IMO 2017 Shortlist N8
Let p be an odd prime number and Zą0 be the set of positive integers. Suppose that a fun tion f : Zą0 ˆ Zą0 Ñ t0,1u sati...
Category: Number Theory
Problem
Let p be an odd prime number and Zą0 be the set of positive integers. Suppose that
a fun tion f : Zą0 ˆ Zą0 Ñ t0,1u satis es the following properties:
• fp1,1q “ 0;
• fpa,bqfpb,aq “ 1 for any pair of relatively prime positive integers pa,bq not both equal to 1; • fpa b,bq “ fpa,bq for any pair of relatively prime positive integers pa,bq.
Prove that
p´1 ÿ
n“1
fpn2
,pq ě
a
2p ´ 2.
(Italy) 12 IMO 2017, Rio de Janeiro