# Mock AIME 1 2010 Problems

## Contents

## Problem 1

Let . Find the number of perfect squares among .

## Problem 2

Find the last three digits of the number of 7-tuples of positive integers such that , that is, divides , divides , divides , divides , divides , divides , and divides 6468.

## Problem 3

Let be a line segment of length , and let be the set of all points such that . Find the last three digits of the largest integer less than the area of .

## Problem 4

A round robin tournament is a tournament in which every player plays every other player exactly once. There is a round robin tournament with 2010 people. In each match, the winner scores one point, and the loser scores no points. There are no ties. Find the last three digits of the greatest possible difference between the first and second highest scores appearing among the players.

## Problem 5

For every integer , the representation of is defined to be the unique sequence of integers , with and such that . We represent as , where if is 0 or 1, and if . For example, . Find the last three digits of the sum of all integers with such that has at least one zero when written in balanced ternary form.

## Problem 6

Find the number of Gaussian integers with magnitude less than 10000 such that there exists a different Gaussian integer such that . (The magnitude of a complex , where and are reals, is defined to be . A Gaussian integer is defined to be a complex number whose real and imaginary parts are both integers.)

## Problem 7

Find the number of positive integers for which there exists a positive integer such that is the square of an integer.

## Problem 8

In the context of this problem, a is a block, a is a block, and a is a block. If is the number of ways George can place one square, two identical dominoes, and three identical trominoes on a chessboard such that no two overlap, find the remainder when is divided by 1000.

## Problem 9

Let and be circles of radii 5 and 7, respectively, and suppose that the distance between their centers is 10. There exists a circle that is internally tangent to both and , and tangent to the line joining the centers of and . If the radius of can be expressed in the form , where , , and are integers, and is not divisible by the square if any prime, find the value of .

## Problem 10

Find the last three digits of the largest possible value of where and are positive reals.

## Problem 11

Let be such that , , and . Let , , and be points such that , , , , , and . If the perimeter of hexagon can be expressed in the form , where and are relatively prime positive integers and is an integer not divisible by the square of any prime, find .

## Problem 12

Suppose , , and for all integers . Find the last three digits of .

## Problem 13

Suppose is inscribed in circle . and are the feet of the altitude from to and to , respectively. Let be the intersection of lines and , let be the point of intersection of and line distinct from , and let be the foot of the perpendicular from to . Given that , , and , and that can be expressed in the form , where and are relatively prime positive integers and is an integer not divisible by the square of any prime, find the last three digits of .

## Problem 14

Let , and let be a subset of with and . For such a set , let denote the number of sets with such that

(i) is a subset of with the same number of elements as ,

(ii) for ,

(iii) for .

Let . What is the smallest positive integer such that is over 9000?

## Problem 15

Let be the set of all integers less than or equal to 2010 such that when its divisors are listed in increasing order, they are alternatingly odd and even. For example, 6 belongs to , since the divisors of 6 are 1, 2, 3, and 6, which are odd, even, odd, and even in that order. Find the last three digits of the largest possible value of , where lies in and denotes the number of divisors of .