@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!

@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!

Markiyan Hirnyk

My answer is a hint to the solution. Final exact solution  you can easily find yourself.

Maximum to be found on the set, which is a broken line on the plane:

solve({(x - 1)*(y - x) >= 0, (7 - y)*(1 - x) >= 0, (x - y)*(y - 7) >= 0, x>=-2, x<=3, y>=0, y<=11});

Mathematica solves this inequality directly:

Mathematica solves this inequality directly:

@Markiyan Hirnyk 

You have polished my idea to perfection!

@Markiyan Hirnyk 

You have polished my idea to perfection!

Carl Love

Your procedure is wrong.  For example tau(24)=8 .  24<30 .

Carl Love

Your procedure is wrong.  For example tau(24)=8 .  24<30 .

To  Markiyan Hirnyk

To find  a smaller set having this property is easy without Maple - just remove any 6 elements of the given set of 16 elements.

I do not know of another type of solutions than those that can be obtained apparent transformations: reflections and rotations.

To  Markiyan Hirnyk

To find  a smaller set having this property is easy without Maple - just remove any 6 elements of the given set of 16 elements.

I do not know of another type of solutions than those that can be obtained apparent transformations: reflections and rotations.