Difference between revisions of "1985 AIME Problems/Problem 3"
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== See also == | == See also == | ||
{{AIME box|year=1985|num-b=2|num-a=4}} | {{AIME box|year=1985|num-b=2|num-a=4}} | ||
+ | * [[AIME Problems and Solutions]] | ||
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[Mathematics competition resources]] | ||
[[Category:Intermediate Complex Numbers Problems]] | [[Category:Intermediate Complex Numbers Problems]] |
Revision as of 14:31, 6 May 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 |