Difference between revisions of "1997 AIME Problems"

(Problem 3)
(Problem 4)
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== Problem 4 ==
 
== Problem 4 ==
 +
Circles of radii 5, 5, 8, and <math>m/n</math> are mutually externally tangent, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
  
 
[[1997 AIME Problems/Problem 4|Solution]]
 
[[1997 AIME Problems/Problem 4|Solution]]

Revision as of 12:33, 11 October 2007

Problem 1

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

Solution

Problem 2

The nine horizontal and nine vertical lines on an $8\times8$ checkeboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

Problem 3

Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit nmber. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

Solution

Problem 4

Circles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

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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