Pigeonhole Sets

Probability Level 5

Find the smallest natural number n n such that for all sets A : = { a 1 , a 2 , . . . , a n } A:=\{a_1,a_2,...,a_n\} there exists a set B : = { b 1 , b 2 , . . . , b 2014 } B:=\{b_1,b_2,...,b_{2014}\} with B ⊆ A B\subseteq A and ∑ i = 1 2014 b i ≡ 0 ( m o d 2014 ) \sum_{i=1}^{2014}b_i\equiv 0\pmod{2014} .


The answer is 4027.

Emmanuel Lasker by Emmanuel Lasker , 25, Italy

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2 solutions

Jon Haussmann
Aug 12, 2014

This is just a specific case of Erdos-Ginzburg-Ziv .

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Sauparna Paul
Jan 12, 2015

Here we can represent any integers as 2014K+r, where k,r are integers. 0<=r<=2013. So,we can take 2013 numbers of form 2014K+r and 2013 numbers of form 2014K-r(excluding r=0). So there are total 2013+2013+1 =4027 (1 for r=0).

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