Difference between revisions of "1985 AIME Problems/Problem 3"
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[[Category:Intermediate Complex Numbers Problems]] | [[Category:Intermediate Complex Numbers Problems]] | ||
Revision as of 15:56, 6 March 2007
Problem
Find 
if 
, 
, and are positive integers which satisfy 
, where 
.
Solution
Expanding out both sides of the given equation we have 
. Two complex numbers are equal if and only if their real parts and imaginary parts are equal, so 
and 
. Since 
are integers, this means is a divisor of 107, which is a prime number. Thus either 
or 
. If , 
so 
, but 
is not divisible by 3, a contradiction. Thus we must have , 
so 
and 
(since we know is positive). Thus 
.
See also
| 1985 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
