Difference between revisions of "1999 AIME Problems"

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== Problem 4 ==
 
== Problem 4 ==
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The two squares shown share the same center <math>\displaystyle O_{}</math> and have sides of length 1. The length of <math>\displaystyle \overline{AB}</math> is <math>\displaystyle 43/99</math> and the area of octagon <math>\displaystyle ABCDEFGH</math> is <math>\displaystyle m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers.  Find <math>\displaystyle m+n.</math>
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[[Image:AIME_1999_Problem_4.png]]
  
 
[[1999 AIME Problems/Problem 4|Solution]]
 
[[1999 AIME Problems/Problem 4|Solution]]

Revision as of 00:45, 22 January 2007

Problem 1

Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.

Solution

Problem 2

Consider the parallelogram with vertices $\displaystyle (10,45),$ $\displaystyle (10,114),$ $\displaystyle (28,153),$ and $\displaystyle (28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $\displaystyle m/n,$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers. Find $\displaystyle m+n.$

Solution

Problem 3

Find the sum of all positive integers $\displaystyle n$ for which $\displaystyle n^2-19n+99$ is a perfect square.

Solution

Problem 4

The two squares shown share the same center $\displaystyle O_{}$ and have sides of length 1. The length of $\displaystyle \overline{AB}$ is $\displaystyle 43/99$ and the area of octagon $\displaystyle ABCDEFGH$ is $\displaystyle m/n,$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers. Find $\displaystyle m+n.$

AIME 1999 Problem 4.png

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different integers $(2^{x})(3^{y})(5^{z})$, where $x, y$, and $z$ take on the values $0, 1, \ldots, 9$. At step i of a 1000-step process, the $i$-th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$-th switch. After step 1000 has been completed, how many switches will be in position $A$?

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

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