Introduction
The floor function, also known as the greatest integer function gives the greatest integer less than or equal to what's given. Basically a "round down" function. The floor of a number x is denoted as
⌊x⌋.
Some other ways of writing the floor of a value x are
⌊x⌋=x−{x}
OR
[x].
Here are some quick examples:
- ⌊3.499⌋=3
- ⌊−3.499⌋=−4
- ⌊10⌋=10
Here are some practice problems.
Practice Problem
(2017 Pui Ching Invitational Mathematics Competition Q3)
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 [5n]=6n.
Practice Problem
(1985 AIME Q10)
How many of the first 1000 positive integers can be expressed in the form
⌊2x⌋+⌊4x⌋+⌊6x⌋+⌊8x⌋, where x is a real number, and ⌊z⌋ denotes the greatest integer less than or equal to z?
Practice Problem
(2023 CEMC Euclid Q10a, Q10b)
For every real number x, define ⌊x⌋ to be equal to the greatest integer less than or equal to x. (We call this the "floor" of x.) For example, ⌊4.2⌋=4,⌊5.7⌋=5,⌊−3.4⌋=−4, and ⌊2⌋=2.
(a) Determine the integer equal to ⌊31⌋+⌊32⌋+⌊53⌋+…+⌊359⌋+⌊360⌋.
(The sum has 60 terms.)
(b) Determine a polynomial p(x) so that for every positive integer m>4,
⌊p(m)⌋=⌊31⌋+⌊32⌋+⌊33⌋+…+⌊3m−2⌋+⌊3m−1⌋
(The sum has m−1 terms.)
A polynomial f(x) is an algebraic expression of the form f(x)=anxn+an−1xn−1+…+a1x+a0 for some integer n≥0 and for some real numbers an,an−1,…,a1,a0.
Final Notes and Tips
- Usually related to the final questions on the Euclid Contest by the CEM.
- Problems usually doesn't require expert knowledge on it, usually tests your ability to problem solve and other algebra/number theory knowledge.
- A useful way of to use the floor function is to rewrite ⌊x⌋=⌊y+k⌋, where y is an integer and k is the leftover value (decimals). This can simplify some problems.
- I would keep in mind what it is.