13. (C)
Rectangle ABCG has area 8×9 = 72, so rectangle FEDG has
area 72-52 = 20.
The length of 
equals DE = 9 - 5 = 4, so the
length of
is
.
Therefore, DE + EF = 4 + 5 = 9.
14. (B) Each team plays 10 games in its own division and 6 games against teams
in
the other division. So each of the 12 teams plays 16 conference games. Because
each game involves two teams , there are 
games scheduled.
15. (C) Because the perimeter of such a triangle is 23, and the sum of the two
equal
side lengths is even, the length of the base is odd. Also, the length of the
base
is less than the sum of the other two side lengths, so it is less than half of
23.
Thus the six possible triangles have side lengths 1, 11, 11; 3, 10, 10; 5, 9, 9;
7, 8, 8; 9, 7, 7 and 11, 6, 6.
16. (D) It is possible for the Martian to pull out at most
4 red, 4 white and 4
blue socks without having a matched set. The next sock it pulls out must be
red, white or blue, which gives a matched set. So the Martian must select
4 × 3 + 1 = 13 socks to be guaranteed a matched set of five socks.
17. (E) Evelyn covered more distance in less time than Briana, Debra and Angela,
so her average speed is greater than any of their average speeds. Evelyn went
almost as far as Carla in less than half the time that it took Carla, so
Evelyn's
average speed is also greater than Carla's.
OR
The ratio of distance to time, or average speed, is indicated by the slope of
the
line from the origin to each runner's point in the graph . Therefore, the line
from the origin with the greatest slope will correspond to the runner with the
greatest average speed. Because Evelyn's line has the greatest slope, she has
the greatest average speed.
18. (C) The smallest three-digit number divisible by 13 is 13×8 = 104, so there
are
seven two- digit multiples of 13. The greatest three-digit number divisible by 13
is 13 × 76 = 988. Therefore, there are 76 - 7 = 69 three-digit numbers divisible
by 13.
OR
Because the integer part of 
is 76, there are
76 multiples of 13 less than or
equal to 999. Because the integer part of 
is
7, there are 7 multiples of 13 less
than or equal to 99. That means there are 76 - 7 = 69 multiples of 13 between
100 and 999.
19. (A)
By the Pythagorean Theorem,
. (Or note that
triangle AEB is similar to a 3-4-5 right triangle, so AE = 3 × 6 = 18.)
Also CF = 24 and 
. The perimeter of the
trapezoid is 50 + 30 + 18 + 50 + 7 + 25 = 180.
20. (A)Write the points where Alice and Bob will stop
after each move and compare
points.
|
Move |
 |
|
Alice. |
|
Bob. |
So Alice and Bob will be together again after six moves.
OR
If Bob does not move and Alice moves 9 + 5 = 14 points or 2 points each time,
they will still be in the same relative position from each other after each
turn.
If Bob does not move, they will be on the same point when Alice first stops on
point 12, where she started. So Alice will have to move 2 steps 6 times to stop
at her starting point.
21. (C) To make a triangle, select as vertices two dots from one row and one
from
the other row. To select two dots in the top row, decide which dot is not used.
This can be done in three ways. There are also three ways to choose one dot to
use from the bottom row. So there are 3 × 3 = 9 triangles with two vertices in
the top row and one in the bottom. Similarly, there are nine triangles with one
vertex in the top row and two in the bottom. This gives a total of 9 + 9 = 18
triangles.
Note. Can you find the four noncongruent triangles?
22. (E) Neither the units of size nor the cost are important in this problem. So
for
convenience, suppose the small size costs $1 and weighs 10 ounces. To determine
the relative value , we compare the cost per unit weight.
So the value, or buy, from best to worst is medium, large
and small, that is
MLS.
23. (B) Reflect the triangle and the semi circle across the hypotenuse
to obtain
a circle inscribed in a square. The circle has area 4π. The radius of a circle
with
area 4π is 2. The side length of the square is 4 and the area of the square is
16.
So the area of the triangle is 8.
24. (B) One way to solve the problem is to work backward,
either dividing by 2 if
the number is even or subtracting 1 if the number is odd.
So if you press 
or 9
keystrokes, you
can reach "200" from "1."
To see that no sequence of eight keystrokes works, begin by noting that of the
four possible sequences of two keystrokes, [×2] [×2] produces the maximum
result. Furthermore, [+1] [×2] produces a result larger than either [×2] [+1]
or [+1] [+1]. So the largest possible result of a sequence of eight keystrokes
is
"256," produced by either
or
The second largest result is "192," produced by
Thus no sequence of eight keystrokes produces a result of
"200."
25. (A) Because the circle and square share the same interior region and the
area
of the two exterior regions indicated are equal, the square and the circle must
have equal area. The area of the square is 22 or 4. Because the area
of both the
circle and the square is 4, 4 = πr2. Solving for r, the radius of the
circle, yields
, so 
.
Note. It is not necessary that the circle and square have the same center.
