2012年AIME II 真题:
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]](https://latex.artofproblemsolving.com/2/8/6/2862b9fac9f2c88c10b30e3908cf4ac1d5f62115.png)
Let be the complex number with
and
such that the distance between
and
is maximized, and let
. Find
.
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