MathDB

7

Part of 2016 South East Mathematical Olympiad

Problems(2)

Easy geometry

Source: China South East Mathematical Olympiad 2016 Grade 10 Prob. 7

7/31/2016
II is incenter of △ABC\triangle{ABC}. The incircle touches BC,CA,ABBC,CA,AB at D,E,FD,E,F, respectively . Let M,N,K=BI,CI,DI∩EFM,N,K=BI,CI,DI \cap EF respectively and BN∩CM=P,AK∩BC=GBN\cap CM=P,AK\cap BC=G. Point QQ is intersection of the perpendicular line to PGPG through II and the perpendicular line to PBPB through PP. Prove that BIBI bisect segment PQPQ.
geometryincenter
find the value of $|P|$

Source: 2016 China South East Mathematical Olympiad Grade 11 Problem 7

8/6/2016
Let A={a3+b3+c3−3abc∣a,b,c∈N}A=\{a^3+b^3+c^3-3abc|a,b,c\in\mathbb{N}\}, B={(a+b−c)(b+c−a)(c+a−b)∣a,b,c∈N}B=\{(a+b-c)(b+c-a)(c+a-b)|a,b,c\in\mathbb{N}\}, P={n∣n∈A∩B,1≤n≤2016}P=\{n|n\in A\cap B,1\le n\le 2016\}, find the value of ∣P∣|P|.
number theory