Find the circle of maximum radius that is tangent to both curves. 
There are no other conditions or restrictions.
Equations of curves

 (x1-1/2)^4+x1*x2+2*x2^2-1=0; 
 (x1-sin(x1))^2+(x2-sin(x2))^2-1=0;

We can say, a test example. Probably a continuation of this topic.
An equation with one variable X, and let 0<X. To try to find such a largest value of X, in the neighborhood of which not a single solution will be missed in the amount of at least 50 solutions in a row. And in this case, the residual value of the equation discrepancy will be no more than 10^-7 .

f :=0.995-cos(0.25*x)*sin(x^3);

Purely for fun, of course.

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 "-".

The problem arose while playing with inscribed circles, as in this post. If anyone is interested, try to find a circle of maximum radius inscribed between the curves
x1^2 + 2*x2^2 - 1 = 0 and (x1 - sin(x1))^2 + (x2 - sin(x2))^2 - 1 = 0.

Curve graphs.

A little continuation of topics 1 and 2. This is a very similar cube from 2, 
 

but with a different equation:

f1 := (x1-sin(x1))^2+(x2-sin(x2))^2+(x3-sin(x3))^2-0.02513144866;
And other point coordinates (-.8283302152, -.8283302152, .8283302152) and (.8283302152, .8283302152, -.8283302152).