2011 CentroAmerican

Part of CentroAmerican

Subcontests

(6)

1

Orthic triangle DEF satisfying 3EF=FD+DE

ABC

be an acute triangle and

D

E

F

be the feet of the altitudes through

A

B

C

respectively. Call

Y

Z

the feet of the perpendicular lines from

B

C

FD

DE

, respectively. Let

F_1

be the symmetric of

F

with respect to

E

E_1

be the symmetric of

E

with respect to

F

3EF=FD+DE

\angle BZF_1=\angle CYE_1

1

Find all possible values of x+y+z

x

y

z

are positive numbers satisfying

x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.

Find all the possible values of

x+y+z

1

Diophantine equation with primes

Find all positive integers

p

q

r

p

q

are prime numbers and

\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.

1

Slipping integers

A slip on an integer

n\geq 2

is an operation that consists in choosing a prime divisor

p

n

n

\frac{n+p^2}{p}.

Starting with an arbitrary integer

n\geq 5

, we successively apply the slip operation on it. Show that one eventually reaches

5

, no matter the slips applied.

1

Equilateral triangles

In a scalene triangle

ABC

D

is the foot of the altitude through

A

E

is the intersection of

AC

with the bisector of

\angle ABC

F

AB

O

the circumcenter of

ABC

X=AD\cap BE

Y=BE\cap CF

Z=CF \cap AD

XYZ

is an equilateral triangle, prove that one of the triangles

OXY

OYZ

OZX

must be equilateral.

1

Flies on a cube

Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?