MathDB

3

Part of 2017 International Zhautykov Olympiad

Problems(2)

Tilings with domino figures

Source: IZHO 2017 day 1 p3

1/16/2017
Rectangle on a checked paper with length of a unit square side being 11 Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with 33 colours such that for any two corners with distance 11 the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.
combinatoricsTilinggeometryrectangle
Tetrahedron inequality

Source: Izho 2017 P6

1/15/2017
Let ABCDABCD be the regular tetrahedron, and M,NM, N points in space. Prove that: AM⋅AN+BM⋅BN+CM⋅CN≥DM⋅DNAM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN
geometry3D geometrytetrahedroninequalities