Difference between revisions of "2016 AIME I Problems/Problem 10"
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Revision as of 16:56, 4 March 2016
A strictly increasing sequence of positive integers ,
,
,
has the property that for every positive integer
, the subsequence
,
,
is geometric and the subsequence
,
,
is arithmetic. Suppose that
. Find
.
Solution
We first create a similar sequence where and
. Continuing the sequence,
Here we can see a pattern; every second term (starting from the first) is a square, and every second term (starting from the third) is the end of a geometric sequence. Similarly, would also need to be the end of a geometric sequence (divisible by a square). We see that
is
, so the squares that would fit in
are
,
,
,
,
, and
. By simple inspection
is the only plausible square, since the other squares don't have enough numbers before them to go all the way back to
.
, so
.
~Iy31n~
See also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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.