# Mock AIME 1 2007-2008 Problems

## Contents

## Problem 1

What is the coefficient of in ?

## Problem 2

The expansion of has 3 consecutive terms with coefficients in the ratio that can be written in the form Find the sum of all possible values of .

## Problem 3

A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?

## Problem 4

If is an odd number, then find the largest integer that always divides the expression

## Problem 5

Let , where . Find .

## Problem 6

A -array is a structured, infinite, collection of numbers. For example, a -array is constructed as follows:

In general, the first entry of each row is times the first entry of the previous row. Then, each succeeding term in a row is times the previous term in the same row. If the sum of all the terms in a -array can be written in the form , where and are relatively prime positive integers, find the remainder when is divided by .

## Problem 7

Consider the following function defined as Find .

## Problem 8

A sequence of ten s and/or s is randomly generated. If the probability that the sequence does not contain two consecutive s can be written in the form , where are relatively prime positive integers, find .

## Problem 9

Let represent the smallest integer that satisfies the following conditions:

- is a perfect square.

- is a perfect cube.

- is a perfect fifth.

How many divisors does have that are not multiples of 10?

## Problem 10

An oreo shop sells different flavors of oreos and different flavors of milk. Alpha and Beta decide to purchase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many ways could they have left the store with 3 products collectively? (A possible purchase is Alpha purchases 1 box of uh-oh oreos and 1 gallon of whole milk while Beta purchases 1 bag of strawberry milkshake oreos).

## Problem 11

is inscribed inside such that lie on , respectively. The circumcircles of have centers , respectively. Also, , and . The length of can be written in the form , where and are relatively prime integers. Find .

## Problem 12

Let and . If , how many integral values of are there such that is a multiple of ?

## Problem 13

Let be a polynomial such that and for such that both sides are defined. Find .

## Problem 14

Points and lie on , with radius , so that is acute. Extend to point so that . Let be the intersection of and such that and . If can be written as , where and are relatively prime and is not divisible by the square of any prime, find .

## Problem 15

The sum can be written in the form , where are trigonometric functions and are degrees . Find .