MathDB

2

Part of 2017 China Team Selection Test

Problems(5)

Common chord bisects segment

Source: China TSTST 3 Day 1 Q2

3/18/2017
Let ABCDABCD be a non-cyclic convex quadrilateral. The feet of perpendiculars from AA to BC,BD,CDBC,BD,CD are P,Q,RP,Q,R respectively, where P,QP,Q lie on segments BC,BDBC,BD and RR lies on CDCD extended. The feet of perpendiculars from DD to AC,BC,ABAC,BC,AB are X,Y,ZX,Y,Z respectively, where X,YX,Y lie on segments AC,BCAC,BC and ZZ lies on BABA extended. Let the orthocenter of △ABD\triangle ABD be HH. Prove that the common chord of circumcircles of △PQR\triangle PQR and △XYZ\triangle XYZ bisects BHBH.
geometrypedal trianglecircumcircleorthocenterChina TST
China Team Selection Test 2017 TST 1 Day 1 Q2

Source: China Shanghai ,Mar 6, 2017

3/7/2017
Let x>1x>1 ,nn be positive integer. Prove that∑k=1n{kx}[kx]<∑k=1n12k−1\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1} Where [kx][kx ] be the integer part of kxkx ,{kx}\{kx \} be the decimal part of kxkx.
inequalitiesalgebraChina TST
Engineers in a conference

Source: China TSTST 2017 Test 2 Day 1 Q2

3/13/2017
20172017 engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations:
i) At least one conversation is in Chinese. ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations.
Show that there exists 673673 engineers such that any two of them conversed with each other in Chinese.
graph theorycombinatorics
g(a_i)=f(a_{i+1})

Source: 2017 China TST 5 P2

4/8/2017
Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number a1,a2,...,a2017a_1,a_2,...,a_{2017} such that g(ai)=f(ai+1)g(a_i)=f(a_{i+1}) for i=1,2,...,2017 where indices taken modulo 2017.
algebrapolynomialabstract algebra
An geometry problem from China TST

Source: China TST 4 Problem 2

3/23/2017
In ΔABC\varDelta{ABC},the excircle of AA is tangent to segment BCBC,line ABAB and ACAC at E,D,FE,D,F respectively.EZEZ is the diameter of the circle.B1B_1 and C1C_1 are on DFDF, and BB1⊥BCBB_1\perp{BC},CC1⊥BCCC_1\perp{BC}.Line ZB1,ZC1ZB_1,ZC_1 intersect BCBC at X,YX,Y respectively.Line EZEZ and line DFDF intersect at HH,ZKZK is perpendicular to FDFD at KK.If HH is the orthocenter of ΔXYZ\varDelta{XYZ},prove that:H,K,X,YH,K,X,Y are concyclic.
geometryTST