Real Value Validation

Algebra Level 3

Each of $X$ , $Y$ and $Z$ is a real number that satisfies both of these equations:

(i) $X + Y = 2$

(ii) $X*Y - Z^2 = 1$

Find the value of $X^5 + Y^5 + Z^5$

1 solution

Danish Ahmed
Aug 11, 2015

If $x + y = 2$ , then the maximum value of $xy$ is when $x = y = 1$ .

So $xy \leq1$

So $z^2 = xy - 1 \leq 0$

So $z$ must be $0$ and $x$ and $y$ must be $1$ .

$x^5 + y^5 + z^5 = 2$