Problem of the Week

Problem 31

Let $O$ be the origin, $N$ be $(0, 1)$ , and $\Sigma$ be the circle with $ON$ as it’s diameter.

Consider a function $\pi$ from points on $\Sigma$ to points on the x-axis such that $\pi(A) =$ the x-coordinate of $A'$ where $NAA'$ are collinear as shown in the diagram.

a) For what point on the circle does $\pi$ of it not exist, and does $\pi$ map unique points to unique points?

b) For two points $A$ and $B$ on $\Sigma$ such that $AB$ is parallel to $ON$ , show that $\angle ONA + \angle ONB = 90^\circ$ .