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Question Number 211157 by RojaTaniya last updated on 30/Aug/24

Commented by Rasheed.Sindhi last updated on 30/Aug/24

$(x, y, z) = (1, 0, 0), (0, 1, 0), (0, 0, 1)$
	Answered by Rasheed.Sindhi last updated on 30/Aug/24
	${ ((x+y+z=1...(i))),((x^2 y+y^2 z+z^2 x=xy^2 +yz^2 +zx^2 ...(ii))),((x^3 +y^2 +z=y^3 +z^2 +x...(iii))) :} Assuming x,y,z>0 (ii)⇒xy(x−y)+yz(y−z)+zx(z−x)=0 xy(x−y)=0∧yz(y−z)=0∧zx(z−x)=0 x−y=0∧y−z=0∧z−x=0 ⇒x=y=z (i)⇒x=y=z=1/3$
	${\begin{cases} x + y + z = 1 . . . (i) \\ x^{2} y + y^{2} z + z^{2} x = {x y}^{2} + {y z}^{2} + {z x}^{2} . . . (i i) \\ x^{3} + y^{2} + z = y^{3} + z^{2} + x . . . (i i i) \end{cases}$ $A s s u m i n g x, y, z > 0$ $(i i) \Rightarrow x y (x - y) + y z (y - z) + z x (z - x) = 0$ $x y (x - y) = 0 \land y z (y - z) = 0 \land z x (z - x) = 0$ $x - y = 0 \land y - z = 0 \land z - x = 0$ $\Rightarrow x = y = z$ $(i) \Rightarrow x = y = z = 1 / 3$
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