MathDB

4

Part of 2024 Caucasus Mathematical Olympiad

Problems(2)

Existence of a line intersecting many segments

Source: Caucasus MO 2024, Juniors P4

3/15/2024
Given a set PP of n>100n>100 points on the plane such that no three of them are collinear, and a set SS of 20n20n distinct segments, each joining a pair of points from PP. Prove that there exists a line not passing through a point from PP and intersecting at least 200200 segments from SS.
combinatorics
Maximal guaranteed sum of numbers in non-adjacent cells

Source: Caucasus MO 2024, Seniors P4

3/15/2024
Yasha writes in the cells of the table 99×9999 \times 99 all positive integers from 1 to 99299^2 (each number once). Grisha looks at the table and selects several cells, among which there are no two cells sharing a common side, and then sums up the numbers in all selected cells. Find the largest sum Grisha can guarantee to achieve.
combinatorics