Difference between revisions of "2002 AMC 10P Problems/Problem 15"
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== Solution == | == Solution == | ||
| − | There are twelve pairs <math>\{ 1, 9 \}</math>, <math>\{ 2, 10 \}</math>, <math>\{ 3, 11 \}</math>, <math>\{ 4, 11 \}</math>, <math>\; \dots \; \{ 12, 20 \}</math> in <math>\{ 1, 2, 3, \; \dots \; , 20 \}</math> that differ by 8. If we take <math>n = 12</math>, it could be that we selected one element from each pair for the subset: the condition may not be fulfilled. | + | There are twelve pairs <math>\{ 1, 9 \}</math>, <math>\{ 2, 10 \}</math>, <math>\{ 3, 11 \}</math>, <math>\{ 4, 11 \}</math>, <math>\; \dots \; \{ 12, 20 \}</math> in <math>\{ 1, 2, 3, \; \dots \; , 20 \}</math> that differ by 8. If we take <math>n = 12</math>, it could be that we selected one element from each pair for the subset: the condition may not be fulfilled. In order to select at least one pair, it is necessary to select <math>\textbf{(D)} \; 13</math> elements (Pigeonhole principle). |
<i>Solution submitted by green_lotus</i> | <i>Solution submitted by green_lotus</i> | ||
Revision as of 10:24, 9 March 2021
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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?
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. In order to select at least one pair, it is necessary to select 
elements (Pigeonhole principle).
Solution submitted by green_lotus
