MathDB

4

Part of 2015 South East Mathematical Olympiad

Problems(2)

Lucky groups

Source: China Southeast Math Olympiad 2015 Day 1 P4

6/6/2017
For any positive integer nn, we have the set Pn={nk∣k=0,1,2,…}P_n = \{ n^k \mid k=0,1,2, \ldots \}. For positive integers a,b,ca,b,c, we define the group of (a,b,c)(a,b,c) as lucky if there is a positive integer mm such that a−1a-1, ab−12ab-12, abc−2015abc-2015 (the three numbers need not be different from each other) belong to the set PmP_m. Find the number of lucky groups.
number theory
2015 China South East MO Grade 11 P4

Source: 2015 China South East MO Grade 11 P4

1/16/2018
Given 88 pairwise distinct positive integers a1,a2,…,a8a_1,a_2,…,a_8 such that the greatest common divisor of any three of them is equal to 11. Show that there exists positive integer n≥8n\geq 8 and nn pairwise distinct positive integers m1,m2,…,mnm_1,m_2,…,m_n with the greatest common divisor of all nn numbers equal to 11 such that for any positive integers 1≤p<q<r≤n1\leq p<q<r\leq n, there exists positive integers 1≤i<j≤81\leq i<j\leq 8 that aiaj∣mp+mq+mra_ia_j\mid m_p+m_q+m_r.
number theory