MathDB

3

Part of 2014 Danube Mathematical Competition

Problems(2)

angle chasing , danube junior geometry 2014

Source: Danube 2014 Junior P3

9/7/2018
Let ABCABC be a triangle with ∠A<90o,AB≠AC\angle A<90^o, AB \ne AC. Denote HH the orthocenter of triangle ABCABC, NN the midpoint of segment [AH][AH], MM the midpoint of segment [BC][BC] and DD the intersection point of the angle bisector of ∠BAC\angle BAC with the segment [MN][MN]. Prove that <ADH=90o<ADH=90^o
geometryright angleorthocentermidpoints
exist n pairwise coprime composite integers form arithmetic progression

Source: Danube 2014 p3

7/22/2019
Given any integer n≥2n \ge 2, show that there exists a set of nn pairwise coprime composite integers in arithmetic progression.
coprimeArithmetic Progressionnumber theoryarithmetic sequence