# 2013 Mock AIME I Problems

## Contents

## Problem 1

Two circles and , each of unit radius, have centers and such that . Let be the midpoint of and let be a circle externally tangent to both and . and have a common tangent that passes through . If this tangent is also a common tangent to and , find the radius of circle .

## Problem 2

Find the number of ordered positive integer triplets such that evenly divides , evenly divides , and .

## Problem 3

Let be the greatest integer less than or equal to , and let . If , compute .

## Problem 4

Compute the number of ways to fill in the following magic square such that:

1. the product of all rows, columns, and diagonals are equal (the sum condition is waived),

2. all entries are *nonnegative* integers less than or equal to ten, and

3. entries CAN repeat in a column, row, or diagonal.

## Problem 5

In quadrilateral , . Also, , and . The perimeter of can be expressed in the form where and are relatively prime, and is not divisible by the square of any prime number. Find .

## Problem 6

Find the number of integer values can have such that the equation has a solution.

## Problem 7

Let be the set of all th primitive roots of unity with imaginary part greater than . Let be the set of all th primitive roots of unity with imaginary part greater than . (A primitive th root of unity is a th root of unity that is not a th root of unity for any .)Let . The absolute value of the real part of can be expressed in the form where and are relatively prime numbers. Find .

## Problem 8

Let and be two perpendicular vectors in the plane. If there are vectors for in the same plane having projections of and along and respectively, then find (Note: and are unit vectors such that and , and the projection of a vector onto is the length of the vector that is formed by the origin and the foot of the perpendicular of onto .)

## Problem 9

In a magic circuit, there are six lights in a series, and if one of the lights short circuit, then all lights after it will short circuit as well, without affecting the lights before it. Once a turn, a random light that isn’t already short circuited is short circuited. If is the expected number of turns it takes to short circuit all of the lights, find .

## Problem 10

Let denote the th triangular number, i.e. . Let and be relatively prime positive integers so that Find .

## Problem 11

Let and be the roots of the equation , and let and be the two possible values of Find .

## Problem 12

In acute triangle , the orthocenter lies on the line connecting the midpoint of segment to the midpoint of segment . If , and the altitude from has length , find .

## Problem 13

In acute , is the orthocenter, is the centroid, and is the midpoint of . It is obvious that , but does not always hold. If , , then the value of which produces the smallest value of such that can be expressed in the form , for squarefree. Compute .

## Problem 14

Let If are its roots, then compute the remainder when is divided by 997.

## Problem 15

Let be the set of integers such that for all integers . Compute the remainder when the sum of the elements in is divided by .