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)