MathDB

Problem 4

Part of 2001 Slovenia National Olympiad

Problems(4)

tiling 8x8 board with Ls (Slovenia National MO 2001 2nd Grade P4)

Source:

4/7/2021
Find the smallest number of squares on an 8×88\times8 board that should be colored so that every LL-tromino on the board contains at least one colored square.
combinatoricsTiling
Game with cutting paper (Slovenia National MO 2001 1st Grade P4)

Source:

4/7/2021
Andrej and Barbara play the following game with two strips of newspaper of length aa and bb. They alternately cut from any end of any of the strips a piece of length dd. The player who cannot cut such a piece loses the game. Andrej allows Barbara to start the game. Find out how the lengths of the strips determine the winner.
game
placing cross-shaped tiles (Slovenia National MO 2001 3rd Grade P4)

Source:

4/7/2021
Cross-shaped tiles are to be placed on a 8×88\times8 square grid without overlapping. Find the largest possible number of tiles that can be placed. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy8zL2EyY2Q4MDcyMWZjM2FmZGFhODkxYTk5ZmFiMmMwNzk0MzZmYmVjLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wNyBhdCA2LjIzLjU4IEFNLnBuZw
combinatorics
points on a circle (Slovenia National MO 2001 4th Grade P4)

Source:

4/7/2021
Let n≥4n\ge4 points on a circle be denoted by 11 through nn. A pair of two nonadjacent points denoted by aa and bb is called regular if all numbers on one of the arcs determined by aa and bb are less than aa and bb. Prove that there are exactly n−3n-3 regular pairs.
geometry