2002 AMC 10P Problems/Problem 15
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Problem
What is the smallest integer 
for which any subset of 
of size must contain two numbers that differ by 8?
![\[\textbf{(A)} \; 2 \quad \textbf{(B)} \; 8 \quad \textbf{(C)} \; 12 \quad \textbf{(D)} \; 13 \quad \textbf{(E)} \; 15\]](http://latex.artofproblemsolving.com/7/f/1/7f1c2e1500e4b158d5d9712ed6040defdc462d91.png)
Solution
There are twelve pairs 
, 
, 
, 
, 
in that differ by 8. If we take 
, it could be that we selected one element from each pair for the subset: the condition may not be fulfilled. the In order to select at least one pair, it is necessary to select 
elements (Pigeonhole principle).
Solution submitted by green_lotus
