# Mock AIME 2 Pre 2005 Problems

## Contents

## Problem 1

Compute the largest integer such that divides .

## Problem 2

is a real number with the property that . Let . Determine the value of .

## Problem 3

In a box, there are green balls, blue balls, red balls, a brown ball, a white ball, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed. This process requires that at most balls be removed. The probability that balls are drawn can be expressed as , where and are relatively prime positive integers. Compute .

## Problem 4

Let . Given that has digits, how many elements of begin with the digit ?

## Problem 5

Let be the set of integers for which , an infinite decimal that has the property that for all positive integers . Given that is prime, how many positive integers are in ? (The are digits.)

## Problem 6

is a scalene triangle. Points , , and are selected on sides , , and respectively. The cevians , , and concur at point . If , , and , determine the area of triangle .

## Problem 7

Anders, Po-Ru, Reid, and Aaron are playing Bridge. After one hand, they notice that all of the cards of the two suits are split between Reid and Po-Ru's hands. Let denote the number of ways cards can be dealt to each player such that this is the case. Determine the remainder obtained when is divided by . (Bridge is a game played with the standard -card deck.)

## Problem 8

Determine the remainder obtained when the expression is divided by .

## Problem 9

Let where and . Determine the remainder obtained when is divided by .

## Problem 10

is a cyclic pentagon with . The diagonals and intersect at . is the foot of the altitude from to . We have , , and . The area of triangle can be expressed as where and are relatively prime positive integers. Determine the remainder obtained when is divided by .

## Problem 11

, , and are the roots of . Let The value of can be written as where and are relatively prime positive integers. Determine the value of .

## Problem 12

is a cyclic quadrilateral with , , , and . Let and denote the circumcenter and intersection of and respectively. The value of can be expressed as , where and are relatively prime positive integers. Determine the remainder obtained when is divided by .

## Problem 13

is a polynomial of minimal degree that satisfies for . The value of can be written as , where and are relatively prime positive integers. Determine .

## Problem 14

Elm trees, Dogwood trees, and Oak trees are to be planted in a line in front of a library such that How many ways can the trees be situated in this manner?

## Problem 15

In triangle , we have , , and . Points , , and are selected on , , and respectively such that , , and concur at the circumcenter of . The value of can be expressed as where and are relatively prime positive integers. Determine .