MathDB

1

Part of 2000 Hungary-Israel Binational

Problems(2)

$(A − A) \cap (B − B)$ is nonempty

Source: 11-th Hungary-Israel Binational Mathematical Competition 2000

4/22/2007
Let AA and BB be two subsets of S={1,2,...,2000}S = \{1, 2, . . . , 2000\} with ∣A∣⋅∣B∣≥3999|A| \cdot |B| \geq 3999. For a set XX , let X−XX-X denotes the set {s−t∣s,t∈X,s≠t}\{s-t | s, t \in X, s \not = t\}. Prove that (A−A)∩(B−B)(A-A) \cap (B-B) is nonempty.
functionpigeonhole principlecombinatorics proposedcombinatorics
partitions of $2000$ (in a sum of positive integers).

Source: 11-th Hungary-Israel Binational Mathematical Competition 2000

4/20/2007
Let SS be the set of all partitions of 20002000 (in a sum of positive integers). For every such partition pp, we define f(p)f (p) to be the sum of the number of summands in pp and the maximal summand in pp. Compute the minimum of f(p)f (p) when p∈S.p \in S .
number theory unsolvednumber theory