Some Great Contest Geometry Problems for 10/20/13
From AMC-12:
1. (2002 #22) Triangle ABC is a right triangle with <ACB as its right angle,
m<ABC = 60°, and AB = 10. Let P be randomly chosen inside ΔABC, and extend
BP to meet AC at D. What is the probability that BD> 5 2 ?
2. (2003 #11) A square and an equilateral triangle have the same perimeter. Let A
be the area of the circle circumscribed about the square and B be the area of the
circle circumscribed about the triangle. Find A/B.
3. (2004 #8) In the figure, <EAB and <ABC are right angles, AB = 4, BC = 6, AE =
8, and AC and BE intersect at D. What is the difference between the areas of
ΔADE and ΔBDC?
E
C
D
8
6
A
4
B
4. (2004 #18) Square ABCD has a side length 2. A semicircle with diameter AB is
constructed inside the square, and the tangent to the semicircle from C intersects
side AD at E. What is the length of CE ?
D
C
E
A
B
5. (2004 #19) Circles A, B, and C are externally tangent to each other and internally
tangent to circle D. Circles B and C are congruent. Circle A has a radius of 1 and
passes through the center of D. What is the radius of circle B?
D
B
A
C
6.
(2004 #22) Three mutually tangent spheres of radius 1 rest on a horizontal plane.
A sphere of radius 2 rests on them. What is the distance from the plane to the top
of the larger sphere?
7. (2005 #14) A circle having center (0, k), with k > 6, is tangent to the lines y = x,
y = -x, and y = 6. What is the radius of this circle?
8. (2005 #16) Eight spheres of radius 1, one per octant, are each tangent to the
coordinate planes. What is the radius of the smallest sphere, centered at the
origin, that contains these eight spheres?
9. (2005 #18) Let A(2, 2) and B(7, 7) be points in the plane. Define R as the region
in the first quadrant consisting of those points C such that ΔABC is an acute
triangle. What is the closest integer to the area of the region R?
10. (2006 #6) The 8 x 18 rectangle ABCD is cut into two congruent hexagons, as
shown, in such a way that the two hexagons can be repositioned without overlap
to form a square. What is y?
C
D
y
A
B
11. (2006 #16) Circles with centers A and B have radii 3 and 8, respectively. A
common internal tangent intersects the circles at C and D, respectively. Lines AB
and CD intersect at E, and AE = 5. What is CD?
D
B
E
A
C
12. (2006 #19) Circles with centers (2, 4) and (14, 9) have radii 4 and 9, respectively.
The equation of a common external tangent to the circles can be written in the
form y = mx + b with m > 0. What is b?
9
(14, 9)
4
(2, 4)
From ARML:
13. (1995T-1) In ABC , side BC is the average of the other two sides. If
AB
cos C 
, compute the numerical value of cos C .
AC
14. (1995T-8) Points A, B, C, and D lie on
the given circle. If AB = 8, AP = 2,
and PC = 4, determine the ratio of the
area of quadrilateral PAEC to the area
of BAE .
15. (1998SR-8) Let T = TNYWR and set
T
K  . In square ABCD, DP = BP =
7
2K, and m 1  m 2  30 . The area
of the square can be written in simplest
form as a  b 3 . Compute a + b in
terms of T.
16. (1997I-2) ABC is inscribed in circle O, the radius of circle O is 12 and
m ABC  30 . A circle with center B is drawn tangent to the line containing
AC . Let R be the region which is within ABC , but outside circle B. Compute
the maximum value for R.
17. (1999 T-1) If AB = 2, BC = 6, AE = 6, BF = 8, CE
= 7, and CF = 7, compute the ratio of the area of
quadrilateral ABDE to the area of CDF .
18. (2000 I-7) The measure of the vertex angle of isosceles triangle ABC is θ and the
sides of the triangle are sin  , sin  , and sin  . Compute the area of ABC .
19. (2000 T-1) In the diagram, the square and the circle
have the same center. If the area inside the square
but outside the circle equals the area outside the
square but inside the circle, compute the ratio of the
side of the square to the radius of the circle.
20. (2000 TieBreaker 1) Points A, B, C, and D are midpoints of the edges of the cube
as shown. A plane passing through AB and CD intersects the cube in a region
whose area is 6 . Compute the surface area of the cube.
Answers and some hints:
1
Area of ABD 2 DA  BC DA 5 3  5 3  3
1.




Area of ABC 1 CA  BC CA
3
5 3
2
2.
27
32
3. 4
4.
5.
6.
5
Remember that tangent segments to a circle are congruent!
2
8
Draw the segment from A through the point of tangency of circles B and C.
9
Where that segment intersects circle A will be the center of circle D. Use a right
triangle relationship and show the radius of circle D that goes through B.
69
Consider the equilateral triangle formed with the centers of the three
3
small spheres as its vertices. Find the centroid of that triangle. Form a right
triangle to find the height of the pyramid with that triangle as base and the center
of the top sphere as the vertex. Add 2 for the additional radius for the top sphere
and 1 for the additional radius for the bottom spheres.
3
7. 6 2  6 Draw a radius to the point of tangency of the circle with the line y = x.
The triangle formed by the y-axis, the line y = x, and that radius is an isosceles
right triangle!
8. 1 3 Use the equivalent of the Pythagorean Theorem for 3D to find the
distance from the origin to the center of the sphere in the octant with all positive
coordinates. This would be 12  12  12  3 . Add on an additional radius to
reach to the outside edge of a sphere, giving 1 3 .
9. 51
Recall that a right triangle is inscribed in a circle. Thus, an acute triangle
would have vertex C outside the circle that has the midpoint of segment AB as its
center. To be acute, you would also need the angles at A and B to be less than
90°. Therefore, find the area of the region inside the large triangle below. Then
subtract the areas of the small triangle and the circle. You would have
2
1
1
2
2
14 14   4  4     4.5  2  4.5  2   51


2
2
10. 6
11.
12.
44
3
The area of the rectangle is 144, so the square must be 12 x 12.
Draw the radii to the points of tangency. The triangles are then similar.
120
Extend the lines in the drawing to the left until they intersect. That point of
119
intersection is  38 ,0 . The slope of the line through the centers is 5 , which
5
12

38
will be the tangent of the angle,  , formed at
,0 . You then need tan( 2 )
5
.

13.
5
7



Use the Law of Cosines and solve the quadratic equation for
c
.
b
14.
5
Use the property of secants to find that DC = 1.
16
Due to congruent angles, we know that
BAE DCE with a ratio of similitude of 8:1.
Using the areas of the three regions to be M, N, and
K, as marked in the figure, we then know that M =
64N. We also know by AA~ Theorem that
PBC PDA with a ratio of similitude of 2:1.
This means that M + K = 4(K + N). Substituting,
M
15

we get M  K  4  K   , so
M  3K and
64 
16

K
5
 .
M 16
6T 2
15.
Draw PE  DC . Use the ratios of special
49
right triangles. Then the area of the square is



2

K 1 3   K 2 4  2 3 


Note: For fun, TNYWR = 28!
16.

T2 42 3
49
.
108
Since m ABC  30 , we know that the

measure of AC is 60 . Then the length of AC =
radius = 12. Let b be the radius of circle B, then
1
12  b and
2
1
A  sector of circle B    b 2
12
A  ABC  
Then, R  6b 

b 2 Find the minimum value for
12
this quadratic function!
17. 1:1 Note: By SSS, AEC  BCF .
18.
19.
8
There are a variety of good methods. Use Law of Cosines to find a value for
25
1
sin  . Or, find the area of the triangle using A  ab cos C and find the area
2
using Heron’s and set those equal to each other to find a value for sin  .
 :1 With the given information, we know that the area of the square equals the
area of the circle.
20. 8 2 See the figure. The region in the plane is a regular hexagon. Label half of
the edge of the cube as s. Then the side of the hexagon is s 2 . So the area of the
2
3
 s 2  3 3 s 2 . So, 3 3 s 2  6 and we get that
hexagon is: A  6 
4
2
s2 
. Since the total surface area of the cube is 24 times that, we are done!
3

 

 