Provide all solutions of
x6=(x-1)6.
Source: Cyprus’s Math. Competition
I could expand things out and brute force it, as in Larry's solution; but there's a more elegant way.
Let w=1/2+i*sqrt(3)/2, the principal complex 6th root of unity.
Now take the 6th root of each side, and on the left side I'll iterate through the powers of w. Let p=0 to 5.
Then w^p*x = x-1. And solve for x we get x=1/(1-w^p).
Then substituting in the values of p we get the solution set
{1/(1-w^0), 1/(1-w^1), 1/(1-w^2), 1/(1-w^3), 1/(1-w^4), 1/(1-w^5)}
Note the first member is division by 0, so must be discarded. Then simplify the rest to get
x = {1/(1/2-i*sqrt(3)/2), 1/(3/2-i*sqrt(3)/2), 1/2, 1/(1/2+i*sqrt(3)/2), 1/(3/2+i*sqrt(3)/2)}
x = {1/2+i*sqrt(3)/2, 1/2+i*sqrt(3)/6, 1/2, 1/2-i*sqrt(3)/6, 1/2-i*sqrt(3)/6}.
Solution
