Difference between revisions of "2010 AIME II Problems"
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== Problem 8 == | == Problem 8 == | ||
− | + | Let <math>N</math> be the number of ordered pairs of nonempty sets <math>\mathcal{A}</math> and <math>\mathcal{B}</math> that have the following properties: | |
+ | |||
+ | <UL> | ||
+ | <LI> <math>\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}</math>,</LI> | ||
+ | <LI> <math>\mathcal{A} \cap \mathcal{B} = \emptyset</math>, <\LI> | ||
+ | <LI> The number of elements of <math>\mathcal{A}</math> is not an element of <math>\mathcal{A}</math>,</LI> | ||
+ | <LI> The number of elements of <math>\mathcal{B}</math> is not an element of <math>\mathcal{B}</math>. | ||
+ | </UL> | ||
+ | |||
+ | Find <math>N</math>. | ||
[[2010 AIME II Problems/Problem 8|Solution]] | [[2010 AIME II Problems/Problem 8|Solution]] |
Revision as of 11:13, 2 April 2010
2010 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
NOTE: THESE ARE THE PROBLEMS FROM THE AIME I. THE PROBLEMS WILL BE UPDATED SHORTLY.
Contents
Problem 1
Let be the greatest integer multiple of
all of whose digits are even and no two of whose digits are the same. Find the remainder when
is divided by
.
Problem 2
A point is chosen at random in the interior of a unit square
. Let
denote the distance from
to the closest side of
. The probability that
is equal to
, where
and
are relatively prime positive integers. Find
.
Problem 3
Let be the product of all factors
(not necessarily distinct) where
and
are integers satisfying
. Find the greatest positive integer
such that
divides
.
Problem 4
Dave arrives at an airport which has twelve gates arranged in a straight line with exactly feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks
feet or less to the new gate be a fraction
, where
and
are relatively prime positive integers. Find
.
Problem 5
Positive numbers ,
, and
satisfy
and
. Find
.
Problem 6
Find the smallest positive integer with the property that the polynomial
can be written as a product of two nonconstant polynomials with integer coefficients.
Problem 7
Let , where
,
, and
are real. There exists a complex number
such that the three roots of
are
,
, and
, where
. Find
.
Problem 8
Let be the number of ordered pairs of nonempty sets
and
that have the following properties:
-
,
-
, <\LI>
- The number of elements of
is not an element of
,
- The number of elements of
is not an element of
.
Find .
Problem 9
Let be the real solution of the system of equations
,
,
. The greatest possible value of
can be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 10
Let be the number of ways to write
in the form
, where the
's are integers, and
. An example of such a representation is
. Find
.
Problem 11
Let be the region consisting of the set of points in the coordinate plane that satisfy both
and
. When
is revolved around the line whose equation is
, the volume of the resulting solid is
, where
,
, and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Problem 12
Let be an integer and let
. Find the smallest value of
such that for every partition of
into two subsets, at least one of the subsets contains integers
,
, and
(not necessarily distinct) such that
.
Note: a partition of is a pair of sets
,
such that
,
.
Problem 13
Rectangle and a semicircle with diameter
are coplanar and have nonoverlapping interiors. Let
denote the region enclosed by the semicircle and the rectangle. Line
meets the semicircle, segment
, and segment
at distinct points
,
, and
, respectively. Line
divides region
into two regions with areas in the ratio
. Suppose that
,
, and
. Then
can be represented as
, where
and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 14
For each positive integer n, let . Find the largest value of n for which
.
Note: is the greatest integer less than or equal to
.
Problem 15
In with
,
, and
, let
be a point on
such that the incircles of
and
have equal radii. Let
and
be positive relatively prime integers such that
. Find
.