Difference between revisions of "1999 AIME Problems"
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== Problem 3 == | == Problem 3 == | ||
| + | Find the sum of all positive integers <math>\displaystyle n</math> for which <math>\displaystyle n^2-19n+99</math> is a perfect square. | ||
[[1999 AIME Problems/Problem 3|Solution]] | [[1999 AIME Problems/Problem 3|Solution]] | ||
Revision as of 00:43, 22 January 2007
Contents
Problem 1
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
Problem 2
Consider the parallelogram with vertices 
and 
A line through the origin cuts this figure into two congruent polygons. The slope of the line is 
where 
and 
are relatively prime positive integers. Find 
Problem 3
Find the sum of all positive integers 
for which 
is a perfect square.
Problem 4
Problem 5
Problem 6
Problem 7
There is a set of 1000 switches, each of which has four positions, called 
, and 
. When the position of any switch changes, it is only from 
to 
, from to 
, from to , or from to . Initially each switch is in position . The switches are labeled with the 1000 different integers 
, where 
, and 
take on the values 
. At step i of a 1000-step process, the 
-th switch is advanced one step, and so are all the other switches whose labels divide the label on the -th switch. After step 1000 has been completed, how many switches will be in position ?
