3

Part of 2009 International Zhautykov Olympiad

Problems(2)

An Outstanding Inequality in an arbitrary hexagon, ZIMO - 09

Source: International Zhautykov Olympiad 2009, day 1, problem 3.

For a convex hexagon

ABCDEF

S

, prove that: AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S

inequalitiesgeometrytrigonometrytrig identitiesLaw of Cosinesgeometry proposed

17*17 table with some cells colored in black, ZIMO - 09

Source: International Zhautykov Olympiad 2009, day 2, problem 6.

17\times 17

n

squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least

6

of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of

n

such that for some initial arrangement of

n

black squares one can paint all squares of the table in black in some steps.

floor functioninequalitiesceiling functioncombinatorics proposedcombinatorics