Difference between revisions of "2022 AMC 10B Problems/Problem 14"
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
| + | == See Also == | ||
| + | {{AMC10 box|year=2022|ab=B|num-b=12|num-a=14}} | ||
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Revision as of 08:01, 18 November 2022
Problem
Suppose that 
is a subset of 
such that the sum of any two (not necessarily distinct)
elements of is never an element of . What is the maximum number of elements may contain?
Solution (Pigeonhole Principle)
Denote by 
the largest number in .
We categorize numbers 
(except 
if is even) into 
groups, such that the 
th group contains two numbers and 
.
Recall that 
and the sum of two numbers in cannot be equal to , and the sum of numbers in each group above is equal to . Thus, each of the above groups can have at most one number in .
Therefore,
Next, we construct an instance of with 
.
Let 
.
Thus, this set is feasible.
Therefore, the most number of elements in is
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
| 2022 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
