Difference between revisions of "2010 AIME I Problems/Problem 12"
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== Problem == | == Problem == | ||
| − | Let <math> | + | Let <math>m \ge 3</math> be an [[integer]] and let <math>S = \{3,4,5,\ldots,m\}</math>. Find the smallest value of <math>m</math> such that for every [[partition]] of <math>S</math> into two subsets, at least one of the subsets contains integers <math>a</math>, <math>b</math>, and <math>c</math> (not necessarily distinct) such that <math>ab = c</math>. |
'''Note''': a partition of <math>S</math> is a pair of sets <math>A</math>, <math>B</math> such that <math>A \cap B = \emptyset</math>, <math>A \cup B = S</math>. | '''Note''': a partition of <math>S</math> is a pair of sets <math>A</math>, <math>B</math> such that <math>A \cap B = \emptyset</math>, <math>A \cup B = S</math>. | ||
Revision as of 07:50, 27 October 2015
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 | |
| 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. 
