Difference between revisions of "Floor function"

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== Problems ==
 
== Problems ==
 
=== 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)
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* 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>.
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(2017 PCIMC)
  
 
===Intermediate Problems ===
 
===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>. (1999-2000 Hong Kong IMO Prelim)
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* 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>.
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(1999-2000 Hong Kong IMO Prelim)
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* What is the units (i.e., rightmost) digit of
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<cmath>\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor</cmath>
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(1986 Putnam Exam, A-2)
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* How many of the first 1000 [[positive integer]]s can be expressed in the form
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<math>\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor</math>,
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where <math>x</math> is a [[real number]], and <math>\lfloor z \rfloor</math> denotes the greatest [[integer]] less than or equal to <math>z</math>?
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[[1985 AIME Problems/Problem 10|(1985 AIME)]]
  
 
=== Olympiad Problems ===
 
=== Olympiad Problems ===
 
* If <math>x</math> is a positive real number, and <math>n</math> is a positive integer, prove that
 
* If <math>x</math> is a positive real number, and <math>n</math> is a positive integer, prove that
 
<cmath>[nx] \geq \frac{[x]}{1} + \frac{[2x]}{2} + \frac{[3x]}{3} + ... + \frac{[nx]}{n},</cmath>
 
<cmath>[nx] \geq \frac{[x]}{1} + \frac{[2x]}{2} + \frac{[3x]}{3} + ... + \frac{[nx]}{n},</cmath>
where <math>[t]</math> denotes the greatest integer less than or equal to <math>t</math>. (1981 USAMO, #5) ([http://www.mathlinks.ro/viewtopic.php?t=174312 Discussion 1]) ([http://www.mathlinks.ro/viewtopic.php?t=101711 Discussion 2])
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where <math>[t]</math> denotes the greatest integer less than or equal to <math>t</math>.
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(1981 USAMO, #5) ([http://www.mathlinks.ro/viewtopic.php?t=174312 Discussion 1]) ([http://www.mathlinks.ro/viewtopic.php?t=101711 Discussion 2])
  
 
* Let <math>[x]</math> denote the integer part of <math>x</math>, i.e., the greatest integer not exceeding <math>x</math>. If <math>n</math> is a positive integer, express as a simple function of <math>n</math> the sum <cmath>\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+...+\left[\frac{n+2^k}{2^{k+1}}\right]+\ldots</cmath>
 
* Let <math>[x]</math> denote the integer part of <math>x</math>, i.e., the greatest integer not exceeding <math>x</math>. If <math>n</math> is a positive integer, express as a simple function of <math>n</math> the sum <cmath>\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+...+\left[\frac{n+2^k}{2^{k+1}}\right]+\ldots</cmath>
(1986 IMO, #6)
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(1968 IMO, #6)
  
 
==See Also==
 
==See Also==

Latest revision as of 21:05, 26 February 2024

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

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, $[2.1]=2$, $[4]=4$ 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)

$\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$,

where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

(1985 AIME)

Olympiad Problems

  • 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 \[\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+...+\left[\frac{n+2^k}{2^{k+1}}\right]+\ldots\]

(1968 IMO, #6)

See Also