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

which maps the real number to the a coordinate in the - plane. Assume throughout that , , , , and are real numbers.

(1) Compute , , , , , and . Sketch a plot of these points, superimposed on the unit circle.

(2) Show that is one-to-one. That is, show that if , then .

(3) Let be the intersection point between the unit circle and the line connecting and . Prove that .

(4) Show that is an ordered pair of rational numbers on the unit circle different from if and only if there is a rational number such that . (This result allows us to deduce that there are infinitely (countably) many rational points on the unit circle.)

According to problem 3, 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 be a “vertical pair” if either , or , or and latex \phi(t)$ are two different points on the same vertical line.
• Let be a “horizontal pair” if either , or and are two different points on the same horizontal line.
• Let be a “diametric pair” if and are two different end points of the same diameter of the circle.

(5) (a) Prove that for all and , is a vertical pair if and only if .
(b) Prove that for all and , is a horizontal pair if and only if .
(c) Determine and prove a relationship between and that is a necessary and sufficient condition for to be a diametric pair.

(6) (a) Suppose that is not a vertical pair. Then, the straight line through them (if , use the tangent line to the circle at that point) intersects the -axis at the point . Find in terms of and , and simplify and prove your answer.
(b) Draw the straight line through the point and , where is the point described in problem (5a). Let denote the point of intersection of this line and the circle. Prove that .

(7) (a) Suppose that is not a horizontal pair. Then, the straight line through them (if , use the tangent line to the circle at that point) intersects the horizontal line at the point . Find in terms of and , and simplify and prove your answer.
(b) Draw the straight line through the point and , where is the point described in problem (6a). Let denote the point of intersection of this line and the circle. Prove that .

(8) Suppose , , , and are distinct real numbers such that and such that the line containing and intersects the line containing and . Find the -coordinate of the intersection point in terms of and only.

(9) Let and be distinct real numbers such that . Given only the unit circle, the - and - axes, the points and , and a straitedge (but no compass), determine a method to construct the point that uses no more than line segments. Prove why the construction works and provide a sketch.

(10) Given only the unit circle, the -and - axes, the point , and a straightedge (but no compass), describe a method to construct the point .