MathDB

2

Part of 2016 All-Russian Olympiad

Problems(3)

Concyclic points

Source: All russian olympiad 2016,Day1,grade 9,P2

5/1/2016
ω\omega is a circle inside angle ∡BAC\measuredangle BAC and it is tangent to sides of this angle at B,CB,C.An arbitrary line ℓ \ell intersects with AB,ACAB,AC at K,LK,L,respectively and intersect with ω\omega at P,QP,Q.Points S,TS,T are on BCBC such that KS∥ACKS \parallel AC and TL∥ABTL \parallel AB.Prove that P,Q,S,TP,Q,S,T are concyclic.(I.Bogdanov,P.Kozhevnikov)
geometryConcyclicgeometry proposed
Line passes through circumcenters is parallel to AD

Source: All russian olympiad 2016,Day1,grade 10,P2

5/1/2016
Diagonals AC,BDAC,BD of cyclic quadrilateral ABCDABCD intersect at PP.Point QQ is onBCBC (betweenBB and CC) such that PQ⊥ACPQ \perp AC.Prove that the line passes through the circumcenters of triangles APDAPD and BQDBQD is parallel to ADAD.(A.Kuznetsov)
geometrycircumcirclecyclic quadrilateralgeometry proposed
circumspheres intersect at one point

Source: All russian olympiad 2016,Day1,grade 11,P2

5/1/2016
In the space given three segments A1A2,B1B2A_1A_2, B_1B_2 and C1C2C_1C_2, do not lie in one plane and intersect at a point PP. Let OijkO_{ijk} be center of sphere that passes through the points Ai,Bj,CkA_i, B_j, C_k and PP. Prove that O111O222,O112O221,O121O212O_{111}O_{222}, O_{112}O_{221}, O_{121}O_{212} andO211O122O_{211}O_{122} intersect at one point. (P.Kozhevnikov)
geometry3D geometrysphere