The median of the list
n, n
+3
, n
+4
, n
+5
, n
+6
, n
+8
, n
+10
, n
+12
, n
+15
is 10. What is the mean?
(A)
4
(B)
6
(C)
7
(D)
10
(E)
11
2001 AMC 10, Problem #1
—
“The middle number in the 9number list is
n
+ 6
.”
Solution
(E)
The middle number in the 9number list is
n
+6
, which is given as 10.
Thus
n
= 4
. Add the terms together to get
9
n
+ 63 = 9
·
4 + 63 = 99
.
Thus the mean is
99
/
9 = 11
.
Difficulty:
Medium
NCTM Standard:
Number and Operations Standard for Grades 9–12: Understand numbers, ways
of representing numbers, relationships among numbers, and number systems.
Mathworld.com Classification:
Probability and Statistics
>
Rank Statistics
>
Median
Calculus and Analysis
>
Special Functions
>
Means
A number
x
is 2 more than the product of its reciprocal
and its additive inverse.
In which interval does the
number lie?
(A)

4
≤
x
≤ 
2
(B)

2
< x
≤
0
(C)
0
< x
≤
2
(D)
2
< x
≤
4
(E)
4
< x
≤
6
2001 AMC 10, Problem #2
—
“The reciprocal of
x
is
1
x
, and the additive inverse of
x
is

x
.”
Solution
(C)
The reciprocal of
x
is
1
x
, and the additive inverse of
x
is

x
.
The
product of these is
(
1
x
)
·
(

x
) =

1
. So
x
=

1 + 2 = 1
, which is in the
interval
0
< x
≤
2
.
Difficulty:
Mediumhard
NCTM Standard:
Number and Operations Standard for Grades 9–12: Understand numbers, ways
of representing numbers, relationships among numbers, and number systems.
Mathworld.com Classification:
Number Theory
>
Arithmetic
>
Multiplication and Division
>
Reciprocal
Algebra
>
Group Theory
>
Group Properties
>
Additive Inverse
The sum of two numbers is
S
. Suppose 3 is added to
each number and then each of the resulting numbers
is doubled. What is the sum of the final two numbers?
(A)
2
S
+ 3
(B)
3
S
+ 2
(C)
3
S
+ 6
(D)
2
S
+ 6
(E)
2
S
+ 12
2001 AMC 10, Problem #3
—
2001 AMC 12, Problem #1
—
“The order of
arithmetic operations
is
add 3
then
double the sum.”
Solution
(E)
Suppose the two numbers are
a
and
b
. Then the desired sum is
2(
a
+ 3) + 2(
b
+ 3) = 2(
a
+
b
) + 12 = 2
S
+ 12
.
Difficulty:
Medium
NCTM Standard:
Number and Operations Standard for Grades 9–12: Understand meanings of
operations and how they relate to one another.
Mathworld.com Classification:
Number Theory
>
Arithmetic
>
General Arithmetic
What is the maximum number for the possible points
of intersection of a circle and a triangle?
(A)
2
(B)
3
(C)
4
(D)
5
(E)
6
2001 AMC 10, Problem #4
—
“The circle can intersect at most two points of each
side of the triangle.”
Solution
(E)
The circle can intersect at most two points of each
side of the triangle, so the number can be no greater
than six.
The figure shows that the number can
indeed be six.
Difficulty:
Mediumeasy
NCTM
Standard:
Geometry
Standard
for
Grades
9–12:
Explore
relationships
(including
congruence and similarity) among classes of two and threedimensional geometric objects, make
and test conjectures about them, and solve problems involving them.
Mathworld.com Classification:
Geometry
>
Plane Geometry
>
Circles
Geometry
>
Plane Geometry
>
Triangles
>
Special Triangles
>
Other Triangles
>
Triangle
How many of the twelve pentominoes pictured below
have at least one line of symmetry?
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 Winter '13
 Kramer
 Decimal, triangle, Numeral system