Difference between revisions of "2016 AIME I Problems/Problem 10"
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==Solution 3== | ==Solution 3== | ||
− | This is not a hard bash. You can try the ratios <math>\frac 2 3</math>, <math>\frac 3 4</math>, and <math>\frac 11 12.</math> Working backwards from <math>11 12,</math> we get <math>504.</math> | + | This is not a hard bash. You can try the ratios <math>\frac 2 3</math>, <math>\frac 3 4</math>, and <math>\frac {11} {12}.</math> Working backwards from <math>11 12,</math> we get <math>504.</math> |
== See also == | == See also == | ||
{{AIME box|year=2016|n=I|num-b=9|num-a=11}} | {{AIME box|year=2016|n=I|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 21:43, 4 February 2019
Contents
[hide]Problem
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 1
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 in the sequence don't have enough elements before them to go all the way back to
while still staying as positive integers.
, so
.
~IYN~
Solution 2
Setting and
, the sequence becomes:
and so forth, with
. Then,
. Keep in mind,
need not be an integer, only
etc. does.
, so only the squares
and
are plausible for
. But when that is anything other than
,
is not an integer. Therefore,
.
Thanks for reading, Rowechen Zhong.
Solution 3
This is not a hard bash. You can try the ratios ,
, and
Working backwards from
we get
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 |
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