MathDB

2

Part of 2018 Danube Mathematical Competition

Problems(2)

danube junior angle chasing 2018 P2

Source: Danube Junior 2018 P2

12/11/2018
Let ABCABC be a triangle such that in its interior there exists a point DD with ∠DAC=∠DCA=30o\angle DAC = \angle DCA = 30^o and ∠DBA=60o \angle DBA = 60^o. Denote EE the midpoint of the segment BCBC, and take FF on the segment ACAC so that AF=2FCAF = 2FC. Prove that DE⊥EFDE \perp EF.
geometryAngle Chasingperpendicularperpendicularity
infinite pairs of (m, n) such m \ n^2+1 and n \m^2+1 ,

Source: Danube 2018 p2

7/22/2019
Prove that there are in finitely many pairs of positive integers (m,n)(m, n) such that simultaneously mm divides n2+1n^2 + 1 and nn divides m2+1m^2 + 1.
number theoryDivisorsinfinitely many