Difference between revisions of "2009 AIME I Problems"
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== Problem 15 == | == Problem 15 == | ||
− | In triangle <math>ABC</math>, <math>AB = 10</math>, <math>BC = 14</math>, and <math>CA = 16</math>. Let <math>D</math> be a point in the interior of <math>\overline{BC}</math>. Let <math>I_B</math> and <math>I_C</math> denote the incenters of triangles <math>ABD</math> and <math>ACD</math>, respectively. The circumcircles of triangles <math>BI_BD</math> and <math>CI_CD</math> meet at distinct points <math>P</math> and <math>D</math>. The maximum possible area of <math>\triangle BPC</math> can be expressed in the form <math>a - b\sqrt {c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers and <math>c</math> is not divisible by the square of any prime. Find <math>a + b + c</math>. | + | In triangle <math>ABC</math>, <math>AB = 10</math>, <math>BC = 14</math>, and <math>CA = 16</math>. Let <math>D</math> be a point in the interior of <math>\overline{BC}</math>. Let points <math>I_B</math> and <math>I_C</math> denote the incenters of triangles <math>ABD</math> and <math>ACD</math>, respectively. The circumcircles of triangles <math>BI_BD</math> and <math>CI_CD</math> meet at distinct points <math>P</math> and <math>D</math>. The maximum possible area of <math>\triangle BPC</math> can be expressed in the form <math>a - b\sqrt {c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers and <math>c</math> is not divisible by the square of any prime. Find <math>a + b + c</math>. |
[[2009 AIME I Problems/Problem 15|Solution]] | [[2009 AIME I Problems/Problem 15|Solution]] |
Revision as of 15:14, 18 August 2020
2009 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
Call a -digit number geometric if it has
distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.
Problem 2
There is a complex number with imaginary part
and a positive integer
such that
Find .
Problem 3
A coin that comes up heads with probability and tails with probability
independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to
of the probability of five heads and three tails. Let
, where
and
are relatively prime positive integers. Find
.
Problem 4
In parallelogram , point
is on
so that
and point
is on
so that
. Let
be the point of intersection of
and
. Find
.
Problem 5
Triangle has
and
. Points
and
are located on
and
respectively so that
, and
is the angle bisector of angle
. Let
be the point of intersection of
and
, and let
be the point on line
for which
is the midpoint of
. If
, find
.
Problem 6
How many positive integers less than
are there such that the equation
has a solution for
?
Problem 7
The sequence satisfies
and
for
. Let
be the least integer greater than
for which
is an integer. Find
.
Problem 8
Let . Consider all possible positive differences of pairs of elements of
. Let
be the sum of all of these differences. Find the remainder when
is divided by
.
Problem 9
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from to
inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were
. Find the total number of possible guesses for all three prizes consistent with the hint.
Problem 10
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from to
in clockwise order. Committee rules state that a Martian must occupy chair
and an Earthling must occupy chair
. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is
. Find
.
Problem 11
Consider the set of all triangles where
is the origin and
and
are distinct points in the plane with nonnegative integer coordinates
such that
. Find the number of such distinct triangles whose area is a positive integer.
Problem 12
In right with hypotenuse
,
,
, and
is the altitude to
. Let
be the circle having
as a diameter. Let
be a point outside
such that
and
are both tangent to circle
. The ratio of the perimeter of
to the length
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 13
The terms of the sequence defined by
for
are positive integers. Find the minimum possible value of
.
Problem 14
For , define
, where
. If
and
, find the minimum possible value for
.
Problem 15
In triangle ,
,
, and
. Let
be a point in the interior of
. Let points
and
denote the incenters of triangles
and
, respectively. The circumcircles of triangles
and
meet at distinct points
and
. The maximum possible area of
can be expressed in the form
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
See also
2009 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2008 AIME II Problems |
Followed by 2009 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
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.