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]]
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== Problem 8 ==
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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>.
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[[2005 AIME II Problems/Problem 8|Solution]]
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== Problem 9 ==
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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>.
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[[2005 AIME II Problems/Problem 9|Solution]]
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== Problem 10 ==
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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>.
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[[2005 AIME II Problems/Problem 10|Solution]]
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== Problem 11 ==
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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

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 $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$.

Solution

Problem 2

For each positive integer k, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is k. For example, $S_3$ is the squence $1,4,7,10 ...$. For how many values of k does $S_k$ contain the term 2005?

Solution

Problem 3

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

Solution

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.

Solution

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.

Solution

Problem 6

Let $P$ be the product of nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P \rfloor$

Solution

Problem 7

In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$ and $m\angle A=m\angle B=60\circ$. Given that $AB=p+\sqrt{q}$, where p and q are positive integers, find $p+q$.

Solution

Problem 8

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

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 $27$ cubes are randomly arranged to form a $3\times 3\times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^b r^c}$ where $p,q,$ and $r$ are distinct primes and $a,b,$ and $c$ are positive integers, find $a+b+c+p+q+r$.

Solution

Problem 10

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

Solution

Problem 11

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

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