Difference between revisions of "2002 AMC 10P Problems/Problem 15"
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What is the smallest integer <math>n</math> for which any subset of <math>\{ 1, 2, 3, \; \dots \; , 20 \}</math> of size <math>n</math> must contain two numbers that differ by 8? | What is the smallest integer <math>n</math> for which any subset of <math>\{ 1, 2, 3, \; \dots \; , 20 \}</math> of size <math>n</math> must contain two numbers that differ by 8? | ||
| − | < | + | <math> \textbf{(A)} \; 2 \quad \textbf{(B)} \; 8 \quad \textbf{(C)} \; 12 \quad \textbf{(D)} \; 13 \quad \textbf{(E)} \; 15 </math> |
== Solution == | == Solution == | ||
Revision as of 11:22, 9 March 2021
\tableofcontents
Problem
What is the smallest integer 
for which any subset of 
of size must contain two numbers that differ by 8?
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
