Difference between revisions of "2010 AIME I Problems"
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== Problem 9 == | == Problem 9 == | ||
− | Let <math>(a,b,c)</math> be | + | Let <math>(a,b,c)</math> be a real solution of the system of equations <math>x^3 - xyz = 2</math>, <math>y^3 - xyz = 6</math>, <math>z^3 - xyz = 20</math>. The greatest possible value of <math>a^3 + b^3 + c^3</math> can be written in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. |
[[2010 AIME I Problems/Problem 9|Solution]] | [[2010 AIME I Problems/Problem 9|Solution]] |
Revision as of 21:38, 22 February 2012
2010 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Maya lists all the positive divisors of . She then randomly selects two distinct divisors from this list. Let
be the probability that exactly one of the selected divisors is a perfect square. The probability
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 2
Find the remainder when is divided by
.
Problem 3
Suppose that and
. The quantity
can be expressed as a rational number
, where
and
are relatively prime positive integers. Find
.
Problem 4
Jackie and Phil have two fair coins and a third coin that comes up heads with probability . Jackie flips the three coins, and then Phil flips the three coins. Let
be the probability that Jackie gets the same number of heads as Phil, where
and
are relatively prime positive integers. Find
.
Problem 5
Positive integers ,
,
, and
satisfy
,
, and
. Find the number of possible values of
.
Problem 6
Let be a quadratic polynomial with real coefficients satisfying
for all real numbers
, and suppose
. Find
.
Problem 7
Define an ordered triple of sets to be minimally intersecting if
and
. For example,
is a minimally intersecting triple. Let
be the number of minimally intersecting ordered triples of sets for which each set is a subset of
. Find the remainder when
is divided by
.
Note: represents the number of elements in the set
.
Problem 8
For a real number , let
denominate the greatest integer less than or equal to
. Let
denote the region in the coordinate plane consisting of points
such that
. The region
is completely contained in a disk of radius
(a disk is the union of a circle and its interior). The minimum value of
can be written as
, where
and
are integers and
is not divisible by the square of any prime. Find
.
Problem 9
Let be a 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
.