Difference between revisions of "2010 AIME I Problems"
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== Problem 14 == | == Problem 14 == | ||
− | For each positive integer n, let <math>f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor</math>. Find the largest value of n for which <math>f(n) \le 300</math>. | + | For each positive integer <math>n,</math> let <math>f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor</math>. Find the largest value of <math>n</math> for which <math>f(n) \le 300</math>. |
'''Note:''' <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>. | '''Note:''' <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>. | ||
Line 78: | Line 78: | ||
== Problem 15 == | == Problem 15 == | ||
− | In <math>\triangle{ABC}</math> with <math>AB = 12</math>, <math>BC = 13</math>, and <math>AC = 15</math>, let <math>M</math> be a point on <math>\overline{AC}</math> such that the incircles of <math>\triangle{ABM}</math> and <math>\triangle{BCM}</math> have equal radii. | + | In <math>\triangle{ABC}</math> with <math>AB = 12</math>, <math>BC = 13</math>, and <math>AC = 15</math>, let <math>M</math> be a point on <math>\overline{AC}</math> such that the incircles of <math>\triangle{ABM}</math> and <math>\triangle{BCM}</math> have equal radii. Then <math>\frac{AM}{CM} = \frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p + q</math>. |
[[2010 AIME I Problems/Problem 15|Solution]] | [[2010 AIME I Problems/Problem 15|Solution]] | ||
== See also == | == See also == | ||
+ | {{AIME box|year=2010|n=I|before=[[2009 AIME II Problems]]|after=[[2010 AIME II Problems]]}} | ||
+ | * [[American Invitational Mathematics Examination]] | ||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:58, 10 August 2020
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
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
denote 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 let
. Find the largest value of
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. Then
, where
and
are relatively prime positive integers. Find
.
See also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2009 AIME II Problems |
Followed by 2010 AIME II Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.