**Definition**

We define the floor function as : greatest integer less than or equal to . e.g., .

**Graph of the floor function**

The function is shown in the image (1st quadrant only)(see the attachment)

This function is discontinuous and indifferentiable at each and every points.

**Some other definitions**

We define “curvy x” as:

We define the ceiling function as: the least integer greater than or equal to ; ie least integer greater than or equal to . For example; .

Hence it is easy to notice that for

**A challenge**

Can you write the function where is approximated to ;i.e.:

…

*Proof*:

Examples

. (alex2008) Prove that .

(solution by Potla)

See this url for proof. (refer to post # 126, 127)

For every ; find the largest , such that:

Hint: Define . Consider different values of to get a recurrence relation. Then set .

.(alex2008 – inequation) Solve the following inequation :

.

(alex2008; concepts of functions needed) Find all the functions such that

(solution by Farenhajt)

Putting we get

Hence

(alex2008)Let be . Show that :

(solution by Farenhajt)

Since , we get , hence

Summing the right-hand inequalities up, we get

and the conclusion follows.

Solve the equation , where is the integer part of the real number .

Solution (makar)

Let

and

and

Now first analyze the case in which there is no solution,there are two such cases:

or

For there is no solution.

there is a solution for only.

Solving for these values of we get

References

http://www.artofproblemsolving.com/Foru … p?t=273532

http://www.artofproblemsolving.com/Foru … p?t=273536

http://www.artofproblemsolving.com/Foru … p?t=273069

http://www.artofproblemsolving.com/Foru … p?t=281136

MY APOLOGISES

Okay; now : PRACTICE.

EXERCISES

2. Prove that we have the following identity:

Where .

( important lemma)The “golden ratio” is defined as . Prove that : ()

(a) or .

(b) 1. (if ) ; and 2.

otherwise.

. (inequality; USAMO 1981) Prove that and we have: .

I leave you all (mainly beginners) to digest these first, and then I will post the solutions. If any of you have some solutions, please post it as a comment to this post.