Difference between revisions of "2018 AIME I Problems/Problem 6"
Mathislife16 (talk | contribs) (→See also) |
Elephant353 (talk | contribs) m (→See also: Corrected misdirected redirection) |
||
Line 6: | Line 6: | ||
== See also == | == See also == | ||
− | {{AIME box|year=2018|n=I|num-b= | + | {{AIME box|year=2018|n=I|num-b=5|num-a=7}} |
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 00:56, 8 March 2018
Problem
Let be the number of complex numbers
with the properties that
and
is a real number. Find the remainder when
is divided by
.
Solution
Let . This simplifies the problem constraint to
. This is true iff
. Let
be the angle
makes with the positive x-axis. Note that there is exactly one
for each angle
. This must be true for
values of
(it may help to picture the reference angle making one orbit from and to the positive x-axis; note every time
). For each of these solutions for
, there are necessarily
solutions for
. Thus, there are
solutions for
, yielding an answer of
.
See also
2018 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.