Olympiad problems

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Found problems: 2

2022 Iran MO (3rd Round), 3

We have $n\ge3$ points on the plane such that no three are collinear. Prove that it's possible to name them $P_1,P_2,\ldots,P_n$ such that for all $1<i<n$, the angle $\angle P_{i-1}P_iP_{i+1}$ is acute.

2024 Korea National Olympiad, 3

Tags: combinatorics , geometric combinatorics

Let $ S $ be a set consisting of $ 2024 $ points on a plane, such that no three points in $ S $ are collinear. A line $ \ell $ passing through two points in $ S $ is called a "weakly balanced line" if it satisfies the following condition: (Condition) The line $ \ell $ divides the plane into two regions, one containing exactly $ 1010 $ points of $ S $, and the other containing exactly $ 1012 $ points of $ S $ (where each region contains no points lying on $ \ell $). Let $ \omega(S) $ denote the number of weakly balanced lines among the lines passing through two points in $ S $. Find the smallest possible value of $ \omega(S) $.