MathDB

5

Part of 2008 239 Open Mathematical Olympiad

Problems(2)

Geometric Inequality

Source: 239 2008 S5

7/28/2020
In the triangle ABCABC, ∠B=120∘\angle{B} = 120^{\circ}, point MM is the midpoint of side ACAC. On the sides ABAB and BCBC, the points KK and LL are chosen such that KL∥ACKL \parallel AC. Prove that MK+ML≥MAMK + ML \geq MA.
geometric inequalityinequalities
Checkered square

Source: 239 2008 J5

7/28/2020
You are given a checkered square, the side of which is n–1n – 1 long and contains n≥10n \geq 10 nodes. A non-return path is a path along edges, the intersection of which with any horizontal or vertical line is a segment, point or empty set, and which does not pass along any edge more than once. What is the smallest number of non-return paths that can cover all the edges? (An edge is a unit segment between adjacent nodes.)
combinatorics