Possible value(s)
Let x , y and z be distinct nonzero numbers such that x x 3 + 1 = y y 3 + 1 = z z 3 + 1 .
What is the value of x 3 + y 3 + z 3 ?
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1 solution
Alternatively, ∑ x 3 = ∑ ( k x − 1 )
Did the same. Nice one.
Thank you for the nice solution :)
Sir, can I ask? How did you transform the systems of equation in the first line to the w 3 − k w + 1 = 0 ? Can you further elaborate? Thanks. :)
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We note that w 3 − k w + 1 = 0 has three roots. We note that x 3 − k x + 1 = 0 , which means that w = x is a root of w 3 − k w + 1 = 0 . Similarly, w = y and w = z are also roots of w 3 − k w + 1 = 0 . Since x , y and z are distinct. Therefore, they are the three roots of w 3 − k w + 1 = 0 .
Let x x 3 + 1 = y y 3 + 1 = z z 3 + 1 = k . ⟹ ⎩ ⎪ ⎨ ⎪ ⎧ x 3 − k x + 1 = 0 y 3 − k y + 1 = 0 z 3 − k z + 1 = 0 .
This means that x , y and z are roots to the equation w 3 − k w + 1 = 0 . By Vieta's formula , ⟹ { x + y + z = 0 x y z = − 1 .
Now, we have
x 3 + y 3 + z 3 = ( x + y + z ) ( x 2 + y 2 + z 2 − x y − y z − z x ) + 3 x y z = 0 + 3 ( − 1 ) = − 3