MathDB

1

Part of 2019 China Team Selection Test

Problems(4)

Cyclic Pentagon with Orthocenters

Source: 2019 China TST Test 1 Day 1 Q1

3/17/2019
ABCDEABCDE is a cyclic pentagon, with circumcentre OO. AB=AE=CDAB=AE=CD. II midpoint of BCBC. JJ midpoint of DEDE. FF is the orthocentre of △ABE\triangle ABE, and GG the centroid of △AIJ\triangle AIJ.CECE intersects BDBD at HH, OGOG intersects FHFH at MM. Show that AM⊥CDAM\perp CD.
geometrycircumcircle
Tangents to a circle and fixed point

Source: China TST 2019 Test 2 Day 1 Q1

3/11/2019
ABAB and ACAC are tangents to a circle ω\omega with center OO at B,CB,C respectively. Point PP is a variable point on minor arc BCBC. The tangent at PP to ω\omega meets AB,ACAB,AC at D,ED,E respectively. AOAO meets BP,CPBP,CP at U,VU,V respectively. The line through PP perpendicular to ABAB intersects DVDV at MM, and the line through PP perpendicular to ACAC intersects EUEU at NN. Prove that as PP varies, MNMN passes through a fixed point.
geometryChinaChina TST2019moving points
Inequality about complex numbers

Source: 2019 China TST Test 3 P1

3/23/2019
Given complex numbers x,y,zx,y,z, with ∣x∣2+∣y∣2+∣z∣2=1|x|^2+|y|^2+|z|^2=1. Prove that: ∣x3+y3+z3−3xyz∣≤1|x^3+y^3+z^3-3xyz| \le 1
inequalitiescomplex numbersalgebraChinaChina TST
Parallelogram Leads to AQ = AR

Source: 2019 China TST Test 4 P1

4/9/2019
Cyclic quadrilateral ABCDABCD has circumcircle (O)(O). Points MM and NN are the midpoints of BCBC and CDCD, and EE and FF lie on ABAB and ADAD respectively such that EFEF passes through OO and EO=OFEO=OF. Let ENEN meet FMFM at PP. Denote SS as the circumcenter of △PEF\triangle PEF. Line POPO intersects ADAD and BABA at QQ and RR respectively. Suppose OSPCOSPC is a parallelogram. Prove that AQ=ARAQ=AR.
geometrycircumcircleangle bisector