@Markiyan Hirnyk 

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@Axel Vogt 

assume(n, posint):

is(n^2 mod n =0);

           FAIL

@Axel Vogt 

assume(n, posint):

is(n^2 mod n =0);

           FAIL

@Joe Riel 

Thanks for the helpful comments and improving of the procedure. Your modification works 2 times faster.

@PatrickT

Here is the translation of this text:

IP addresses in China and some other countries are not currently served, with the exception of images placed on authorized sites. Contact the administrator of radikal.ru, to add your resource to the list of authorized sites. 

@PatrickT

Here is the translation of this text:

IP addresses in China and some other countries are not currently served, with the exception of images placed on authorized sites. Contact the administrator of radikal.ru, to add your resource to the list of authorized sites. 

Yes, you're absolutely right! I am from Russia and I'm keeping the images on the radical.ru, and then insert on  the forum, giving references. I do not know why you do not see these pictures on the forum, I see everything well. Here is the same picture, inserted by a method that you have recommended. Do you see it? For some reason, the picture is not inserted after the text, but in front of it, despite the fact that the cursor was after the text.

Yes, you're absolutely right! I am from Russia and I'm keeping the images on the radical.ru, and then insert on  the forum, giving references. I do not know why you do not see these pictures on the forum, I see everything well. Here is the same picture, inserted by a method that you have recommended. Do you see it? For some reason, the picture is not inserted after the text, but in front of it, despite the fact that the cursor was after the text.

Unfortunately, Maple does not solve even the simplest equations with function  frac .

For example:

solve(frac(2*x)=1/2) assuming x>=2, x<=3;

allvalues(%);

But your equation is easily solved by hand. Consider a solution on the interval  x=95..96 . For all values ​​of the argument in this segment function   f(x)=x*floor(x))  is a linear function  and it is  f(x)=95*x . So it takes integer values ​​at the points  x=95, 95+1/95, 95+2/95, ...  and so on. The fractional part is equal to  1/2  in the middles between these values.

Unfortunately, Maple does not solve even the simplest equations with function  frac .

For example:

solve(frac(2*x)=1/2) assuming x>=2, x<=3;

allvalues(%);

But your equation is easily solved by hand. Consider a solution on the interval  x=95..96 . For all values ​​of the argument in this segment function   f(x)=x*floor(x))  is a linear function  and it is  f(x)=95*x . So it takes integer values ​​at the points  x=95, 95+1/95, 95+2/95, ...  and so on. The fractional part is equal to  1/2  in the middles between these values.

@Markiyan Hirnyk 

What you have written, it is not interesting, because it's easy to find the exact values ​​of all the roots of your equation in any interval. They will always be rational. For the interval  95<=x<=96  all the roots are

seq(18051/190+(n-1)/95, n=1..95);

This sequence is an arithmetic progression with difference  1/95 .

@Markiyan Hirnyk 

What you have written, it is not interesting, because it's easy to find the exact values ​​of all the roots of your equation in any interval. They will always be rational. For the interval  95<=x<=96  all the roots are

seq(18051/190+(n-1)/95, n=1..95);

This sequence is an arithmetic progression with difference  1/95 .

@Markiyan Hirnyk 

You see that  Maple can not solve a much simpler example! It is therefore important to combine human and machine capabilities.

@Markiyan Hirnyk 

You see that  Maple can not solve a much simpler example! It is therefore important to combine human and machine capabilities.

@Markiyan Hirnyk 

I already answered your original question (What is the number of all the solutions of the equation frac(x*floor(x)) = 1/2 belonging to RealRange(1,100)? ) . Gotten the exact result and specified the idea of solution. Why do you think my answer is partial?

Your new question a lot more difficult. If the function  f(x)=frac(x*floor(x))-1/2  is the same, then the problem can be solved. For an arbitrary fuction  f  question is too general. In any case, create a new topic, in which accurately define a new question!