MathDB

4

Part of 2017 China Northern MO

Problems(2)

China Northern Mathematical Olympiad 2017, Problem 4

Source: China Northern Mathematical Olympiad 2017

7/29/2017
Let QQ be a set of permutations of 1,2,...,1001,2,...,100 such that for all 1≤a,b≤1001\leq a,b \leq 100, aa can be found to the left of bb and adjacent to bb in at most one permutation in QQ. Find the largest possible number of elements in QQ.
graph theorycombinatorics
2017 CNMO Grade 11 P4

Source: 2017 China Northern MO, Grade 11, Problem 4

2/24/2020
Positive intenger n≥3n\geq3. a1,a2,⋯ ,ana_1,a_2,\cdots,a_n are nn positive intengers that are pairwise coprime, satisfying that there exists k1,k2,⋯ ,kn∈{−1,1},∑i=1nkiai=0k_1,k_2,\cdots,k_n\in\{-1,1\}, \sum_{i=1}^{n}k_ia_i=0. Are there positive intengers b1,b2,⋯ ,bnb_1,b_2,\cdots,b_n, for any k∈Z+k\in\mathbb{Z}_+, b1+ka1,b2+ka2,⋯ ,bn+kanb_1+ka_1,b_2+ka_2,\cdots,b_n+ka_n are pairwise coprime?
number theory