EisatoponAI: ▪CentroAmerican Mathematical Olympiad 2012

Ημέρα 1η

1. Find all positive integers that are equal to

times the sum of its digits.

2. Let

be the circumcircle of the acute triangle

. Let

be the midpoint of the minor arc

. The parallel to

through

cuts

and

at points

and

, respectively. Let

and

. Show that

passes through the midpoint of

if and only if

3. Let

be real numbers that satisfy

and

Show that

Ημέρα 2η

1. Trilandia is a very unusual city. The city has the shape of an equilateral triangle of side lenght 2012. The streets divide the city into several blocks that are shaped like equilateral triangles of side lenght 1. There are streets at the border of Trilandia too. There are 6036 streets in total. The mayor wants to put sentinel sites at some intersections of the city to monitor the streets. A sentinel site can monitor every street on which it is located. What is the smallest number of sentinel sites that are required to monitor every street of Trilandia?

2. Alexander and Louise are a pair of burglars. Every morning, Louise steals one third of Alexander's money, but feels remorse later in the afternoon and gives him half of all the money she has. If Louise has no money at the beginning and starts stealing on the first day, what is the least positive integer amount of money Alexander must have so that at the end of the 2012th day they both have an integer amount of money?

3. Let

be a triangle with