf.png)
and ![$[5.7]=5$](//latex.artofproblemsolving.com/2/4/d/24dc8bc3d862f61f4b7f4955a2553dd72c752ff1.png)
. How many positive integers 
satisfy the equation ![$\left[\frac{n}{5}\right]=\frac{n}{6}$](//latex.artofproblemsolving.com/5/6/1/561c423e60a1a773205ba852a4c3d74f297bd5a3.png)
. (2017 PCIMC)
![$[2.1]=2$](http://latex.artofproblemsolving.com/8/4/5/845a60595be566ae18cc6ab4335a3229209eec59.png)
, ![$[4]=4$](http://latex.artofproblemsolving.com/3/b/c/3bcdeec719a406f28bd765d6cb61d4b6af47f7bf.png)
and ![$[5.7]=5$](http://latex.artofproblemsolving.com/2/4/d/24dc8bc3d862f61f4b7f4955a2553dd72c752ff1.png)
. How many positive integers 
satisfy the equation ![$\left[\frac{n}{5}\right]=\frac{n}{6}$](http://latex.artofproblemsolving.com/5/6/1/561c423e60a1a773205ba852a4c3d74f297bd5a3.png)
. (2017 PCIMC)Intermediate Problems
- Find the integer satisfying
![$\left[\frac{n}{1!}\right]+\left[\frac{n}{2!}\right]+...+\left[\frac{n}{10!}\right]=1999$](//latex.artofproblemsolving.com/c/c/d/ccd9d2e757785d313225dddf2953e096aace4141.png)
. Here denotes the greatest integer less than or equal to .
![$\left[\frac{n}{1!}\right]+\left[\frac{n}{2!}\right]+...+\left[\frac{n}{10!}\right]=1999$](http://latex.artofproblemsolving.com/c/c/d/ccd9d2e757785d313225dddf2953e096aace4141.png)
. Here Olympiad Problems
- If is a positive real number, and is a positive integer, prove that
![\[[nx] \geq \frac{[x]}{1} + \frac{[2x]}{2} + \frac{[3x]}{3} + ... + \frac{[nx]}{n},\]](http://latex.artofproblemsolving.com/5/2/c/52c05af83d2354182afba4ce3f78434072f5944d.png)
where ![$[t]$](http://latex.artofproblemsolving.com/8/a/8/8a86702dc60b9db916798b1115a6b6822bf828b9.png)
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
![\[\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+...+\left[\frac{n+2^k}{2^{k+1}}\right]+\ldots\]](//latex.artofproblemsolving.com/6/b/a/6ba05e683d271bf4c948c7f9b9420f104be804e1.png)
![\[\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+...+\left[\frac{n+2^k}{2^{k+1}}\right]+\ldots\]](http://latex.artofproblemsolving.com/6/b/a/6ba05e683d271bf4c948c7f9b9420f104be804e1.png)
(1986 IMO, #6)
