Difference between revisions of "2008 AIME II Problems"
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== Problem 13 == | == Problem 13 == | ||
− | A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let <math>R</math> be the region outside the hexagon, and let <math>S = {\ | + | A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let <math>R</math> be the region outside the hexagon, and let <math>S = \{\frac{1}{z}|z \in R\}</math>. Then the area of <math>S</math> has the form <math>a\pi + \sqrt{b}</math>, where <math>a</math> and <math>b</math> are positive integers. Find <math>a + b</math>. |
[[2008_AIME_II_Problems/Problem_13|Solution]] | [[2008_AIME_II_Problems/Problem_13|Solution]] |
Revision as of 20:12, 3 April 2008
Contents
Problem 1
Let , where the additions and subtractions alternate in pairs. Find the remainder when
is divided by
.
Problem 2
Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the -mile mark at exactly the same time. How many minutes has it taken them?
Problem 3
A block of cheese in the shape of a rectangular solid measures cm by
cm by
cm. Ten slices are cut from the cheese. Each slice has a width of
cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?
Problem 4
There exist unique nonnegative integers
and
unique integers
(
) with each
either
or
such that
Find
.
Problem 5
In trapezoid with
, let
and
. Let
,
, and
and
be the midpoints of
and
, respectively. Find the length
.
Problem 6
The sequence is defined by
The sequence
is defined by
Find
.
Problem 7
Let ,
, and
be the three roots of the equation
Find
.
Problem 8
Let . Find the smallest positive integer
such that
is an integer.
Problem 9
A particle is located on the coordinate plane at . Define a move for the particle as a counterclockwise rotation of
radians about the origin followed by a translation of
units in the positive
-direction. Given that the particle's position after
moves is
, find the greatest integer less than of equal to
.
Problem 10
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Problem 11
In triangle ,
, and
. Circle
has radius
and is tangent to
and
. Circle
is externally tangent to circle
and is tangent to
and
. No point of circle
lies outside of
. The radius of circle
can be expressed in the form
,where
,
, and
are positive integers and
is the product of distinct primes. Find
.
Problem 12
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Problem 13
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let be the region outside the hexagon, and let
. Then the area of
has the form
, where
and
are positive integers. Find
.
Problem 14
Let and
be positive real numbers with
. Let
be the maximum possible value of
for which the system of equations
has a solution
satisfying
and
. Then
can be expressed as a fraction
, where
and
are relatively prime positive integers. Find
.
Problem 15
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