# Mock AIME 4 2005-2006/Problems

http://www.mathlinks.ro/Forum/latexrender/pictures/2ea0ce047615395691113a82d6c190b3.gif

## Contents

## Problem 1

1. A 5-digit number is leet if and only if the sum of the first 2 digits, the sum of the last 2 digits and the middle digit are equal. How many 5-digit leet numbers exist? Solution

## Problem 2

2. Qin Shi Huang wants to count the number of warriors he has to invade China. He puts his warriors into lines with the most people such that they have even length. The people left over are the remainder. He makes 2 lines, with a remainder of 1, 3 lines with a remainder of 2, 4 lines with a remainder of 3, 5 lines with a remainder of 4, and 6 lines with a remainder of 5. Find the minimum number of warriors that he has. Solution

## Problem 3

3. is a regular tetrahedron. Tetrahedron is formed by connecting the centers of the faces of . Generally, a new tetrahedron is formed by connecting the centers of the faces of . is the volume of tetrahedron . where and are coprime positive integers, find the remainder when is divided by . Solution

## Problem 4

4. Let . Let be the product of the roots. How many digits are does have where denotes the greatest integer less than or equal to ? Solution

## Problem 5

5. A parabola is rotated degrees clockwise about the origin to . This image is translated upward to . Point , , and is in Quadrant I, on . If the area of is at a maximum, it is where , and are integers and is square free, find . Solution

## Problem 6

6. Define a sequence and for all positive integers . Find the remainder when is divided by . Solution

## Problem 7

7. is a function that satisfies for all defined . Suppose that the sum of the zeros of where and are coprime positive integers, find . Solution

## Problem 8

8. is a solution to . Suppose that find where is the greatest integer less than or equal to . Solution

## Problem 9

9. Zeus, Athena, and Posideon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and stay for 3 hours. All three hours does not need to fall within 12 pm to 12 am. If any of the 2 gods see each other during 12 pm to 12 am, it will be a good day. The probability of it being a good day is where and are coprime positive integers, find . Solution

## Problem 10

10. Define . Find the remainder when is divided by . Solution

## Problem 11

11. is isosceles with . A point lies inside the triangle such that , , and . Let be the area of . Find the remainder when is divided by . Solution

## Problem 12

12. There exists a line with points ,, with in between and . Point , not on the line is such that , , with . Construct on ray such that and . Point is on ray such that . If where and are integers, then find . Solution

## Problem 13

13. is isosceles with base . Construct on segment such that . Construct on such that . Contiue this pattern: construct with on segment and with on segment . The points do not coincide and . Suppose is the last point you can construct on the perimeter of the triangle. Find the remainder when is divided by . Solution

## Problem 14

14. is the probability that if you flip a fair coin, heads will occur before tails. If where and are relatively prime positive integers, find the remainder when is divided by . Solution

## Problem 15

15. A regular 61-gon with verticies , , ,... is inscribed in a circle with a radius of . Suppose . If where and are coprime positive integers, find the remainder when is divided by . Solution