7

Part of 2021 239 Open Mathematical Olympiad

Problems(2)

Hard and nice combinatorial geometry

Source: 239MO-2021

n

lines on the plane, they divide the plane onto several bounded or bounded polygonal regions. Define the rank of a region as the number of vertices on its boundary (a vertex is a point which belongs to at least two lines). Prove that the sum of squares of ranks of all regions does not exceed

10n^2

geometrycombinatorial geometry

Combinatorics about a grid with numbers

Source: 239 MO 2021 (8-9).7

Given is a grid with

2

120

columns, such that each cell has a number from the set

1, 2, ..., 120

. It is known that in each column, the upper number in it is smaller than the lower number, and in each row, the numbers are in non-strict increasing order from left to right. Prove that the number of these tables is multiple of

239

combinatoricscatalan