Difference between revisions of "2022 AMC 10B Problems/Problem 14"
(Created page with "==Problem== Suppose that <math>S</math> is a subset of <math>\left\{ 1, 2, 3, \cdots , 25 \right\}</math> such that the sum of any two (not necessarily distinct) elements of...") |
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Thus, this set is feasible. | Thus, this set is feasible. | ||
Therefore, the most number of elements in <math>S</math> is | Therefore, the most number of elements in <math>S</math> is | ||
− | \boxed{\textbf{(B) 13}}. | + | <math>\boxed{\textbf{(B) 13}}</math>. |
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) |
Revision as of 17:38, 17 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)