2008 AMC 10 B, Problem #22—“Note that there are6!/(3!2!1!) = 60distinguishableorders of the beads on the line.”
Solution
Answer (C):
There are
6!
/
(3!2!1!) = 60
distinguishable orders of the
beads on the line. To meet the required condition, the red beads must be
placed in one of four configurations: positions 1, 3, and 5, positions 2, 4,
and 6, positions 1, 3, and 6, or positions 1, 4, and 6. In the first two cases,
the blue bead can be placed in any of the three remaining positions. In the
last two cases, the blue bead can be placed in either of the two adjacent
remaining positions.
In each case, the placement of the white beads is
then determined. Hence there are
2
·
3 + 2
·
2 = 10
orders that meet the
required condition, and the requested probability is
10
60
=
1
6
.
Difficulty:
Mediumhard
NCTM Standard:
Data Analysis and Probability Standard: understand and apply basic concepts
of probability.
Mathworld.com Classification:
Probability and Statistics
>
Probability
>
Probability
A rectangular floor measures
a
feet by
b
feet, where
a
and
b
are positive integers with
b > a
. An artist paints
a rectangle on the floor with the sides of the rectangle
parallel to the sides of the floor. The unpainted part
of the floor forms a border of width 1 foot around the
painted rectangle and occupies half the area of the
entire floor. How many possibilities are there for the
ordered pair
(
a, b
)
?
(A)1(B)2(C)3(D)4(E)5
2008 AMC 10 B, Problem #23—2008 AMC 12 B, Problem #16—“Because the area of the border is half the area of thefloor, the same is true of the painted rectangle.Thepainted rectangle measuresa2byb2feet.”
Solution
Answer (B):
Because the area of the border is half the area of the
floor, the same is true of the painted rectangle.
The painted rectangle
measures
a

2
by
b

2
feet. Hence
ab
= 2(
a

2)(
b

2)
, from which
0 =
ab

4
a

4
b
+ 8
. Add 8 to each side of the equation to produce
8 =
ab

4
a

4
b
+ 16 = (
a

4)(
b

4)
.
Because the only integer factorizations of 8 are
8 = 1
·
8 = 2
·
4 = (

4)
·
(

2) = (

8)
·
(

1)
,
and because
b > a >
0
, the only possible ordered pairs satisfying this
equation for
(
a

4
, b

4)
are
(1
,
8)
and
(2
,
4)
. Hence
(
a, b
)
must be one
of the two ordered pairs
(5
,
12)
, or
(6
,
8)
.
Difficulty:
Hard
NCTM Standard:
Geometry Standard: analyze properties and determine attributes of two and
threedimensional objects.
Mathworld.com Classification:
Geometry
>
Plane Geometry
>
Rectangles
>
Rectangle
A
B
C
D
M
A
B
C
D
O
Quadrilateral
ABCD
has
AB
=
BC
=
CD
,
∠
ABC
= 70
◦
, and
∠
BCD
= 170
◦
.
What is the
degree measure of
∠
BAD
?
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 Winter '13
 Kramer
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