# Mock AIME 1 2006-2007/Problems

## Contents

## Problem 1

has positive integer side lengths of ,, and . The angle bisector of hits at . If , and the maximum value of where and are relatively prime positive intgers, find . (Note that denotes the area of ).

## Problem 2

Let be the sum of the digits of a positive integer . is the set of positive integers such that for all elements in , we have that and . If is the number of elements in , compute .

## Problem 3

Let have , , and . If where is an integer, find the remainder when is divided by .

## Problem 4

has all of its vertices on the parabola . The slopes of and are and , respectively. If the x-coordinate of the triangle's centroid is , find the area of .

## Problem 5

Let be a prime and satisfy for all integers . is the greatest integer less than or equal to . If for fixed , there exists an integer such that:

then . If there is no such , then . If , find the sum: .

## Problem 6

Let and be two parabolas in the cartesian plane. Let be the common tangent of and that has a rational slope. If is written in the form for positive integers where . Find .

## Problem 7

Let have and . Point is such that and . Construct point on segment such that . and are extended to meet at . If where and are positive integers, find (note: denotes the area of ).

## Problem 8

Let be a convex pentagon with , , , and . If where and are relatively prime positive integers, find .

## Problem 9

Let be a geometric sequence for with and . Let denote the infinite sum: . If the sum of all distinct values of is where and are relatively prime positive integers, then compute the sum of the positive prime factors of .

## Problem 10

In , , , and have lengths , , and , respectively. Let the incircle, circle , of touch , , and at , , and , respectively. Construct three circles, , , and , externally tangent to the other two and circles , , and are internally tangent to the circle at , , and , respectively. Let circles , , , and have radii , , , and , respectively. If where and are positive integers, find .

## Problem 11

Let be the set of strings with only 0's or 1's with length such that any 3 adjacent place numbers sum to at least 1. For example, works, but does not. Find the number of elements in .

## Problem 12

Let be a positive integer with a first digit four such that after removing the first digit, you get another positive integer, , that satisfies . Find the number of possible values of between and .

## Problem 13

Let , , and be geometric sequences with different common ratios and let for all integers . If , , , , , and , find .

## Problem 14

Three points , , and are fixed such that lies on segment , closer to point . Let and where and are positive integers. Construct circle with a variable radius that is tangent to at . Let be the point such that circle is the incircle of . Construct as the midpoint of . Let denote the maximum value for fixed and where . If is an integer, find the sum of all possible values of .

## Problem 15

Let be the set of integers . An element (in) is chosen at random. Let denote the sum of the digits of . The probability that is divisible by 11 is where and are relatively prime positive integers. Compute the last 3 digits of