MathDB

3

Part of 2014 Middle European Mathematical Olympiad

Problems(2)

concurrent lines incenter

Source: Middle European Mathematical Olympiad I-3

9/20/2014
Let ABCABC be a triangle with AB<ACAB < AC and incentre II. Let EE be the point on the side ACAC such that AE=ABAE = AB. Let GG be the point on the line EIEI such that ∠IBG=∠CBA\angle IBG = \angle CBA and such that EE and GG lie on opposite sides of II.
Prove that the line AIAI, the line perpendicular to AEAE at EE, and the bisector of the angle ∠BGI\angle BGI are concurrent.
geometryincentersymmetryangle bisectorperpendicular bisectorgeometry proposed
MEMOrable rectangles

Source: Middle European Mathematical Olympiad T-3

9/21/2014
Let KK and LL be positive integers. On a board consisting of 2K×2L2K \times 2L unit squares an ant starts in the lower left corner square and walks to the upper right corner square. In each step it goes horizontally or vertically to a neighbouring square. It never visits a square twice. At the end some squares may remain unvisited. In some cases the collection of all unvisited squares forms a single rectangle. In such cases, we call this rectangle MEMOrable.
Determine the number of different MEMOrable rectangles.
Remark: Rectangles are different unless they consist of exactly the same squares.
geometryrectanglecombinatorics proposedcombinatorics