MathDB

2

Part of 2016 Tuymaada Olympiad

Problems(2)

Covering

Source: Tuymaada 2016, Senior P2, Junior P3

7/22/2016
A cube stands on one of the squares of an infinite chessboard. On each face of the cube there is an arrow pointing in one of the four directions parallel to the sides of the face. Anton looks on the cube from above and rolls it over an edge in the direction pointed by the arrow on the top face. Prove that the cube cannot cover any 5×55\times 5 square.
combinatorics
Easy geo problem

Source: Tuymaada 2016. junior league\ P2

7/22/2016
The point DD on the altitude AA1AA_1 of an acute triangle ABCABC is such that ∠BDC=90∘\angle BDC=90^\circ; HH is the orthocentre of ABCABC. A circle with diameter AHAH is constructed. Prove that the tangent drawn from BB to this circle is equal to BDBD.
geometry proposedgeometry