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
Find the number of ordered pairs of positive integers that satisfy
and
.
Problem 5
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is where
and
are positive integers, and
and
are relatively prime, find
Problem 6
The cards in a stack of cards are numbered consecutively from 1 through
from top to bottom. The top
cards are removed, kept in order, and form pile
The remaining cards form pile
The cards are then restacked by taking cards alternately from the tops of pile
and
respectively. In this process, card number
becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles
and
are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. Find the number of cards in the magical stack in which card number 131 retains its original position.
Problem 8
How many different arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?
Problem 9
Given a nonnegative real number , let
denote the fractional part of
; that is,
, where
denotes the greatest integer less than or equal to
. Suppose that
is positive,
, and
. Find the value of
.
Problem 10
Let be the set of points whose coordinates
and
are integers that satisfy
and
Two distinct points are randomly chosen from
The probability that the midpoint of the segment they determine also belongs to
is
where
and
are relatively prime positive integers. Find
Problem 11
In a rectangular array of points, with 5 rows and columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through
the second row is numbered
through
and so forth. Five points,
and
are selected so that each
is in row
Let
be the number associated with
Now renumber the array consecutively from top to bottom, beginning with the first column. Let
be the number associated with
after the renumbering. It is found that
and
Find the smallest possible value of
Problem 12
A sphere is inscribed in the tetrahedron whose vertices are and
The radius of the sphere is
where
and
are relatively prime positive integers. Find
Problem 9
The system of equations
has two solutions and
. Find
.
Problem 12
Given a function for which
![$f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)$](http://latex.artofproblemsolving.com/e/d/3/ed3b71f245528b2172017648ad46875af6955664.png)
holds for all real what is the largest number of different values that can appear in the list
?
Problem 14
There are complex numbers that satisfy both
and
. These numbers have the form
, where
and angles are measured in degrees. Find the value of
.
Problem 13
In triangle the medians
and
have lengths 18 and 27, respectively, and
. Extend
to intersect the circumcircle of
at
. The area of triangle
is
, where
and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 14
The perimeter of triangle is
, and the angle
is a right angle. A circle of radius
with center
on
is drawn so that it is tangent to
and
. Given that
where
and
are relatively prime positive integers, find
.
Problem 13
Let be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when
is divided by 1000.
Problem 14
The decimal representation of where
and
are relatively prime positive integers and
contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of
for which this is possible.