MathDB

5

Part of 2018 Saint Petersburg Mathematical Olympiad

Problems(3)

Arrows in hexagon

Source: St Petersburg Olympiad 2018, Grade 11, P5

6/21/2018
Regular hexagon is divided to equal rhombuses, with sides, parallels to hexagon sides. On the three sides of the hexagon, among which there are no neighbors, is set directions in order of traversing the hexagon against hour hand. Then, on each side of the rhombus, an arrow directed just as the side of the hexagon parallel to this side. Prove that there is not a closed path going along the arrows.
geometryrhombus
Inscribed quadrilateral

Source: St Petersburg Olympiad 2018, Grade 10, P5

6/22/2018
ABCDABCD is inscribed quadrilateral. Line, that perpendicular to BDBD intersects segments ABAB and BCBC and rays DA,DCDA,DC at P,Q,R,SP,Q,R,S . PR=QSPR=QS. MM is midpoint of PQPQ. Prove that AM=CMAM=CM
geometry
Triangle and points

Source: St Petersburg Olympiad 2018, Grade 9, P5

7/13/2018
Can we draw △ABC\triangle ABC and points X,YX,Y, such that AX=BY=ABAX=BY=AB, BX=CY=BC BX = CY = BC, CX=AY=CACX = AY = CA?
geometry