Difference between revisions of "Mock AIME 1 Pre 2005 Problems"
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== Problem 14 == | == Problem 14 == | ||
Wally's Key Company makes and sells two types of keys. Mr. Porter buys a total of 12 keys from Wally's. Determine the number of possible arrangements of My. Porter's 12 new keys on his keychain (rotations are considered the same and any two keys of the same type are identical). | Wally's Key Company makes and sells two types of keys. Mr. Porter buys a total of 12 keys from Wally's. Determine the number of possible arrangements of My. Porter's 12 new keys on his keychain (rotations are considered the same and any two keys of the same type are identical). | ||
+ | |||
+ | Note: The problem is meant to be interpreted so that if you cannot produce one arrangement from another by rotation, then the two arrangements are different, even if you can produce one from the other from a combination of rotation and reflection. | ||
[[Mock AIME 1 Pre 2005 Problems/Problem 14|Solution]] | [[Mock AIME 1 Pre 2005 Problems/Problem 14|Solution]] |
Revision as of 14:23, 22 March 2012
Contents
Problem 1
Let denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when
is divided by
.
Problem 2
If , then the largest possible value of
can be written as
, where
and
are relatively prime positive integers. Determine
.
Problem 3
and
are collinear in that order such that
and
. If
can be any point in space, what is the smallest possible value of
?
Problem 4
When is divided by
, a remainder of
is obtained. Determine the value of
.
Problem 5
Let and
be the two real values of
for which
The smaller of the two values can be expressed as
, where
and
are integers. Compute
.
Problem 6
A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.
Problem 7
Let denote the number of permutations of the
-character string AAAABBBBBCCCCCC such that
- None of the first four letter is an A.
- None of the next five letters is a B.
- None of the last six letters is a C.
Find the remainder when is divided by
.
Problem 8
, a rectangle with
and
, is the base of pyramid
, which has a height of
. A plane parallel to
is passed through
, dividing
into a frustum
and a smaller pyramid
. Let
denote the center of the circumsphere of
, and let
denote the apex of
. If the volume of
is eight times that of
, then the value of
can be expressed as
, where
and
are relatively prime positive integers. Compute the value of
.
Problem 9
and
are three non-zero integers such that
and
Compute
.
Problem 10
is a regular heptagon inscribed in a unit circle centered at
.
is the line tangent to the circumcircle of
at
, and
is a point on
such that triangle
is isosceles. Let
denote the value of
. Determine the value of
.
Problem 11
Let denote the value of the sum
Determine the remainder obtained when
is divided by
.
Problem 12
is a rectangular sheet of paper.
and
are points on
and
respectively such that
. If
is folded over
,
maps to
on
and
maps to
such that
. If
and
, then the area of
can be expressed as
square units, where
and
are integers and
is not divisible by the square of any prime. Compute
.
Problem 13
A sequence obeys the recurrence
for any integers
. Additionally,
and
. Let
can be expressed as
for two relatively prime positive integers
and
. Determine the value of
.
Problem 14
Wally's Key Company makes and sells two types of keys. Mr. Porter buys a total of 12 keys from Wally's. Determine the number of possible arrangements of My. Porter's 12 new keys on his keychain (rotations are considered the same and any two keys of the same type are identical).
Note: The problem is meant to be interpreted so that if you cannot produce one arrangement from another by rotation, then the two arrangements are different, even if you can produce one from the other from a combination of rotation and reflection.
Problem 15
Triangle has an inradius of
and a circumradius of
. If
, then the area of triangle
can be expressed as
, where
and
are positive integers such that
and
are relatively prime and
is not divisible by the square of any prime. Compute
.