Difference between revisions of "Mock AIME 2 2006-2007 Problems"
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== Problem 7 == | == Problem 7 == | ||
− | A right circular cone of base radius <math>\displaystyle 17</math>cm and slant height <math>\displaystyle 34</math>cm is given. <math>\displaystyle P</math> is a point on the circumference of the base and the shortest path from <math>\displaystyle P</math> around the cone and back is drawn (see diagram). If the | + | A right circular cone of base radius <math>\displaystyle 17</math>cm and slant height <math>\displaystyle 34</math>cm is given. <math>\displaystyle P</math> is a point on the circumference of the base and the shortest path from <math>\displaystyle P</math> around the cone and back is drawn (see diagram). If the length of this path is <math>\displaystyle m\sqrt{n},</math> where <math>\displaystyle m</math> and <math>\displaystyle n</math> are relatively prime positive integers, find <math>\displaystyle m+n.</math> |
− | [[Image: | + | [[Image:Mock_AIME_2_2007_Problem8.jpg]] |
[[Mock_AIME_2_2006-2007/Problem_7|Solution]] | [[Mock_AIME_2_2006-2007/Problem_7|Solution]] |
Revision as of 14:09, 25 July 2006
Contents
Problem 1
A positive integer is called a dragon if it can be partitioned into four positive integers and
such that
Find the smallest dragon.
Problem 2
The set consists of all integers from
to
inclusive. For how many elements
in
is
an integer?
Problem 3
Let be the sum of all positive integers
such that
is a perfect square. Find the remainder when
is divided by
Problem 4
Let be the smallest positive integer for which there exist positive real numbers
and
such that
. Compute
.
Problem 5
Given that and
find
Problem 6
If and
find
Problem 7
A right circular cone of base radius cm and slant height
cm is given.
is a point on the circumference of the base and the shortest path from
around the cone and back is drawn (see diagram). If the length of this path is
where
and
are relatively prime positive integers, find
Problem 8
The positive integers satisfy
and
for
. Find the last three digits of
.
Problem 9
In right triangle
Cevians
and
intersect at
and are drawn to
and
respectively such that
and
If
where
and
are relatively prime and
has no perfect square divisors excluding
find
Problem 10
Find the number of solutions, in degrees, to the equation where
Problem 11
Find the sum of the squares of the roots, real or complex, of the system of simultaneous equations
Problem 12
In quadrilateral
and
If
,
and the area of
is
where
are relatively prime positive integers, find
Note*: and
refer to the areas of triangles
and
Problem 13
In his spare time, Richard Rusczyk shuffles a standard deck of 52 playing cards. He then turns the cards up one by one from the top of the deck until the third ace appears. If the expected (average) number of cards Richard will turn up is where
and
are relatively prime positive integers, find
Problem 14
In triangle ABC, and
Given that
,
and
intersect at
and are an angle bisector, median, and altitude of the triangle, respectively, compute the length of
Problem 15
A cube is composed of
unit cubes. The faces of
unit cubes are colored red. An arrangement of the cubes is
if there is exactly
red unit cube in every
rectangular box composed of
unit cubes. Determine the number of
colorings.