The Art of Problem Solving Online Classes

Introduction to Geometry
Challenge Set 7
www.artofproblemsolving.com

Instructions

Try to do as many of these problems as you can. Show your work, and if you can’t solve a
problem, please show us what you tried on the problem. If you get stuck, remember the general
problem solving technique of making up a similar simpler problem and solving that one first.
There are two groups of problems in this assignment separated by a horizontal line (you’ll see it
when you get to it). You should be able to solve almost all of the problems in the first group. The
problems below the line are especially challenging and will require more thought. Do your best,
but don’t expect to solve all of them.
When you write up and submit your solutions, please follow these instructions:

1. You can write your solutions by hand or by computer. If you write them by hand, please
write NEATLY in pen on only one side of the paper.

2. You can upload your solutions from the My Classes page. You can also email, fax, or mail
your solutions. If you upload or email them, they must be in a single PDF file. Do not email
us multiple files, an MS Word document, or a list of numerical answers. Make sure you keep
a copy of the solutions for yourself! Once we file your solutions, you will get a confirmation
email and it will appear on your My Classes page.

Email: classes@artofproblemsolving.com
Fax: (619) 659-8146
Mail: AoPS Incorporated
PO Box 2185
Alpine, CA 91903-2185

3. A completed cover sheet must be included with your solutions. A sample cover sheet follows
these instructions. All the information on this sheet must be included on your cover sheet.
You may use the sample cover sheet as your cover sheet.

4. Put your answers in order.

1
 The Art of Problem Solving Online Classes
Introduction to Geometry
Challenge Set 7
www.artofproblemsolving.com

Tips for mathematical writing in general

• Write in complete sentences. An equation by itself is not a sentence. For example, suppose
you want to prove that for all positive a, we have 2a > 0.

BAD GOOD
a>0 For positive a, we have a > 0.
2a > a > 0 This means 2a = a + a > a + 0 > 0.

• Show your work. The following rule of thumb is useful: if you show your solution to a
classmate do you think your classmate would understand it without having to do any extra
work?

• Make it obvious what your final answers are. A good way to do this is to put boxes around
them.

More suggestions can be found in Art of Problem Solving’s article How to Write a Solution:

http://www.artofproblemsolving.com/Resources/articles.php?page=howtowrite

You are encouraged to work with your classmates. Get together in the classroom during free
class time to work with others (post on the message board if you don’t know what this means).
However, if you work with others, please credit them appropriately when turning in your solutions.
(In other words, say with whom you worked, and on which problems you worked together.)

2
 The Art of Problem Solving Online Classes
Introduction to Geometry
Challenge Set 7
www.artofproblemsolving.com

Cover Sheet

Class Name:

Username:

Class ID:

User ID:
(Your Class ID and User ID can be found in the “My Classes” section of the website)

Challenge Set Number:

3
 The Art of Problem Solving Online Classes
Introduction to Geometry
Challenge Set 7
www.artofproblemsolving.com

Problems

1. We cut a regular pentagon out of a piece of cardboard, and then place the pentagon back in
the cardboard.

(a) How many different ways can we place the pentagon back in the cardboard, if we are
allowed to rotate but not reflect the pentagon?

(b) How many different ways can we place the pentagon back in the cardboard, if we are
allowed to rotate and reflect the pentagon?

2. What point is the centroid of the triangle with vertices (−2, 6), (8, 0), and (6, 12)?

3. A circle is tangent to the y-axis at the point (0, 2) and passes through the point (8, 0), as shown
below. Find the radius of the circle.
y

(0, 2)
x
(8, 0)

4. OP QRST U V W XY Z is a regular dodecagon. A rotation of θ degrees about U maps X to R.

6. Two lines ` and m intersect at an angle of 28◦ . Let A be a point inside the acute angle formed
by ` and m. Let B and C be the reflections of A in lines ` and m, respectively. Find ∠BOC
and ∠BAC.

4
 The Art of Problem Solving Online Classes
Introduction to Geometry
Challenge Set 7
www.artofproblemsolving.com

7. A square sheet of paper has area 6 cm2 . The front is white and the back is shaded. When the
sheet is folded so that point A rests on the diagonal as shown, the visible shaded area is equal
to the visible white area. How many centimeters is A0 from its original position, A? Hints: 2

A0

8. The lines 2x + y = 4, 2x + 5y = 10, x = 0, and y = 0 enclose a quadrilateral, as shown below.

The following problems are extra challenging. Do your best, but don’t expect to solve all of
them.
9. Equilateral 4ABC has centroid G. Triangle A0 B 0 C 0 is the image of triangle ABC upon a
dilation with center G and scale factor −2/3.

(a) Find [A0 B 0 C 0 ]/[ABC].

(b) Let K be the area of the region that is within both triangles. Find K/[ABC].

10. Use analytic geometry to prove that the altitudes of any triangle are concurrent.

11. The center of the cue ball on my rectangular pool table is directly above point A on the table.
I wish to bounce the cue ball off a rail such that after it bounces off the rail, the center of the
ball will pass directly over point B on the table. The radius of the cue ball is 1 in. A is 6 inches
from the rail, and B is 9 inches from the rail. A is 5 inches from B. How far from the nearest
point on the rail to point B do I want the cue ball to hit the rail? Hints: 1, 3

5
 The Art of Problem Solving Online Classes
Introduction to Geometry
Challenge Set 7
www.artofproblemsolving.com

12. In this problem, we will prove that the set of all points that have the same power with respect
to two circles is a straight line. This line is called the radical axis of the two circles.
To tackle this problem, we will define the power of a point to be negative if the point is
inside the circle, and positive if the point is outside the circle. In other words, if a line through a
point X inside the circle intersects the circle at A and B, then the power of X is −(XA)(XB).
If X is outside the circle and a line through X intersects the circle at C and D, then the power
of X is (XC)(XD).

(a) Suppose OP = t, and that we have a circle with radius r centered at O. Find an expression
in terms of r and t for the power of point P with respect to O.

(b) Suppose circle C1 is centered at the origin and has radius r, and circle C2 is centered at
(c, 0) and has radius s. Show that all the points that are on the radical axis of these two
circles lie on the same vertical line.

(c) Does every point on the vertical line from part (b) have the same power with respect to
two circles?

(d) As an extra challenge, consider the three circles shown below, in which each pair of circles
intersects at two points. For each pair of circles, we have drawn the chord that is common
to those two circles. Notice that the three chords appear to be concurrent. Prove that
this is the case.

13. Lines k and m intersect at point X, forming a 30◦ angle. Segment AB is reflected over line k
to produce A0 B 0 , and the resulting image is reflected over line m to produce A00 B 00 . Describe a
single transformation (with proof!) that maps AB to A00 B 00 . Hints: 4

6
 The Art of Problem Solving Online Classes
Introduction to Geometry
Challenge Set 7
www.artofproblemsolving.com

Below are hints to the problems. Work on the problems for a while before resorting to the hints.
Note that the hint numbers do not correspond to the problem numbers!

1. Make sure you track the path of the center of the ball, not the edge of the ball. You shouldn’t
be aiming at the reflection of B over the rail, because the ball will bounce before the center
reaches the rail!

2. What other portions of the diagram have area equal to the shaded area?

3. When the ball bounces off the rail, how far will the center be from the rail?

4. How is AX related to A00 X?

c 2010 AoPS Incorporated. All rights reserved.