MathDB

2

Part of 2024 Tuymaada Olympiad

Problems(2)

Inequality with sizes of hedgehogs

Source: Tuymaada 2024 Juniors P2

7/9/2024
We will call a hedgehog a graph in which one vertex is connected to all the others and there are no other edges; the number of vertices of this graph will be called the size of the hedgehog. A graph GG is given on nn vertices (where n>1n > 1). For each edge ee, we denote by s(e)s(e) the size of the maximum hedgehog in graph GG, which contains this edge. Prove the inequality (summation is carried out over all edges of the graph GG): ∑e1s(e)⩽n2.\sum_e \frac{1}{s(e)} \leqslant \frac{n}{2}. Proposed by D. Malec, C. Tompkins
inequalitiescombinatorics
Moving chess king game

Source: 2024 Tuymaada Senior P2

7/9/2024
Chip and Dale play on a 100×100100 \times 100 table. In the beginning, a chess king stands in the upper left corner of the table. At each move the king is moved one square right, down or right-down diagonally. A player cannot move in the direction used by his opponent in the previous move. The players move in turn, Chip begins. The player that cannot move loses. Which player has a winning strategy?
combinatoricsgameGame TheoryCombinatorial gamesTuymaada