2

Part of 2024 239 Open Mathematical Olympiad

Problems(2)

Rich knight adds coins

Source: 239 MO 2024 S2

A rich knight has a chest and a lot of coins, so every day he puts into the chest some quantity of coins - among the numbers

1, 2, \ldots, 100

. If there exist two days on which he added equal quantities of coins (say,

k

coins) and he has added in total at most

100k

coins on the days between these two days, he stops putting coins into the chest. Prove that this will necessarily happen eventually.

Intersecting segments

Source: 239 MO 2024 J2

2n

points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose

n

drawn segments having no common ends?

combinatoricscombinatorial geometry