2010 AIME I Problems/Problem 12
Problem
Let be an integer and let
. Find the smallest value of
such that for every partition of
into two subsets, at least one of the subsets contains integers
,
, and
(not necessarily distinct) such that
.
Note: a partition of is a pair of sets
,
such that
,
.
Solution
We claim that is the minimal value of
. Let the two partitioned sets be
and
; we will try to partition
and
such that the
condition is not satisfied. Without loss of generality, we place
in
. Then
must be placed in
, so
must be placed in
, and
must be placed in
. Then
cannot be placed in any set, so we know
is less than or equal to
.
For , we can partition
into
and
, and in neither set are there values where
(since
and
). Thus
.
See Also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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