The following example appeared in Maxima mailing list (Maxima could not do it). Here we solve it using FriCAS. First we form system of equations:

(1) -> -- Form system of equations
eqn1 := a1*x1^5 + a2*x2^5 + a3*x3^5 = 1 / 6

(1)

eqn2 := a1*x1^4 + a2*x2^4 + a3*x3^4 = 1 / 5

(2)

eqn3 := a1*x1^3 + a2*x2^3 + a3*x3^3 = 1 / 4

(3)

eqn4 := a1*x1^2 + a2*x2^2 + a3*x3^2 = 1 / 3

(4)

eqn5 := a1*x1 + a2*x2 + a3*x3 = 1 / 2

(5)

eqn6 := a1 + a2 + a3 = 1

(6)

eqns := [eqn1, eqn2, eqn3, eqn4, eqn5, eqn6];

Now we solve it

solve(eqns, [a1, a2, a3, x1, x2, x3])

(7)

The solution above is not fully explicit, in first solotion we need to solve quadratic equation in x2 (so this solution really represents two solutions, one for each solution to the quadratic equation). Similarly, second and third solution contain quadratic equation in x3. Usualy such form of solution is convenient for further computations. But one can also get more explicit version of solution using 'radicalSolve':

radicalSolve(eqns, [a1, a2, a3, x1, x2, x3])

(8)

Note: not all equations have solutions in radicals and FriCAS can find solutions in radicals only in relatively simple cases. But system above has simple solution in radicals and FriCAS can find it.