MathDB

3

Part of 2018 China Team Selection Test

Problems(4)

Incenter lies on line

Source: China TST 1 Day 1 Q3

1/2/2018
Circle ω\omega is tangent to sides ABAB,ACAC of triangle ABCABC at DD,EE respectively, such that D≠BD\neq B, E≠CE\neq C and BD+CE<BCBD+CE<BC. FF,GG lies on BCBC such that BF=BDBF=BD, CG=CECG=CE. Let DGDG and EFEF meet at KK. LL lies on minor arc DEDE of ω\omega, such that the tangent of LL to ω\omega is parallel to BCBC. Prove that the incenter of △ABC\triangle ABC lies on KLKL.
geometryTST
How to Play a Infinite Writing-Number Game

Source: 2018 China Team Selection Test 2 Problem 3

1/8/2018
Two positive integers p,q∈Z+p,q \in \mathbf{Z}^{+} are given. There is a blackboard with nn positive integers written on it. A operation is to choose two same number a,aa,a written on the blackboard, and replace them with a+p,a+qa+p,a+q. Determine the smallest nn so that such operation can go on infinitely.
combinatoricsTSTChina TST
2018 China TST 3 Day 1 Q3

Source: Mar 20, 2018

3/27/2018
Prove that there exists a constant C>0C>0 such that H(a1)+H(a2)+⋯+H(am)≤C∑i=1miaiH(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i} holds for arbitrary positive integer mm and any mm positive integer a1,a2,⋯ ,ama_1,a_2,\cdots,a_m, where H(n)=∑k=1n1k.H(n)=\sum_{k=1}^{n}\frac{1}{k}.
inequalitiesalgebraChina TST
Concyclic with circumcenters

Source: China TST 4 2018 Day 1 Q3

3/27/2018
In isosceles △ABC\triangle ABC, AB=ACAB=AC, points D,E,FD,E,F lie on segments BC,AC,ABBC,AC,AB such that DE∥ABDE\parallel AB, DF∥ACDF\parallel AC. The circumcircle of △ABC\triangle ABC ω1\omega_1 and the circumcircle of △AEF\triangle AEF ω2\omega_2 intersect at A,GA,G. Let DEDE meet ω2\omega_2 at K≠EK\neq E. Points L,ML,M lie on ω1,ω2\omega_1,\omega_2 respectively such that LG⊥KG,MG⊥CGLG\perp KG, MG\perp CG. Let P,QP,Q be the circumcenters of △DGL\triangle DGL and △DGM\triangle DGM respectively. Prove that A,G,P,QA,G,P,Q are concyclic.
geometrycircumcircle