Floor function

[lime]

Looking forward to it. As a hint you might consider to delineate the graphs first.

[Luma]

Originally Posted by lime

Looking forward to it. As a hint you might consider to delineate the graphs first.

OK I'll consider the hint after I get back from work and format my laptop
Keep on posting these questions

[kinghus]

I'll give this a shot. Numbers can be split into an integer part, and a decimal part, in this case floor(x) is equal to the integer part and {x} the decimal part. Lets call these two parts 'a' and 'b' respectively.
So x = a + b
(1): x^2 = a^2 + 2ab + b^2
but x^2 - [x^2] = {x}^2 --> x^2 = [x^2] + {x}^2 which is the same as:
(2): x^2 = a^2 + b^2

The question is asking when does equation (1) equal (2). The answer is when 2ab=0. Or whenever the integer part and/or the decimal part equals zero. On the given interval, this occurs for 1.0, 2.0, 3.0 etc ...
The answer is 100.

[lime]

Incorrect.

[Luma]

@Lime
I sketched x² and [x²] in the interval [1,100] and shaded the difference but I still can't see the answer

[lime]

Luma, be careful with using "[" and "]" in posting message.

Let's look at it another way: how many solutions will we have if we reduce interval to [1, 2] ?

[Luma]

Originally Posted by lime

Let's look at it another way: how many solutions will we have if we reduce interval to [1, 2] ?

again I sketched the graph, check it, I worry I made something wrong:
http://img8.imageshack.us/img8/5114/34956825.jpg

[lime]

What color corresponds to which function on the graph?

[Luma]

blue: x² (parabola)
red: [x²] (floor) however, I made approximation.
dark blue lines to mark the interval [1,2]

[lime]

Both graphs are correct. But I think to solve it you should have drawn another functions:

f(x) = x^2 - [x^2]
g(x) = {x}^2

and then find points of their intersection.