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 3
Let be a regular
and
be a regular
such that each interior angle of
is
as large as each interior angle of
. What's the largest possible value of
?
Problem 5
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will be the resulting product?
Problem 4
In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio ?
Problem 9
Suppose that and that
where
is in lowest terms. Find
Problem 8
For any sequence of real numbers , define
to be the sequence
, whose
term is
. Suppose that all of the terms of the sequence
are
, and that
. Find
.
Problem 7
Three numbers, ,
,
, are drawn randomly and without replacement from the set
. Three other numbers,
,
,
, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let
be the probability that, after a suitable rotation, a brick of dimensions
can be enclosed in a box of dimensions
, with the sides of the brick parallel to the sides of the box. If
is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Problem 12
Let be a tetrahedron with
,
,
,
,
, and
, as shown in the figure. Let
be the distance between the midpoints of edges
and
. Find
.
Problem 11
Twelve congruent disks are placed on a circle of radius 1 in such a way that the twelve disks cover
, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the form
, where
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 12
Rhombus is inscribed in rectangle
so that vertices
,
,
, and
are interior points on sides
,
,
, and
, respectively. It is given that
,
,
, and
. Let
, in lowest terms, denote the perimeter of
. Find
.
Problem 10
Euler's formula states that for a convex polyhedron with vertices,
edges, and
faces,
. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its
vertices,
triangular faces and
pentagonal faces meet. What is the value of
?
Problem 13
Let be a subset of
such that no two members of
differ by
or
. What is the largest number of elements
can have?
Problem 14
Given a positive integer , it can be shown that every complex number of the form
, where
and
are integers, can be uniquely expressed in the base
using the integers
as digits. That is, the equation
![$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$](http://latex.artofproblemsolving.com/f/6/4/f6491a5fef34ba26ef6657aae299c94700b40a0e.png)
is true for a unique choice of non-negative integer and digits
chosen from the set
, with
. We write
![$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$](http://latex.artofproblemsolving.com/5/6/a/56aa9ca7c641491fbd454fff558b7a8fb8918ced.png)
to denote the base expansion of
. There are only finitely many integers
that have four-digit expansions
![$k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.$](http://latex.artofproblemsolving.com/6/0/f/60f64fe15a0ae3ffb0c11f59e745a2bd2b665a3d.png)
Find the sum of all such .
Problem 14
The rectangle below has dimensions
and
. Diagonals
and
intersect at
. If triangle
is cut out and removed, edges
and
are joined, and the figure is then creased along segments
and
, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.
Problem 15
Define a positive integer to be a factorial tail if there is some positive integer
such that the decimal representation of
ends with exactly
zeroes. How many positive integers less than
are not factorial tails?
Problem 14
A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form , for a positive integer
. Find
.