MathDB

3

Part of 2018 China Second Round Olympiad

Problems(2)

Expressing integers as difference of two elements

Source: China second round 2018 (A) Q3

5/5/2019
Let n,k,mn,k,m be positive integers, where k≥2k\ge 2 and n≤m<2k−1knn\le m < \frac{2k-1}{k}n. Let AA be a subset of {1,2,…,m}\{1,2,\ldots ,m\} with nn elements. Prove that every integer in the range (0,nk−1)\left(0,\frac{n}{k-1}\right) can be expressed as a−ba-b, where a,b∈Aa,b\in A.
combinatorics
A=\{1,2,\ldots,n\} maxX&gt;minY

Source: China second round 2018 (B) Q3

6/22/2019
Let set A={1,2,…,n},A=\{1,2,\ldots,n\} , and X,YX,Y be two subsets (not necessarily distinct) of A.A. Define that maxX\textup{max} X and minY\textup{min} Y represent the greatest element of XX and the least element of Y,Y, respectively. Determine the number of two-tuples (X,Y)(X,Y) which satisfies maxX>minY.\textup{max} X>\textup{min} Y.
combinatorics