A friend of mine showed this to me a few days ago: x+y+z=0 x^{3}+y^{3}+z^{3}=3 x^{5}+y^{5}+z^{5}=15 x^{2}+y^{2}+z^{2}=? An answer must be supported with justification (so that I know that you didn't guess and get lucky).
Unless you can demonstrate that there is a clever shortcut to this problem which would raise it to the level of a brainteaser, this is just a set of simultaneous equations which is easily solvable with, for example, the Wolfram Alpha website.
Ok...that was prety hard to solve...it took many hours. I hope you have an easier way to get the answer (tricky): Spoiler x= 2 cos (20) y= 2 cos (140) z= 2 cos (260) the answer to the question is 6
OK, I got it...it´s really simple to get the answer to x2+y2+z2=?.... Spoiler the answer is 6. and you just have to set values to x (positive) y(negative) and z(negative) and calcutale the proportion like: set x=3 y=-2 z=-1, then ^2 = 14 --> ? ^3 = 18 --> 3 ^5 = 210 --> 15 so 14 . 18 . 15 = ? . 3 . 210 ? = 6 That´s the tricky way...