@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.

@Axel Vogt

 There are infinitely many rational solutions. All of these solutions can be obtained as follows: set any rational values ​​to variables  x  and  y  and solve the resulting equation for  z .

@Axel Vogt

 There are infinitely many rational solutions. All of these solutions can be obtained as follows: set any rational values ​​to variables  x  and  y  and solve the resulting equation for  z .

If you fix all of these errors, the procedure is correct. Since you find  the zeros not of the function but its derivative, so the first argument of the procedure should be the derivative (or correct the text of the procedure appropriately):

prcNewton(x->cos(x^2)*2*x+1, 1.0);

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