MathDB

3

Part of 2015 China Second Round Olympiad

Problems(2)

Angle is twice of another

Source: China Second Round 2015 (A) Q3

5/5/2016
PP is a point on arc \overarc{BC} of the circumcircle of △ABC\triangle ABC not containing AA, KK lies on segment APAP such that BKBK bisects ∠ABC\angle ABC. The circumcircle of △KPC\triangle KPC meets AC,BDAC,BD at D,ED,E respectively. PEPE meets ABAB at FF. Prove that ∠ABC=2∠FCB\angle ABC=2\angle FCB.
geometrycircumcircle
Infinitely many triplets

Source: China Second Round 2015 (B) Q3

5/5/2016
Prove that there exist infinitely many positive integer triples (a,b,c)(a,b,c>2015)(a,b,c)(a,b,c>2015) such that a∣bc−1,b∣ac+1,c∣ab+1. a|bc-1, b|ac+1, c|ab+1.
number theory