The Unapologetic Mathematician

Mathematics for the interested outsider

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).

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May 21, 2009 Posted by | Uncategorized | 2 Comments