4

Part of 2022 May Olympiad

Problems(2)

numbers at vertices of triangle / tetrahedron, consecutive and gcd related

Source: 2022 May Olympiad L2 p4

a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order? b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?

geometrycombinatorics

4 squares on a chessboard

Source: 2022 May Olympiad L1 p4

Ana and Bruno have an

8 \times 8

checkered board. Ana paints each of the

64

squares with some color. Then Bruno chooses two rows and two columns on the board and looks at the

4

squares where they intersect. Bruno's goal is for these

4

squares to be the same color. How many colors, at least, must Ana use so that Bruno can't fulfill his objective? Show how you can paint the board with this amount of colors and explain because if you use less colors then Bruno can always fulfill his goal.

combinatoricswinning strategy