Difference between revisions of "2009 AIME I Problems/Problem 6"
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~IceMatrix | ~IceMatrix | ||
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+ | ==Video Solution== | ||
+ | https://youtu.be/kALrIDMR0dg | ||
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+ | ~Shreyas S | ||
== See also == | == See also == | ||
{{AIME box|year=2009|n=I|num-b=5|num-a=7}} | {{AIME box|year=2009|n=I|num-b=5|num-a=7}} | ||
+ | [[Category:Intermediate Number Theory Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 18:23, 18 June 2020
Problem
How many positive integers less than
are there such that the equation
has a solution for
? (The notation
denotes the greatest integer that is less than or equal to
.)
Solution
First, must be less than
, since otherwise
would be at least
which is greater than
.
Because must be an integer, we can do some simple case work:
For ,
as long as
. This gives us
value of
.
For ,
can be anything between
to
excluding
Therefore, . However, we got
in case 1 so it got counted twice.
For ,
can be anything between
to
excluding
This gives us
's
For ,
can be anything between
to
excluding
This gives us
's
For ,
can be anything between
to
excluding
This gives us
's
Since must be less than
, we can stop here and the answer is
possible values for
.
Alternatively, one could find that the values which work are to get the same answer.
Video Solution
Mostly the above solution explained on video: https://www.youtube.com/watch?v=2Xzjh6ae0MU&t=11s
~IceMatrix
Video Solution
~Shreyas S
See also
2009 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.