MathDB

5

Part of 2015 China Team Selection Test

Problems(2)

Adding weights to numbers such that they sum to 0

Source: 2015 China Tst 1 Day 2 Q2

3/14/2015
FIx positive integer nn. Prove: For any positive integers a,b,ca,b,c not exceeding 3n2+4n3n^2+4n, there exist integers x,y,zx,y,z with absolute value not exceeding 2n2n and not all 00, such that ax+by+cz=0ax+by+cz=0
number theoryDiophantine equation
subsets of subset has same sum

Source: 2015 China TST 2 Day 2 Q2

3/19/2015
Set SS to be a subset of size 6868 of {1,2,...,2015}\{1,2,...,2015\}. Prove that there exist 33 pairwise disjoint, non-empty subsets A,B,CA,B,C such that ∣A∣=∣B∣=∣C∣|A|=|B|=|C| and ∑a∈Aa=∑b∈Bb=∑c∈Cc\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c
combinatoricscombinatorics proposed