MathDB

3

Part of 2016 Middle European Mathematical Olympiad

Problems(2)

concyclic with two vertices and the circumcenter

Source: MEMO 2016 I3

8/24/2016
Let ABCABC be an acute triangle such that ∠BAC>45∘\angle BAC > 45^{\circ} with circumcenter OO. A point PP is chosen inside triangle ABCABC such that A,P,O,BA, P, O, B are concyclic and the line BPBP is perpendicular to the line CPCP. A point QQ lies on the segment BPBP such that the line AQAQ is parallel to the line POPO.
Prove that ∠QCB=∠PCO\angle QCB = \angle PCO.
geometrygeometry proposedCircumcenterperpendicular lines
houses in the shade on 8x8 board

Source: MEMO 2016 T3

8/25/2016
A 8×88 \times 8 board is given, with sides directed north-south and east-west. It is divided into 1×11 \times 1 cells in the usual manner. In each cell, there is most one house. A house occupies only one cell.
A house is in the shade if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade.
Find the maximal number of houses that can be placed on the board such that no house is in the shade.
combinatoricscombinatorics proposedboard