Is it true that $x(1-x)=y(1-y)$ if and only if $x=y$? Consider two numbers $0\leq x\leq 1$ and $0\leq y\leq 1$. Is it true that $x(1-x)=y(1-y)$ if and only if $x=y$? If not, can you give a counterexample?

$$x(1-x) = y(1-y) \implies -x^2+x-y(1-y)=0 \implies x^2-x+y(1-y)=0\text{.}$$
By the quadratic formula,
$$x = \dfrac{1\pm \sqrt{1-4(1)y(1-y)}}{2}=\dfrac{1 \pm \sqrt{1-4y(1-y)}}{2} = \dfrac{1\pm \sqrt{1-4y+4y^2}}{2} = \dfrac{1 \pm |2y-1|}{2}\text{,}$$
so any pair $\left(\dfrac{1 + |2y-1|}{2}, y\right)$ or $\left(\dfrac{1 - |2y-1|}{2}, y\right)$ for any $y \in \mathbb{R}$ will work. Obviously, $x = y$ isn't necessary.
