5

Part of 2018 May Olympiad

Problems(2)

coloring 1 of 10 colors of circle May Olympiad (Olimpiada de Mayo) 2018 L2 P5

Source:

Each point on a circle is colored with one of

10

colors. Is it true that for any coloring there are

4

points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

combinatoricsColoringcombinatorial geometry

no 2, 3, 4, 5 in squares of 5x5 board , 2018 May Olympiad L1 p5

Source:

In each square of a

5 \times 5

board one of the numbers

2, 3, 4

5

is written so that the the sum of all the numbers in each row, in each column and on each diagonal is always even. How many ways can we fill the board?
Clarification. A

5\times 5

board has exactly

18

diagonals of different sizes. In particular, the corners are size

1