MathDB

3

Part of 2006 International Zhautykov Olympiad

Problems(2)

Good board filled with 0s and 1s

Source: Kazakhstan international contest 2006, Problem 3

1/22/2006
Let m≥n≥4 m\geq n\geq 4 be two integers. We call a m×n m\times n board filled with 0's or 1's good if 1) not all the numbers on the board are 0 or 1; 2) the sum of all the numbers in 3×3 3\times 3 sub-boards is the same; 3) the sum of all the numbers in 4×4 4\times 4 sub-boards is the same. Find all m,n m,n such that there exists a good m×n m\times n board.
combinatorics proposedcombinatorics
Convex hexagon with AD=BC+EF, BE=AF+CD, CF=DE+AB

Source: Kazakhstan international contest 2006, Problem 6

1/22/2006
Let ABCDEF ABCDEF be a convex hexagon such that AD \equal{} BC \plus{} EF, BE \equal{} AF \plus{} CD, CF \equal{} DE \plus{} AB. Prove that: \frac {AB}{DE} \equal{} \frac {CD}{AF} \equal{} \frac {EF}{BC}.
inequalitiesgeometryparallelogramEulergeometric transformationreflectiontriangle inequality