2009 AMC 12 B, Problem #7
—
“Represent the price at the end of March with the price
at the beginning of January.”
Solution
Answer (B):
Let
p
denote the price at the beginning of January. The
price at the end of March was
(1
.
2)(0
.
8)(1
.
25)
p
= 1
.
2
p
. Because the price
at the end of April was
p
, the price decreased by
0
.
2
p
during April, and the
percent decrease was
x
= 100
·
0
.
2
p
1
.
2
p
=
100
6
≈
16
.
7
.
To the nearest integer,
x
is 17.
Difficulty:
Medium
NCTM Standard:
Algebra Standard: use symbolic algebra to represent and explain mathematical
relationships.
Mathworld.com Classification:
Number Theory
>
Arithmetic
>
Fractions
>
Percent
Segment
BD
and
AE
intersect at
C
, as shown,
AB
=
BC
=
CD
=
CE
, and
∠
A
=
5
2
∠
B
. What is
the degree measure of
∠
D
?
A
B
C
D
E
(A)
52
.
5
(B)
55
(C)
57
.
5
(D)
60
(E)
62
.
5
2009 AMC 10 B, Problem #9
—
“Notice that
4
ABC
and
4
CDE
are isosceles.”
Solution
Answer (A):
Because
4
ABC
is isosceles,
∠
A
=
∠
C
. Because
∠
A
=
5
2
∠
B
, we have
5
2
∠
B
+
5
2
∠
B
+
∠
B
= 180
◦
, so
∠
B
= 30
◦
.
Therefore
∠
ACB
=
∠
DCE
= 75
◦
.
Because
4
CDE
is isosceles,
2
∠
D
+ 75
◦
=
180
◦
, so
∠
D
= 52
.
5
◦
.
Difficulty:
MediumNCTM Standard:Geometry Standard: analyze properties and determine attributes of two andthreedimensional objects.Mathworld.com Classification:Geometry>Plane Geometry>Triangles>Special Triangles>Other Triangles>Isosceles Triangle
A flagpole is originally 5 meters tall. A hurricane snaps
the flagpole at a point
x
meters above the ground
so that the upper part, still attached to the stump,
touches the ground 1 meter away from the base. What
is
x
?
(A)2.0(B)2.1(C)2.2(D)2.3(E)2.4
2009 AMC 10 B, Problem #10
—“Draw a figure to represent the situation.”
Solution
Answer (E):
Let
x
be the height of the stump. Then
5

x
is the height
of the snapped part, now forming the hypotenuse of a right triangle. By
the Pythagorean Theorem,
x
2
+ 1
2
= (5

x
)
2
=
x
2

10
x
+ 25
from which
x
= 2
.
4
.
x
5
x
1
Difficulty:
Medium
NCTM Standard:
Geometry Standard: analyze properties and determine attributes of two and
threedimensional objects.
Mathworld.com Classification:
Geometry
>
Plane Geometry
>
Triangles
>
Special Triangles
>
Other Triangles
>
Right Triangle
How many 7 digit palindromes (numbers that read the
same backward as forward) can be formed using the
digits 2, 2, 3, 3, 5, 5, 5 ?
2009 AMC 10 B, Problem #11
—“Because the digit 5 appears three times, 5 must bethe middle digit of any such palindrome.”
Solution
Answer (A):
Because the digit 5 appears three times, 5 must be the
middle digit of any such palindrome. In the first three digits each of 2, 3,
and 5 must appear once and the order in which they appear determines
the last three digits. Since there are
3! = 6
ways to order three distinct
digits the number of palindromes is 6.
Difficulty:
Medium
NCTM Standard:
Number and Operations Standard:
compare and contrast the properties of
numbers and number systems, including the rational and real numbers, and understand complex
numbers as solutions to quadratic equations that do not have real solutions.
Mathworld.com Classification:
Number Theory
>
Special Numbers
>
Palindromic Numbers
>
Palindromic Number
Distinct points
A
,
B
,
C
, and
D
lie on a line, with
AB
=
BC
=
CD
= 1
.
Points
E
You've reached the end of your free preview.
Want to read all 25 pages?
 Winter '13
 Kramer
 triangle