This system is from the last century. This one has already appeared several times on the forum.
It is clear that nowadays it is not difficult to find several of its solutions, especially with the help of Maple. But in this case we are talking about a complete solution in real numbers. Of course, there are techniques that practically allow us to talk about such a solution, only without theoretical justification. 
The simple idea of ​​transforming it from transcendental to polynomial and getting a complete solution on a theoretical basis did not lead to a good result: it turned out that this is too complicated work for the  Isolate procedure. Perhaps I made a mistake somewhere, or (and) my old PC is too weak.
(The values ​​of the constants are not very important.)

restart; with(RootFinding):
 b1 := 114.069^2; b2 := 109.2389^2; b3 := 103.892^2; b4 := 99.76348^2; b5 := 97.24296^2; 
f1 := x1^2+x2^2+x3^2+2*(x1*x2*cos(x4)+x2*x3*cos(x5)+x1*x3*cos(x4+x5))-b1; 
f2 := x1^2+x2^2+x3^2+2*(x1*x2*cos(2*x4)+x2*x3*cos(2*x5)+x1*x3*cos(2*(x4+x5)))-b2; 
f3 := x1^2+x2^2+x3^2+2*(x1*x2*cos(3*x4)+x2*x3*cos(3*x5)+x1*x3*cos(3*(x4+x5)))-b3; 
f4 := x1^2+x2^2+x3^2+2*(x1*x2*cos(4*x4)+x2*x3*cos(4*x5)+x1*x3*cos(4*(x4+x5)))-b4; 
f5 := x1^2+x2^2+x3^2+2*(x1*x2*cos(5*x4)+x2*x3*cos(5*x5)+x1*x3*cos(5*(x4+x5)))-b5; 
f := seq(cat(f, i), i = 1 .. 5):
# fsolve([seq(f[i], i = 1 .. 5)]);
for i to 5 do 
F[i] := expand(f[i]);
F[i] := subs(cos(x4) = x4, cos(x5) = x5, F[i]); 
F[i] := subs(sin(x4) = sqrt(-x4^2+1), sin(x5) = sqrt(-x5^2+1), F[i]);
F[i] := subs(sqrt(-x4^2+1) = y4, sqrt(-x5^2+1) = y5, F[i]); 
F[i] := collect(F[i], [y4, y5]); 
F[i] := subs(y4 = sqrt(-x4^2+1), y5 = sqrt(-x5^2+1), F[i]); 
F[i] := op(1, F[i])^2-(sum(op(k, F[i]), k = 2 .. nops(F[i])))^2 
end do:
for i to 5 do 
F[i] 
end do;
#fsolve([seq(F[i], i = 1 .. 5)]);
#T := Isolate([seq(F[i], i = 1 .. 5)], [x1, x2, x3, x4, x5]): j := nops(T);
# T;

Edited: "+" to "-".

I tried connecting Maple to Jupyter Notebook, but it seems that it is not keen on outputting everything within the for loop; it only outputs the last item.

for i from 1 to 24 do
    print(i);
end do;

I've noticed that the Maple interface is always white, and after a long time, it causes eye strain. I wonder if it's possible to adjust the interface color, similar to Mathematica. My system is Windows 11.

Currently, I often use VSCode as an alternative, but some mathematical symbols don't display very clearly.

I have noticed similar discussions, such as this one, but I don’t know how to do it

In the attachment are three plotting issues that might be (partially) related.

 
Jagged shading     

Loss of color and incomplete clipping

The last one is the most severe because it takes color away when a plotobject is inside another object.

How can this be fixed? Explanations always welcome.

Shading_artefacs_and_lost_labels_and_color.mw

 I don't know how to solve with maple the following inequality with respect to the real variable x, n being an integer
Please tell me the procedure.

(f(x,n)>0 and (f(x,n)<0

f := (x, n) -> 2^n/(1 + n*2^n*x^2) - 2*x^2*(2^n)^2*n/(1 + n*2^n*x^2)^2

I have been working on solving a system of PDEs and plotting the results. However I've been encountered issues with setting the spacestep and timestep parameters.

First of all, i attempted to plot graphs t=0..2 using the animate command,

with spacestep = 5*10^-4 x 1/151,          timestep=1/1000

However, as shown in the image, an error ocurred indicating that calculations could not proceed after 1.57 sec.

Secondly, i kept the same spacestep but changed the timestep to 1/100 for plotting graphs 
t=0..2. This time, the graphs were plotted without any issues.

I initially beleved that larger spacesteps or timesteps would produce more accurate data. However, in my case, simply increasing these parameters did not work.

I would greatly appreciate if you could provide guidance on what factors should be considered when setting timestep and spacestep parameters, or any experiences in resolving similar issues.

So recently I noticed Youtube (owned by Google) has slowly been chipping away at what you could have for free in the past (1080p vidoes easily downloadable) to 720p then recently removed for pro users only at a monthly price. 

Now I noticed all the maple video content can be watched on youtube.  I no longer see links to download the video for offline viewing later from maplesoft (I think they used to be there) they just seem only available on youtube - and now since youtube is becoming less easily available as saveable videos offline it's almost impossible to save a maplesoft video for offline.  I have a question that only recently just started pondering.  Since I can't download the video from maplesoft - there's no download link anymore- seems only available on youtube.  Have all the training videos, recorded webinars, etc..  become the property of youtube and then by parent company become also property rights of google?

Just wondering, since the digital universe in the last couple of years is changing so rapidly, and google / youtube basically becoming a monopolizing giant, dictating the way videos can be watched.  I mean are maplesoft videos on maplesoft servers, or youtube servers? 

Thought I'd throw this out there for thoughts and discussion. 

Looks like this

From: dsolve_numeric_output_with_extra_brackets.mw

which was derived from a 2d document where I removed all unnecessary overhead.

When the option output=listprocedure is removed, this extra argument is not shown any more.

With that extra argument substitution of a solution into the corresponding ode(s) is not possible.

Any idea why that is? How to avoid or remove it?

Given a graph, as in the example below, I need all of them to have the same size, for example, 7 cm horizontally and 4 cm vertically. Is there a command to standardize the entire graph?

I have to follow certain dimensions that Maple usually does not show when adjusting the graph size with the mouse.

with(plots);

NMC := [0, ln(2)], [6, ln(2)];
plot([NMC], color = black);

Thank you so much!

What we are looking for is a flat polygon with N corners and side lengths 1,..., N, which has a maximum surface area.
BTW:
1.) I am a Maple newbie and would like to get to know the diversity and power of the Maple world. For this reason, I occasionally post tasks here from various areas of mathematics and of varying difficulty levels in order to learn from the answers. Of course, I know the solutions to these tasks. So these are not homework.
2.) I have difficulty subsequently inserting text at any point in working worksheets. I keep getting error messages because "or" or ";" is missing. What am I doing wrong?

I am running a code that loops through some 500 cases, each one considered separate, each working with a matrix of rationals 100x30 to 5000x900 or possibly larger dimensions. The matrices are not preserved and variables are reused. A concerning pattern emerges as more and more memory is slowly added until all ram (32GB) is spent and this makes no sense. Observing kernelopts(heaps); the number of entries increases steadily. Using gc() a few entries will occasionally disappear, but gc() itself takes longer with each loop.

What can I use to manage this condition?
Also why is there an increasing number of "Thread-Local" entries among the heaps?

I currently have problems uploading

Determine the smallest positive integer that is the sum of six distinct positive integers, where the sum of each selection of five of the six summands is a different square number.

hi

i am new user for maple

Can Olympiad geometry problems be solved in Maple?

I want to display in 3d the following objects

• Animated plottools objects (e.g. a spinning cube)
• Static plot objects (e.g. surface plot)/li>
• Time dependent spacecurves (to trace the path of objects)

Technically I could

• create a sequence of time stamps
• generate for each time stamp a plot structure using plots:-display for all the above objects
• display the sequence of plot structures with plots:-display with the option insequence.

or

• create separate plots with plots:-animate for each time dependent object
• plot with plot3d all static objects
• display all plots with plots:-display

or

• defining a procedure containing all plotting commands as a function of time
• create a background plot of all static objects using plots:-display
• animate the procedure with a plots:-animate call with the background option

Without having any experience, my perference would be the last way. However I intend to generate some plot coordinates with dsolve,numeric. This would mean that at each point in time to be plotted, dsolve starts integration from the inital conditions (which I have not planned to update animation frame by animation frame)

Any recommensations or thoughts on this?