MathDB

3

Part of 2015 China National Olympiad

Problems(2)

x+y in B iff x,y in A

Source: China Mathematical Olympiad 2015 Q3

12/20/2014
Let n≥5n \geq 5 be a positive integer and let AA and BB be sets of integers satisfying the following conditions:
i) ∣A∣=n|A| = n, ∣B∣=m|B| = m and AA is a subset of BB ii) For any distinct x,y∈Bx,y \in B, x+y∈Bx+y \in B iff x,y∈Ax,y \in A
Determine the minimum value of mm.
absolute valuecombinatorics proposedcombinatorics
Erdos Discrepancy Problem Variation

Source: China Mathematical Olympiad 2015 Q6

12/21/2014
Let a1,a2,...a_1,a_2,... be a sequence of non-negative integers such that for any m,nm,n ∑i=12main≤m. \sum_{i=1}^{2m} a_{in} \leq m. Show that there exist k,dk,d such that ∑i=12kaid=k−2014. \sum_{i=1}^{2k} a_{id} = k-2014.
combinatoricsalgebra