# 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

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

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