Difference between revisions of "2021 AIME I Problems/Problem 7"
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Revision as of 19:58, 11 March 2021
Problem
Find the number of pairs of positive integers with
such that there exists a real number
satisfying
Solution
Since ,
means that each of
and
must be exactly
. Then
and
must be cycles away, or the difference between them must be multiple of
. If
is
, then
can be
. Like this, the table below can be listed:
Range of ![]() |
Number of Possible ![]() | |
---|---|---|
Case 1 | ![]() |
![]() |
Case 2 | ![]() |
![]() |
Case 3 | ![]() |
![]() |
Case 4 | ![]() |
![]() |
Case 5 | ![]() |
![]() |
Case 6 | ![]() |
![]() |
Case 7 | ![]() |
![]() |
Case 8 | ![]() |
![]() |
In total, there are possible solutions.
However the answer is , where is the last possible solution?
~Interstigation
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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.