MathDB

Problem 4

Part of 2001 Croatia National Olympiad

Problems(4)

existence of points inside polygon with integral differences

Source: Croatia MO 2001 2nd Grade P4

4/22/2021
On the coordinate plane is given a polygon P\mathcal P with area greater than 11. Prove that there exist two different points (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) inside the polygon P\mathcal P such that x1−x2x_1-x_2 and y1−y2y_1-y_2 are both integers.
geometry
partitioning a board into l-trinominoes

Source: Croatia MO 2001 1st Grade P4

4/22/2021
Find all possible values of nn for which a rectangular board 9×n9\times n can be partitioned into tiles of the shape: https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8wLzdjM2Y4ZmE0Zjg1YWZlZGEzNTQ1MmEyNTc3ZjJkNzBlMjExYmY1LnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0yMiBhdCA1LjEzLjU3IEFNLnBuZw==
combinatorics
subset of 100 positive integers whose product is a square

Source: Croatia MO 2001 3rd Grade P4

4/22/2021
Let SS be a set of 100100 positive integers less than 200200. Prove that there exists a nonempty subset TT of SS the product of whose elements is a perfect square.
number theory
0s and 1s in square board, maximum number of 1s

Source: Croatia MO 2001 4th Grade P4

4/22/2021
Suppose that zeros and ones are written in the cells of an n×nn\times n board, in such a way that the four cells in the intersection of any two rows and any two columns contain at least one zero. Prove that the number of ones does not exceed n2(1+4n−3)\frac n2\left(1+\sqrt{4n-3}\right).
combinatoricsgame