# The Unapologetic Mathematician

## ARML Scrimmage Power Question

I helped the Howard County and Baltimore County ARML teams practice tonight by joining the group of local citizens and team alumni to field a scrimmage team. As usual, my favorite part is the power question. It follows, as printed, but less the (unnecessary) diagrams:

Consider the function

$\displaystyle\phi(t)=\left(\frac{2t}{t^2+1},\frac{t^2-1}{t^2+1}\right)=(x,y)$

which maps the real number $t$ to the a coordinate in the $x$-$y$ plane. Assume throughout that $q$, $r$, $s$, $t$, and $u$ are real numbers.

(1) Compute $\phi(1)$, $\phi(1/2)$, $\phi(2)$, $\phi(-1)$, $\phi(-1/2)$, and $\phi(-2)$. Sketch a plot of these points, superimposed on the unit circle.

(2) Show that $\phi$ is one-to-one. That is, show that if $\phi(s)=\phi(t)$, then $s=t$.

(3) Let $(x_\phi,y_\phi)$ be the intersection point between the unit circle and the line connecting $(0,1)$ and $(t,0)$. Prove that $\phi(t)=(x_\phi,y_\phi)$.

(4) Show that $(x,y)$ is an ordered pair of rational numbers on the unit circle different from $(0,1)$ if and only if there is a rational number $t$ such that $\phi(t)=(x,y)$. (This result allows us to deduce that there are infinitely (countably) many rational points on the unit circle.)

According to problem 3, $\phi(t)$ is a particular geometric mapping of a single point on the real line to the unit circle. Now, we will be concerned with the relationship between the pairs of points, which will lead to a way of doing arithmetic by geometry. Use these definitions:

• Let $\left\{\phi(s),\phi(t)\right\}$ be a “vertical pair” if either $s=t=1$, or $s=t=-1$, or $st\neq0$ and $\phi(s) and$latex \phi(t)\$ are two different points on the same vertical line.
• Let $\left\{\phi(s),\phi(t)\right\}$ be a “horizontal pair” if either $s=t=0$, or $\phi(s)$ and $\phi(t)$ are two different points on the same horizontal line.
• Let $\left\{\phi(s),\phi(t)\right\}$ be a “diametric pair” if $\phi(s)$ and $\phi(t)$ are two different end points of the same diameter of the circle.

(5) (a) Prove that for all $s$ and $t$, $\left\{\phi(s),\phi(t)\right\}$ is a vertical pair if and only if $st=1$.
(b) Prove that for all $s$ and $t$, $\left\{\phi(s),\phi(t)\right\}$ is a horizontal pair if and only if $s=-t$.
(c) Determine and prove a relationship between $s$ and $t$ that is a necessary and sufficient condition for $\left\{\phi(s),\phi(t)\right\}$ to be a diametric pair.

(6) (a) Suppose that $\left\{\phi(s),\phi(t)\right\}$ is not a vertical pair. Then, the straight line through them (if $\phi(s)=\phi(t)$, use the tangent line to the circle at that point) intersects the $y$-axis at the point $(0,b)$. Find $b$ in terms of $s$ and $t$, and simplify and prove your answer.
(b) Draw the straight line through the point $(1,0)$ and $(0,b)$, where $(0,b)$ is the point described in problem (5a). Let $\phi(u)$ denote the point of intersection of this line and the circle. Prove that $u=st$.

(7) (a) Suppose that $\left\{\phi(s),\phi(t)\right\}$ is not a horizontal pair. Then, the straight line through them (if $\phi(s)=\phi(t)$, use the tangent line to the circle at that point) intersects the horizontal line $y=1$ at the point $(a,1)$. Find $a$ in terms of $s$ and $t$, and simplify and prove your answer.
(b) Draw the straight line through the point $(0,-1)$ and $(a,1)$, where $(a,1)$ is the point described in problem (6a). Let $\phi(u)$ denote the point of intersection of this line and the circle. Prove that $u=s+t$.

(8) Suppose $q$, $r$, $s$, and $t$ are distinct real numbers such that $qr=st$ and such that the line containing $\phi(q)$ and $\phi(s)$ intersects the line containing $\phi(r)$ and $\phi(t)$. Find the $y$-coordinate of the intersection point in terms of $s$ and $t$ only.

(9) Let $s$ and $t$ be distinct real numbers such that $st>0$. Given only the unit circle, the $x$- and $y$- axes, the points $\phi(s)$ and $\phi(t)$, and a straitedge (but no compass), determine a method to construct the point $\phi(\sqrt{st})$ that uses no more than $5$ line segments. Prove why the construction works and provide a sketch.

(10) Given only the unit circle, the $x$-and $y$- axes, the point $(1,1)$, and a straightedge (but no compass), describe a method to construct the point $\left(-\frac{2\sqrt{3}}{3},0\right)$.