MathDB

3

Part of 2017 Middle European Mathematical Olympiad

Problems(2)

Minimum number of bad lamps on a board

Source: MEMO 2017 T3

8/25/2017
There is a lamp on each cell of a 2017×20172017 \times 2017 board. Each lamp is either on or off. A lamp is called bad if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board? (Two lamps are neighbours if their respective cells share a side.)
combinatorics
Convex pentagon

Source: MEMO 2017 I3

8/25/2017
Let ABCDEABCDE be a convex pentagon. Let PP be the intersection of the lines CECE and BDBD. Assume that ∠PAD=∠ACB\angle PAD = \angle ACB and ∠CAP=∠EDA\angle CAP = \angle EDA. Prove that the circumcentres of the triangles ABCABC and ADEADE are collinear with PP.
geometry