Difference between revisions of "2012 AIME II Problems"
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<center><asy> | <center><asy> | ||
− | pair | + | pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); |
− | pair | + | pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; |
− | + | pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); | |
− | + | draw(S1--S2--S3--S4--cycle); | |
− | + | draw(Sp1--Sp2--Sp3--Sp4--cycle); | |
− | + | draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); | |
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− | draw( | ||
− | draw( | ||
− | draw( | ||
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</asy></center> | </asy></center> | ||
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== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year=2012|n=II|before=[[2012 AIME I Problems]]|after=[[2013 AIME I Problems]]}} | ||
+ | |||
* [[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 12:16, 13 March 2020
2012 AIME II (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
Find the number of ordered pairs of positive integer solutions to the equation
.
Problem 2
Two geometric sequences and
have the same common ratio, with
,
, and
. Find
.
Problem 3
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements.
Problem 4
Ana, Bob, and Cao bike at constant rates of meters per second,
meters per second, and
meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point
on the south edge of the field. Cao arrives at point
at the same time that Ana and Bob arrive at
for the first time. The ratio of the field's length to the field's width to the distance from point
to the southeast corner of the field can be represented as
, where
,
, and
are positive integers with
and
relatively prime. Find
.
Problem 5
In the accompanying figure, the outer square has side length
. A second square
of side length
is constructed inside
with the same center as
and with sides parallel to those of
. From each midpoint of a side of
, segments are drawn to the two closest vertices of
. The result is a four-pointed starlike figure inscribed in
. The star figure is cut out and then folded to form a pyramid with base
. Find the volume of this pyramid.
![[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]](http://latex.artofproblemsolving.com/2/8/6/2862b9fac9f2c88c10b30e3908cf4ac1d5f62115.png)
Problem 6
Let be the complex number with
and
such that the distance between
and
is maximized, and let
. Find
.
Problem 7
Let be the increasing sequence of positive integers whose binary representation has exactly
ones. Let
be the 1000th number in
. Find the remainder when
is divided by
.
Problem 8
The complex numbers and
satisfy the system
Find the smallest possible value of
.
Problem 9
Let and
be real numbers such that
and
. The value of
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 10
Find the number of positive integers less than
for which there exists a positive real number
such that
.
Note: is the greatest integer less than or equal to
.
Problem 11
Let , and for
, define
. The value of
that satisfies
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 12
For a positive integer , define the positive integer
to be
-safe if
differs in absolute value by more than
from all multiples of
. For example, the set of
-safe numbers is
. Find the number of positive integers less than or equal to
which are simultaneously
-safe,
-safe, and
-safe.
Problem 13
Equilateral has side length
. There are four distinct triangles
,
,
, and
, each congruent to
,
with
. Find
.
Problem 14
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when
is divided by
.
Problem 15
Triangle is inscribed in circle
with
,
, and
. The bisector of angle
meets side
at
and circle
at a second point
. Let
be the circle with diameter
. Circles
and
meet at
and a second point
. Then
, where
and
are relatively prime positive integers. Find
.
See also
2012 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2012 AIME I Problems |
Followed by 2013 AIME I 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.