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
.