2

Part of 2017 Czech-Polish-Slovak Match

Problems(2)

largest number of coloured squares in board of 4 kinds

Source: Czech-Polish-Slovak Match 2017 day 2 P2

{4n^2}

unit squares of a

{2n \times 2n}

{(n \ge 1) }

has been colored blue or red. A set of four different unit squares of the board is called pretty if these squares can be labeled

{A,B,C,D}

in such a way that

{A}

{B}

lie in the same row,

{C}

{D}

lie in the same row,

{A}

{C}

lie in the same column,

{B}

{D}

lie in the same column,

{A}

{D}

{B}

{C}

are red. Determine the largest possible number of different pretty sets on such a board.
(Poland)

Squaresboardcombinatoricsmaximum

equal angles, interior point of triangle, concyclic

Source: Czech-Polish-Slovak Match 2017 day 1 P2

{\omega}

be the circumcircle of an acute-angled triangle

{ABC}

{D}

lies on the arc

{BC}

{\omega}

not containing point

{A}

{E}

lies in the interior of the triangle

{ABC}

, does not lie on the line

{AD}

, and satisfies

{\angle DBE =\angle ACB}

{\angle DCE = \angle ABC}

{F}

be a point on the line

{AD}

such that lines

{EF}

{BC}

are parallel, and let

{G}

{\omega}

{A}

{AF = FG}

. Prove that points

{D,E, F,G}

lie on one circle.
(Slovakia)

Concyclicequal anglesgeometrycircle