User:Rowechen
Hey how did you get to this page? If you aren't me then I have to say hello. If you are me then I must be pretty conceited to waste my time looking at my own page. If you aren't me, seriously, how did you get to this page? This is pretty cool. Well, nice meeting you! I'm going to stop wasting my time typing this up and do some math. Gtg. Bye.
Here's the AIME compilation I will be doing:
Contents
Problem 2
A circle with diameter of length 10 is internally tangent at
to a circle of radius 20. Square
is constructed with
and
on the larger circle,
tangent at
to the smaller circle, and the smaller circle outside
. The length of
can be written in the form
, where
and
are integers. Find
.
Problem 6
For how many pairs of consecutive integers in is no carrying required when the two integers are added?
Problem 6
The graphs of the equations
![$y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,$](http://latex.artofproblemsolving.com/1/4/2/1420d57ceea649c3bb54cffc8b0aae09465d2483.png)
are drawn in the coordinate plane for These 63 lines cut part of the plane into equilateral triangles of side
. How many such triangles are formed?
Problem 8
Let be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of
so that the union of the two subsets is
? The order of selection does not matter; for example, the pair of subsets
,
represents the same selection as the pair
,
.
Problem 7
For certain ordered pairs of real numbers, the system of equations
![$ax+by=1\,$](http://latex.artofproblemsolving.com/9/4/b/94b5a3e0b1a726c775259827de5d37bb20d31b62.png)
![$x^2+y^2=50\,$](http://latex.artofproblemsolving.com/d/5/b/d5b1e8d558c3818bdd87c0b06261db4f6a0977fe.png)
has at least one solution, and each solution is an ordered pair of integers. How many such ordered pairs
are there?
Problem 8
The points ,
, and
are the vertices of an equilateral triangle. Find the value of
.
Problem 12
The vertices of are
,
, and
. The six faces of a die are labeled with two
's, two
's, and two
's. Point
is chosen in the interior of
, and points
,
,
are generated by rolling the die repeatedly and applying the rule: If the die shows label
, where
, and
is the most recently obtained point, then
is the midpoint of
. Given that
, what is
?
Problem 11
Ninety-four bricks, each measuring are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes
or
or
to the total height of the tower. How many different tower heights can be achieved using all 94 of the bricks?
Problem 12
Pyramid has square base
congruent edges
and
and
Let
be the measure of the dihedral angle formed by faces
and
Given that
where
and
are integers, find
Problem 10
Find the smallest positive integer solution to .
Problem 13
Let be the integer closest to
Find
Problem 14
In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form where
and
are positive integers and
is not divisible by the square of any prime number. Find
Problem 13
In triangle ,
,
, and
. There is a point
for which
bisects
, and
is a right angle. The ratio
can be written in the form , where
and
are relatively prime positive integers. Find
.
Problem 15
In parallelogram let
be the intersection of diagonals
and
. Angles
and
are each twice as large as angle
and angle
is
times as large as angle
. Find the greatest integer that does not exceed
.
Problem 13
If is a set of real numbers, indexed so that
its complex power sum is defined to be
where
Let
be the sum of the complex power sums of all nonempty subsets of
Given that
and
where
and
are integers, find