## 20269 Reputation

15 years, 348 days

## The proof?...

@Mariusz Iwaniuk  Your solution uses the first few terms of the sequence  [k, f(k)] . But it does not prove that this will be true for any  k . The same result can be obtained using the first   values, since the polynomial of degree  <=n  is uniquely found of values at  n+1 points:

```restart;
f := k -> sum((-1)^i*(k - i + 1)^(2*k + 4)/(i!*(2*k - i + 2)!), i = 0 .. k);
L := [seq([k,f(k)], k = 0 .. 3)];
factor(CurveFitting:-PolynomialInterpolation(L, k));
```

## Bug in your version Maple...

@bstuan  I am using Maple 2018.2. You didn't specify which version of Maple you are using. This can be seen from the picture you provided, your Maple is missing one solution. This is reason for the error message, because you can't link to the second solution. So to find the missing solution try the other code below where I added another system where  cos(Pi/4)  is replaced with cos(3*Pi/4) :

```restart:
local D:
np:=<A,B,C>: nq:=<1,-1,0>: P:=A*x+B*y+C*z+D=0: M:=[1,0,0]: N:=[0,0,-1]:
Eq1:=eval(P,[x,y,z]=~M);
Eq2:=eval(P,[x,y,z]=~N);
Eq31:=(np.nq)/sqrt(np.np)/sqrt(nq.nq)=cos(Pi/4) assuming real;
Eq32:=(np.nq)/sqrt(np.np)/sqrt(nq.nq)=cos(3*Pi/4) assuming real;
Sol1:=solve({Eq1,Eq2,Eq31});
Sol2:=solve({Eq1,Eq2,Eq32});
simplify(eval(P,Sol1)/B); # The first solution
simplify(eval(P,Sol2)/B); # The second solution
```

## Parametric equations...

Here is another way to draw a similar curve using parametric equations. This method makes it easy to animate this curve:

```plots:-animate(plot,[[sin(t)^3, 13*cos(t)/15-cos(2*t)/3-2*cos(3*t)/15-cos(4*t)/15, t = -Pi .. a], color=red, thickness=3, scaling=constrained], a=-Pi..Pi, frames=90, axes=none, paraminfo=false);
```

## ?...

@JAMET  But you have already received instructions from  vv  on what to do so that there is no confusion between  parabola and ellipse.

## Code?...

@bstuan Submit your code here (as text not an image) or upload your worksheet here. You may have forgotten to download the LinearAlgebra package when using the  CrossProduct  command. Should be the  LinearAlgebra:- CrossProduct  command.

## expand...

@vv  For these examples, the  expand command does the work. But everyone would like the simplify command to do the same.

```expand(tan(x+k*Pi))    assuming   k::integer;
expand(sin(x+2*k*Pi)) assuming   k::integer;```

## ......

@Emrah Akyar It doesn't matter what names you use. If you make this assignment  H:=G , then why can't you immediately name the original graph as  , and call the new modified graph through  G . Your code will become shorter and simpler.

## Iterator...

@nm  For large values  n>20  it is better to use the commands of the  Iterator  package: Iterator:-Permute  or  Iterator:-CartesianProduct .

## Dotted lines...

@GEEMIC Yes. Just replace non-strict inequalities with strict ones and change the  axes  option:

```restart;
A:=plot(1/x, x=0..3, 0..3, thickness=2):
B:=plots:-inequal({x>0,x<1}, x=0..3,y=0..3, optionsfeasible = [color = "LightBlue"]):
plots:-display(A,B, axes=frame);```

## Thanks...

@vv  Thank you. If we add up the areas of all triangles, we get for each of the two solutions  x+y+z+1003 = 1922 , 1922+100 = 2022

Everyone can see my solution at the link
http://math.hashcode.ru/questions/237436/%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F-%D1%91%D0%BB%D0%BA%D0%B0-%D0%B2-%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%D0%B0%D1%85

@mehdibaghaee I don't know how you got it, you probably used palettes to enter these expressions. I advise you to use 1Dmath input instead of palettes and 2Dmath input, then such problems will disappear.

## ......

@Dkunb  I do not have Maple 2021 (only Maple 2018 and older), so I cannot check how the code works in your version of Maple.

## ?...

@Dkunb  Strange, in Maple 2018 this code works as expected. Did you do  restart  in the beginning? Your screenshot doesn't show it.

## Numerically only...

@jud  vv have already answered you that this problem can probably only be solved numerically.

## OK...

@vv  Thank you. I found the reason for the disappearance of this solution. I missed one condition in ListTools:-Categorize command. If we add it, then the tetrahedron you specified appears. I will now edit my answer.

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