MathDB

1

Part of 2017 International Zhautykov Olympiad

Problems(2)

Concyclic points

Source: IZHO 2017 day 1 p1

1/14/2017
Let ABCABC be a non-isosceles triangle with circumcircle ω\omega and let H,MH, M be orthocenter and midpoint of ABAB respectively. Let P,QP,Q be points on the arc ABAB of ω\omega not containing CC such that ∠ACP=∠BCQ<∠ACQ\angle ACP=\angle BCQ < \angle ACQ.Let R,SR,S be the foot of altitudes from HH to CQ,CPCQ,CP respectively. Prove that thé points P,Q,R,SP,Q,R,S are concyclic and MM is the center of this circle.
geometry
Sequnce of positive integers

Source: IZHO 2017 day 2 p4

1/15/2017
Let (an)(a_n) be sequnce of positive integers such that first kk members a1,a2,...,aka_1,a_2,...,a_k are distinct positive integers, and for each n>kn>k, number ana_n is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers a1,a2,...,an−1a_1,a_2,...,a_{n-1}. Prove that an=2an−1a_n=2a_{n-1} for all sufficently large nn.
algebranumber theoryInteger sequenceSequence