MathDB

3

Part of 2019 China Second Round Olympiad

Problems(2)

Tricky Sequence With NT Properties

Source: 2019 China Second Round P3

9/8/2019
Let mm be an integer where ∣m∣≥2|m|\ge 2. Let a1,a2,⋯a_1,a_2,\cdots be a sequence of integers such that a1,a2a_1,a_2 are not both zero, and for any positive integer nn, an+2=an+1−mana_{n+2}=a_{n+1}-ma_n.
Prove that if positive integers r>s≥2r>s\ge 2 satisfy ar=as=a1a_r=a_s=a_1, then r−s≥∣m∣r-s\ge |m|.
China second roundalgebranumber theory
Geometry in 2019 China Second Round(B)

Source: 2019 China Second Round(B) P3

9/8/2019
Point A,B,C,D,EA,B,C,D,E lie on a line in this order, such that BC=CD=AB⋅DE,BC=CD=\sqrt{AB\cdot DE}, PP doesn't lie on the line, and satisfys that PB=PD.PB=PD. Point K,LK,L lie on the segment PB,PD,PB,PD, respectively, such that KCKC bisects ∠BKE,\angle BKE, and LCLC bisects ∠ALD.\angle ALD. Prove that A,K,L,EA,K,L,E are concyclic.
geometry