2022 AMC 10B Problems/Problem 14
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.