MathDB

5

Part of 2009 Saint Petersburg Mathematical Olympiad

Problems(3)

Geometry with circles

Source: St Petersburg Olympiad 2009, Grade 11, P5

8/30/2017
OO -circumcenter of ABCDABCD. ACAC and BDBD intersect in EE, ADAD and BCBC in FF. X,YX,Y - midpoints of ADAD and BCBC. O1O_1 -circumcenter of EXYEXY. Prove that OF∥O1EOF \parallel O_1E
geometrycircumcircle
Chess rooks

Source: St Petersburg Olympiad 2009, Grade 9, P6

8/30/2017
Call a set of some cells in infinite chess field as board. Set of rooks on the board call as awesome if no one rook can beat another, but every empty cell is under rook attack. There are awesome set with 20082008 rooks and with 20102010 rooks. Prove, that there are awesome set with 20092009 rooks.
combinatorics
Altitudes and sides

Source: St Petersburg Olympiad 2009, Grade 9, P5

8/30/2017
ABCABC is acute-angled triangle. AA1,BB1,CC1AA_1,BB_1,CC_1 are altitudes. X,YX,Y - midpoints of AC1,A1CAC_1,A_1C. XY=BB1XY=BB_1. Prove that one side of ABCABC in 2\sqrt{2} greater than other side.
geometry