Difficulty:
10: Medium
12: Medium Easy
SMP-CCSS:
2
CCSS-M:
None
9
How
many
ordered
pairs
of
positive
integers
(
M, N
)
satisfy
the
equation
M
6
=
6
N
?
(A)
6
(B)
7
(C)
8
(D)
9
(E)
10
2012 AMC 10B, Problem #10—
“Cross multiply and consider divisors.”
Solution
Answer (D):
Multiplying the given equation by
6
N
gives
MN
= 36
. The divisors of 36 are 1, 2, 3, 4, 6, 9, 12,
18, and 36. Each of these divisors can be paired with a divisor to make a product of 36. Hence there are 9 ordered
pairs
(
M, N
)
.
Difficulty:
Medium
SMP-CCSS:
7
CCSS-M:
4.OA4, A-APR.7
10
A dessert chef prepares the dessert for every day of a week
starting with Sunday. The dessert each day is either cake, pie,
ice cream, or pudding. The same dessert may not be served
two days in a row. There must be cake on Friday because of a
birthday. How many different dessert menus for the week are
possible?
(A)
729
(B)
972
(C)
1024
(D)
2187
(E)
2304
2012 AMC 10B, Problem #11—
2012 AMC 12B, Problem #8—
“Consider how many choices of desserts are possible on Saturday
and Thursday.”
Solution
Answer (A):
There are 3 choices for Saturday (anything except cake) and for the same reason 3 choices for Thursday.
Similarly there are 3 choices for Wednesday, Tuesday, Monday, and Sunday (anything except what was to be served
the following day). Therefore there are
3
6
= 729
possible dessert menus.
OR
If any dessert could be served on Friday, there would be 4 choices for Sunday and 3 for each of the other six days.
There would be a total of
4
·
3
6
dessert menus for the week, and each dessert would be served on Friday with equal
frequency. Because cake is the dessert for Friday, this total is too large by a factor of 4. The actual total is
3
6
= 729
.
Difficulty:
10: Medium Hard
12: Medium Hard
SMP-CCSS:
2
CCSS-M:
None
11
Point
B
is due east of point
A
. Point
C
is due north of point
B
.
The distance between points
A
and
C
is
10
√
2
meters,
and
∠
BAC
= 45
◦
. Point
D
is 20 meters due north of point
C
. The distance
AD
is between which two integers?
(A)
30 and 31
(B)
31 and 32
(C)
32 and 33
(D)
33
and 34
(E)
34 and 35
2012 AMC 10B, Problem #12—
“What type of triangle is
4
ABC
?”
Solution
Answer (B):
Note that
∠
ABC
= 90
◦
, so
4
ABC
is a
45
–
45
–
90
◦
triangle. Because hypotenuse
AC
= 10
√
2
,
the legs of
4
ABC
have length 10.
Therefore
AB
= 10
and
BD
=
BC
+
CD
= 10 + 20 = 30
.
By the
Pythagorean Theorem,
AD
=
p
10
2
+ 30
2
=
√
1000
.
Because
31
2
= 961
and
32
2
= 1024
, it follows that
31
< AD <
32
.
A
B
C
D
10
10
10
√
2
20
Difficulty:
Medium
SMP-CCSS:
5
CCSS-M:
G-SRT.8
12
It takes Clea 60 seconds to walk down an escalator when it is
not operating, and only 24 seconds to walk down the escalator
when it is operating. How many seconds does it take Clea to
ride down the operating escalator when she just stands on it?
(A)
36
(B)
40
(C)
42
(D)
48
(E)
52
2012 AMC 10B, Problem #13—
2012 AMC 12B, Problem #9—
“Assign variables to the rate of walking and the rate of the moving
escalator, then compare times and rates for the same distance.”
Solution
Answer (B):
Let
x
be Clea’s rate of walking and
r
be the rate of the moving escalator. Because the distance is
constant,
24(
x
+
r
) = 60
x
. Solving for
r
yields
r
=
3
2
x
. Let
t
be the time required for Clea to make the escalator
trip while just standing on it. Then
rt
= 60
x
, so
3
2
xt
= 60
x
. Therefore
t
= 40
seconds.
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