# Mock AIME 3 Pre 2005 Problems

## Contents

## Problem 1

Three circles are mutually externally tangent. Two of the circles have radii and . If the area of the triangle formed by connecting their centers is , then the area of the third circle is for some integer . Determine .

## Problem 2

Let denote the number of digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when is divided by . (Repeated digits are allowed.)

## Problem 3

A function is defined for all real numbers . For all non-zero values , we have

Let denote the sum of all of the values of for which . Compute the integer nearest to .

## Problem 4

and are complex numbers such that

Compute .

## Problem 5

In Zuminglish, all words consist only of the letters and . As in English, is said to be a vowel and and are consonants. A string of and is a word in Zuminglish if and only if between any two there appear at least two consonants. Let denote the number of -letter Zuminglish words. Determine the remainder obtained when is divided by .

## Problem 6

Let denote the value of the sum

can be expressed as , where and are positive integers and is not divisible by the square of any prime. Determine .

## Problem 7

is a cyclic quadrilateral that has an inscribed circle. The diagonals of intersect at . If and then the area of the inscribed circle of can be expressed as , where and are relatively prime positive integers. Determine .

## Problem 8

Let denote the number of -tuples of real numbers such that and

Determine the remainder obtained when is divided by .

## Problem 9

is an isosceles triangle with base . is a point on and is the point on the extension of past such that is right. If and , then can be expressed as , where and are relatively prime positive integers. Determine .

## Problem 10

is a sequence of positive integers such that

for all integers . Compute the remainder obtained when is divided by if .

## Problem 11

is an acute triangle with perimeter . is a point on . The circumcircles of triangles and intersect and at and respectively such that and . If , then the value of can be expressed as , where and are relatively prime positive integers. Compute .

## Problem 12

Determine the number of integers such that and is divisible by .

## Problem 13

Let denote the value of the sum

Determine the remainder obtained when is divided by .

## Problem 14

Circles and are centered on opposite sides of line , and are both tangent to at . passes through , intersecting again at . Let and be the intersections of and , and and respectively. and are extended past and intersect and at and respectively. If and , then the area of triangle can be expressed as , where and are positive integers such that and are coprime and is not divisible by the square of any prime. Determine .

## Problem 15

Let denote the value of the sum

The value of can be expressed as , where and are relatively prime positive integers. Compute .