2024 AMC 10A Problems/Problem 20
Problem
Let 
be a subset of 
such that the following two conditions hold:
- If 
and 
are distinct elements of , then 
- If and are distinct odd elements of , then 
.
What is the maximum possible number of elements in ?
Solution 1
All lists are organized in ascending order:
By listing out the smallest possible elements of subset 
we can find that subset starts with 
It is easily noticed that the elements of the subset "loop around" every 3 elements, specifically adding 10 each time. This means that there will be 
or 
whole loops in the subset implying that there will be 
elements in S. However, we have undercounted, as we did not count the remainder that resulted from 
With a remainder of 
we can fit 
more elements into the subset namely 
and 
resulting in a total of 
or 
elements in subset .
$-weihou0
NOTE:
To prove that this is the best we can do, consider adding each element one by one, for the first element, say n. If n is greater than 2, we can choose n - 2 which is always better. Therefore, n = 1 or n = 2.
If n = 2 was optimal, then choose it, then the set of usable numbers in$ (Error compiling LaTeX. Unknown error_msg)S
S
Q
Q
Q
Q
S
Q
S$. By induction we see that our list would be {2,6,10,14,18,.....2022} which only gives 506 elements which is sub-optimal. Therefore, we can conclude that n = 1 is optimal, and we proceed as the solution above..
See also
| 2024 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 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 | ||
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