# Difference between revisions of "2005 AIME II Problems"

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[[2005 AIME II Problems/Problem 7|Solution]] | [[2005 AIME II Problems/Problem 7|Solution]] | ||

+ | == Problem 8 == | ||

+ | The equation '''<math>2^{333x-2}+2^{111x+2}=2^{222x+1}+1</math>''' has three real roots. Given that their sum is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. | ||

+ | |||

+ | [[2005 AIME II Problems/Problem 8|Solution]] | ||

+ | == Problem 9 == | ||

+ | Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The <math>27</math> cubes are randomly arranged to form a <math>3\times 3\times 3</math> cube. Given the probability of the entire surface area of the larger cube is orange is <math>\frac{p^a}{q^b r^c}</math> where <math>p,q,</math> and <math>r</math> are distinct primes and <math>a,b,</math> and <math>c</math> are positive integers, find <math>a+b+c+p+q+r</math>. | ||

+ | |||

+ | [[2005 AIME II Problems/Problem 9|Solution]] | ||

+ | == Problem 10 == | ||

+ | Triangle <math>ABC</math> lies in the Cartesian Plane and has an area of 70. The coordinates of <math>B</math> and <math>C</math> are <math>(12,19)</math> and <math>(23,20),</math> respectively, and the coordinates of <math>A</math> are <math>(p,q).</math> The line containing the median to side <math>BC</math> has slope <math>-5</math>. Find the largest possible value of <math>p+q</math>. | ||

+ | |||

+ | [[2005 AIME II Problems/Problem 10|Solution]] | ||

+ | == Problem 11 == | ||

+ | A semicircle with diameter <math>d</math> is contained in a square whose sides have length 8. Given the maximum value of <math>d</math> is <math>m-\sqrt{n}</math>, find <math>m+n</math>. |

## Revision as of 11:56, 5 July 2006

## Contents

## Problem 1

Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle with radius . Let be the area of the region inside circle and outside of the six circles in the ring. Find .

## Problem 2

For each positive integer *k*, let denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is *k*. For example, is the squence . For how many values of *k* does contain the term 2005?

## Problem 3

How many positive integers have exactly three proper divisors, each of which is less than 50?

## Problem 4

The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.

## Problem 5

Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distunguishable arrangements of the 8 coins.

## Problem 6

Let be the product of nonreal roots of . Find

## Problem 7

In quadrilateral , , , and . Given that , where *p* and *q* are positive integers, find .

## Problem 8

The equation has three real roots. Given that their sum is where and are relatively prime positive integers, find .

## Problem 9

Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The cubes are randomly arranged to form a cube. Given the probability of the entire surface area of the larger cube is orange is where and are distinct primes and and are positive integers, find .

## Problem 10

Triangle lies in the Cartesian Plane and has an area of 70. The coordinates of and are and respectively, and the coordinates of are The line containing the median to side has slope . Find the largest possible value of .

## Problem 11

A semicircle with diameter is contained in a square whose sides have length 8. Given the maximum value of is , find .