Difference between revisions of "1997 AIME Problems"
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== Problem 5 == | == Problem 5 == | ||
| + | The number <math>r</math> can be expressed as a four-place decimal <math>0.abcd,</math> where <math>a, b, c,</math> and <math>d</math> represent digits, any of which could be zero. It is desired to approximate <math>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to <math>r</math> is <math>\frac 27.</math> What is the number of possible values for <math>r</math>? | ||
[[1997 AIME Problems/Problem 5|Solution]] | [[1997 AIME Problems/Problem 5|Solution]] | ||
Revision as of 11:40, 11 October 2007
Contents
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?
Problem 2
The nine horizontal and nine vertical lines on an 
checkeboard form 
rectangles, of which 
are squares. The number 
can be written in the form 
where 
and 
are relatively prime positive integers. Find 
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?
Problem 4
Circles of radii 5, 5, 8, and 
are mutually externally tangent, where and are relatively prime positive integers. Find
Problem 5
The number can be expressed as a four-place decimal 
where 
and 
represent digits, any of which could be zero. It is desired to approximate by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to is 
What is the number of possible values for ?
