2025 AMC 10A Problems/Problem 21: Difference between revisions
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( | A set of numbers is called <imath>sum</imath>-<imath>free</imath> if whenever <imath>x</imath> and <imath>y</imath> are (not necessarily distinct) elements of the set, <imath>x+y</imath> is not an element of the set. For example, <imath>\{1,4,6\}</imath> and the empty set are sum-free, but <imath>\{2,4,5\}</imath> is not. What is the greatest possible number of elements in a sum-free subset of <imath>\{1,2,3,...,20\}</imath>? | ||
<imath>\textbf{(A) } 8 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12</imath> | |||
==Solution 1== | ==Solution 1== | ||
Revision as of 16:09, 6 November 2025
A set of numbers is called 
-
if whenever 
and 
are (not necessarily distinct) elements of the set, 
is not an element of the set. For example, 
and the empty set are sum-free, but 
is not. What is the greatest possible number of elements in a sum-free subset of 
?
Solution 1
Let our subset be 
If we add any one element from the set 
we will have to remove at least one element from our current subset. Hence, the size of our set cannot exceed 
~Tacos_are_yummy_1
Solution 2
Let our subset be 
Since odd numbers + odd numbers will always sum to an even number, this subset holds true. Leo, the addition of any even number will result in a violation of the rule, so the maximum number of elements is
~KMS888
See also
| 2025 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 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 | ||
| 2025 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 17 |
Followed by Problem 19 |
| 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 12 Problems and Solutions | |
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