2013 Olympic Revenge

Part of Olympic Revenge

Subcontests

(5)

1

Problem 5, Olympic Revenge 2013

n

lamps clockwise numbered from

1

n

on a circle.
Let

\xi

to be a configuration where

0 \le \ell \le n

random lamps are turned on. A cool procedure consists in perform, simultaneously, the following operations: for each one of the

\ell

lamps which are turned on, we verify the number of the lamp; if

i

is turned on, a signal of range

i

is sent by this lamp, and it will be received only by the next

i

lamps which follow

i

, turned on or turned off, also considered clockwise. At the end of the operations we verify, for each lamp, turned on or turned off, how many signals it has received. If it was reached by an even number of signals, it remains on the same state(that is, if it was turned on, it will be turned on; if it was turned off, it will be turned off). Otherwise, it's state will be changed.
The example in attachment, for

n=4

, ilustrates a configuration where lamps

2

4

are initially turned on. Lamp

2

sends signal only for the lamps

3

4

4

sends signal for lamps

1

2

3

4

. Therefore, we verify that lamps

1

2

received only one signal, while lamps

3

4

received two signals. Therefore, in the next configuration, lamps

1

4

will be turned on, while lamps

2

3

will be turned off.
Let

\Psi

to be the set of all

2^n

possible configurations, where

0 \le \ell \le n

random lamps are turned on. We define a function

f: \Psi \rightarrow \Psi

\xi

is a configuration of lamps, then

f(\xi)

is the configurations obtained after we perform the cool procedure described above.
Determine all values of

n

f

1

Problem 4, Olympic Revenge 2013

Find all triples

(p,n,k)

of positive integers, where

p

is a Fermat's Prime, satisfying

p^n + n = (n+1)^k

.
Observation: a Fermat's Prime is a prime number of the form

2^{\alpha} + 1

\alpha

positive integer.

1

Problem 3, Olympic Revenge 2013

a,b,c,d

to be non negative real numbers satisfying

ab+ac+ad+bc+bd+cd=6

\dfrac{1}{a^2+1} + \dfrac{1}{b^2+1} + \dfrac{1}{c^2+1} + \dfrac{1}{d^2+1} \ge 2

1

Problem 2, Olympic Revenge 2013

ABC

to be an acute triangle. Also, let

K

L

to be the two intersections of the perpendicular from

B

with respect to side

AC

with the circle of diameter

AC

K

B

L

X

Y

are the two intersections of the perpendicular from

C

with respect to side

AB

with the circle of diamter

AB

X

C

Y

. Prove that the intersection of

XL

KY

BC

1

Problem 1, Olympic Revenge 2013

n

to be a positive integer. A family

\wp

[i, j]

0 \le i < j \le n

i

j

integers is considered happy if, for any

I_1 = [i_1, j_1] \in \wp

I_2 = [i_2, j_2] \in \wp

I_1 \subset I_2

i_1 = i_2

j_1 = j_2

.
Determine the maximum number of elements of a happy family.