3
Part of 1971 Bundeswettbewerb Mathematik
Problems(2)
Acute triangle
Source: Spanish 2001; JBMO TST (2006?) Moldova, problem 1; German BWM 1970/71, 1st round, problem 3
2/27/2006
Given five segments such that any three of them can be used to form a triangle. Show that at least one of these triangles is acute-angled.
Alternative formulation: Five segments have lengths such that any three of them can be sides of a triangle. Prove that there exists at least one acute-angled triangle among these triangles.
geometry solvedgeometry
One-way roads between cities
Source: Bundeswettbewerb Mathematik 1971, round 2 problem 3
6/14/2006
Between any two cities of a country there is only one one-way road. Show that there is a city from that every other city can be reached directly or by going over only one intermediate city.
I'm sure it was posted before but couldn't find it.
induction