4

Part of 1997 Estonia National Olympiad

Problems(4)

red anb black squars in a 3nx3n grid

Source: 1997 Estonia National Olympiad Final Round grade 9 p4

3n \times 3n

grid, each square is either black or red. Each red square not on the edge of the grid has exactly five black squares among its eight neighbor squares.. On every black square that not at the edge of the grid, there are exactly four reds in the adjacent squares box. How many black and how many red squares are in the grid?

combinatoricsgridsquare tableColoring

n points, circle passing through 3 points and have not n-3 interior ones

Source: 1997 Estonia National Olympiad Final Round grade 11 p4

n\ge 3

distinct points in the plane. Is it always possible to find a circle which passes through three of the points and contains none of the remaining points (a) inside the circle. (b) inside the circle or on its boundary?

circlescombinatoricscombinatorial geometry

19 lines in the plane dividing the plane into exactly 9$ pieces

Source: 1997 Estonia National Olympiad Final Round grade 12 p4

19

lines in the plane dividing the plane into exactly

97

pieces. (a) Prove that among these pieces there is at least one triangle. (b) Show that it is indeed possible to place

19

lines in the above way.

combinatoricslinescombinatorial geometry

game with two piles of m and n candies, winning strategy

Source: 1997 Estonia National Olympiad Final Round grade 10 p4

Mari and Yuri play the next play. At first, there are two piles on the table, with

m

n

candies, respectively. At each turn, players eats one pile of candy from the table and distribute another pile of candy into two non-empty parts ,. Everything is done in turn and wins the player who can no longer share the pile (when there is only one candy left). Which player will win if both use the optimal strategy and Mari makes the first move?

game strategygamecombinatoricswinning strategy