MathDB

5

Part of 2010 Saint Petersburg Mathematical Olympiad

Problems(2)

Pyramid geometry

Source: St Petersburg Olympiad 2010, Grade 11, P5

9/14/2017
SABCDSABCD is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane. ABCDABCD is base of pyramid and ACAC and BDBD intersects at PP, where SPSP is height of pyramid. Prove that AC⊥BDAC \perp BD
geometry3D geometrypyramid
Roads and cities

Source: St Petersburg Olympiad 2010, Grade 9, P5

9/14/2017
There are 20102010 cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Ministry destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly 1111 different cities?
combinatorics