2016 IberoAmerican

Part of IberoAmerican

Subcontests

(6)

1

Bishops in a chessboard

Determine the maximum number of bishops that we can place in a

8 \times 8

chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.
Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the

2

main diagonals and the ones parallel to those ones.

1

A, E and F are collinear

The circumferences

C_1

C_2

cut each other at different points

A

K

. The common tangent to

C_1

C_2

K

C_1

B

C_2

C

P

be the foot of the perpendicular from

B

AC

Q

be the foot of the perpendicular from

C

AB

E

F

are the symmetric points of

K

with respect to the lines

PQ

BC

, respectively, prove that

A, E

F

1

Last 2k digits

k

be a positive integer and

a_1, a_2,

\cdot \cdot \cdot

, a_k

digits. Prove that there exists a positive integer

n

such that the last

2k

2^n

are, in the following order,

a_1, a_2,

\cdot \cdot \cdot

, a_k , b_1, b_2,

\cdot \cdot \cdot

, b_k

, for certain digits

b_1, b_2,

\cdot \cdot \cdot

, b_k

1

Five points lie on a circle

ABC

be an acute triangle and

\Gamma

its circumcircle. The lines tangent to

\Gamma

B

C

P

M

be a point on the arc

AC

that does not contain

B

M \neq A

M \neq C

K

be the point where the lines

BC

AM

R

be the point symmetrical to

P

with respect to the line

AM

Q

the point of intersection of lines

RA

PM

J

be the midpoint of

BC

L

be the intersection point of the line

PJ

and the line through

A

PR

L, J, A, Q,

K

all lie on a circle.

1

Prime numbers in an equation

Find all prime numbers

p,q,r,k

pq+qr+rp = 12k+1

1

Iberoamerican 2016 problem $2$

Find all positive real numbers

(x,y,z)

x = \frac{1}{y^2+y-1}

y = \frac{1}{z^2+z-1}

z = \frac{1}{x^2+x-1}