MathDB

4

Part of 2019 239 Open Mathematical Olympiad

Problems(2)

Truth or lie at a table

Source: 239 2019 S4

7/31/2020
There are n>1000n>1000 people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the 2020 people sitting clockwise from where I sit there are as many knights as among the 2020 people seated counterclockwise from where I sit”. For what nn could this happen?
combinatoricsknights and knaves
20x20 treasure map

Source: 239 2019 J4

7/31/2020
A 20×2020 \times 20 treasure map is glued to a torus. A treasure is hidden in a cell of this map. We can ask questions about 1×41\times 4 or 4×14 \times 1 rectangles so that we find out if there is a treasure in this rectangle or not. The answers to all questions are absolutely true, but they are given only after all rectangles we want to ask are set. What is the least amount of questions needed to be asked so that we can be sure to find the treasure? (If you describe the position of the cells in a torus with numbers (i,j)(i, j) of row and column, 1≤i,j≤201 \leq i, j \leq 20, then two cells are neighbors, if and only if two of the coordinates they have are the same, and the other two differ by 11 mod 2020.)
combinatorics