2012 CentroAmerican

Part of CentroAmerican

Subcontests

(3)

2

Sum of 1/a+b = 1

a,b,c

be real numbers that satisfy

\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c} =1

ab+bc+ac >0

a+b+c - \frac{abc}{ab+bc+ac} \ge 4

Parallel to angle bisector

ABC

be a triangle with

AB < BC

E

F

AC

AB

BF = BC = CE

, both on the same halfplane as

A

with respect to

BC

G

be the intersection of

BE

CF

H

be a point in the parallel through

G

AC

HG = AF

H

C

in opposite halfplanes with respect to

BG

\angle EHG = \frac{\angle BAC}{2}

2

Least amount of money

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?

Line passes through midpoint

\gamma

be the circumcircle of the acute triangle

ABC

P

be the midpoint of the minor arc

BC

. The parallel to

AB

P

BC, AC

\gamma

R,S

T

, respectively. Let

K \equiv AP \cap BT

L \equiv BS \cap AR

KL

passes through the midpoint of

AB

CS = PR

2

700 times the sum of its digits

Find all positive integers that are equal to

700

times the sum of its digits.

Intersections of lines in equilateral triangle

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?