Let us denote by  x  the angle between the sides  a  and  b . Using the law of cosines, we express the side  c . By writing the doubled area of ​​the triangle in two ways, it is easy to find the height  h  dropped to  c . Using the similarity of triangles, we find the side of the inscribed square  d  as a function of  x .

restart;
a:=3: b:=4: 
c:=sqrt(a^2+b^2-2*a*b*cos(x)): 
h:=solve(h*c=a*b*sin(x), h);
d:=solve((h-d)/h=d/c, d); # The target function 
Optimization:-Maximize(d, {x>=0, x<=Pi}); # The answer

Explanation: Triangle DBE is similar to triangle ABC with a similarity coefficient = BP/BQ

The recursive procedure named  P  finds the list of values  x[n], y[n]  ​​for given  x0, y0, n .
As an example, the first 11 values ​​were found and the first 4 points were plotted:
 

Download proc1.mw

Edited.

Download intervals.mw

restart;
l1:=<a,b,c>: l2:=<a,b,c>:
if LinearAlgebra:-Equal(l1, l2) then 
print("Equal Vectors");
end if:

# or
if andmap(t->t=0, l1-l2) then 
print("Equal Vectors");
end if:

To plot vectors, you can use the plots:-arrow command:

Download arrow.mw

Download MSNL_new.mw

You must specify the coordinates of the center  c  and write down the parametric equations of the circle not in vector form, but coordinate-wise:

restart;
c:=[1,2]:
plots:-animate(plot, [[c[1]+r*cos(t),c[2]+r*sin(t),t=0..6.3], scaling=constrained],r=5..10);

If the number  n  is large, then finding all its divisors can take an unacceptably long time. In this case, a simple enumeration will be much more efficient. In the example below, we find all divisors of  n  that do not exceed the number  N .

restart;
n:=10^100+1: N:=10^6:
{seq(`if`(irem(n,k)=0,k,NULL), k=1..N)};

{1, 73, 137, 401, 1201, 1601, 10001, 29273, 54937, 87673, 116873, 164537, 219337, 481601, 642001}

I think it would be more useful for a beginner to solve these examples manually, rather than with Maple. If he still needs to solve with Maple, then it would be useful for a beginner to first master such important functions from the Maple core as diff , int , plot . It is also important to understand the difference between specifying a function with the  ->  (arrow) and using the  unapply  command. The solution below uses these tools:

restart;
# Example (a)
 f := x->1/x^2: 
 a := 1: b := 10:
 F := unapply(int(f(t),t=a..x), x) assuming x>a;
 diff(F(x),x);
 is(%=f(x)); # Check
 A := int(f(x), x=a..b);
 B := F(b) - F(a);
 is(A = B); # Check
plot([f, F], 1 ..10, color=[red,blue], legend=[f(x),F(x)]);

Edit.  The number pi=3.14... should be coded in Maple as  Pi  not  pi  (in example (b)).

To solve this, we use vectors and a well-known formula that expresses the area of ​​a triangle through the coordinates of the vectors of its constituent sides:

restart;
local D:
A:=<0,0>: B:=<b,c>: C:=<a+b,c>: D:=<a,0>: 
AD:=D-A: DC:=C-D: M:=AD+2/5*DC: N:=(B+M)/2: BC:=AD:
BM:=M-B: AN:=N-A: 
area_AND:=1/2*LinearAlgebra:-Determinant(<AD | AN>):
area_BCM:=1/2*LinearAlgebra:-Determinant(<BM | BC>):
area_BCM/area_AND; 

First we set the coordinates of the points. Then we find the coordinates of the point  F  as the intersection of the lines  AC  and  EB . We find the area of ​​the triangle using the LinearAlgebra:-CrossProduct  command:

restart;
local D;
A, B, C, D, E := <0,0>, <0,b>, <a,b>, <a,0>, <a/2,0>:
solve({y=b/a*x, y=b-b/(a/2)*x}, {x,y});
F:=eval(<x,y>, %);
v1:=F-E; v2:=C-E;
v:=LinearAlgebra:-CrossProduct(<v1[1],v1[2],0>,<v2[1],v2[2],0>);
S:=1/2*sqrt(v[1]^2+v[2]^2+v[3]^2) assuming positive; # The area of the triangle EFC
simplify(S, {a*b=24}); # The answer

Download Download_2024-09-11_Has_Select_Question_new.mw

I think in Maple the most natural way to solve this particular problem is to use the  select  command:

restart;
L:=combinat:-permute([1,2,3,4]):
select(t->ListTools:-Search(2,t)<ListTools:-Search(3,t), L);

[[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3], [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1], [4, 1, 2, 3], [4, 2, 1, 3], [4, 2, 3, 1]]

Here is a simple calculation of the volume of this body and its drawing. Your body is like a torus. First we calculate the volume of this "torus" without a hole, and then simply subtract the volume of the hole. For drawing we use parametric equations of the surface of this body. For clarity, a cutout of 1/4 of the body is made.

restart;
f:=y->y^2-3: g:=y->y-y^2:
int(Pi*(2-f(y))^2, y=-1..3/2)-int(Pi*(2-g(y))^2, y=-1..3/2); # The volume of the solid
evalf(%);
M:=<cos(t),-sin(t); sin(t),cos(t)>:
P1:=plot3d([convert(M.<f(z)-2,0>, list)[1]+2, convert(M.<f(z)-2,0>, list)[2], z], t=-Pi/2..Pi, z=-1..3/2, color="Blue", scaling=constrained, axes=normal):
P2:=plot3d([convert(M.<g(z)-2,0>, list)[1]+2, convert(M.<g(z)-2,0>, list)[2], z], t=-Pi/2..Pi, z=-1..3/2, color="Red", scaling=constrained, axes=normal):
L:=plottools:-line([2,0,-2.9],[2,0,2.9], color=green, linestyle=3, thickness=2):
plots:-display(P1, P2, L, view=[-4.3..8.1,-6.1..6.1,-2.1..3.1]);

Download Count_operations_new.mw