Difference between revisions of "2010 AIME I Problems/Problem 14"
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== Problem == | == Problem == |
Revision as of 15:45, 9 August 2018
Problem
Problem
For each positive integer n, let . Find the largest value of n for which
.
Note: is the greatest integer less than or equal to
.
Solution
Solution 1
Observe that is strictly increasing in
. We realize that we need
terms to add up to around
, so we need some sequence of
s,
s, and then
s.
It follows that . Manually checking shows that
and
. Thus, our answer is
.
Solution 2
Because we want the value for which , the average value of the 100 terms of the sequence should be around
. For the value of
to be
,
. We want kn to be around the middle of that range, and for k to be in the middle of 0 and 100, let
, so
, and
.
, so we want to lower
. Testing
yields
, so our answer is still
.
Solution 3
For any where the sum is close to
, all the terms in the sum must be equal to
,
or
. Let
be the number of terms less than or equal to
and
be the number of terms equal to
(also counted in
). With this definition of
and
the total will be
, from which
. Now
is the smallest integer
for which
or
, thus
Similarly,
Therefore, Since we want the largest possible
, the answer is
.
See Also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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.