2018 lberoAmerican

Part of IberoAmerican

Subcontests

(6)

1

Isogonal lines in constructed triangle

ABC

be an acute triangle with

AC > AB > BC

. The perpendicular bisectors of

AC

AB

BC

D

E

respectively. Let

P

Q

be points on lines

AC

AB

respectively, both different from

A

AB = BP

AC = CQ

K

be the intersection of lines

EP

DQ

M

be the midpoint of

BC

\angle DKA = \angle EKM

1

Maximums and minimums in permutation

n

be a positive integer. For a permutation

a_1, a_2, \dots, a_n

1, 2, \dots, n

b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j

We say that the permutation

a_1, a_2, \dots, a_n

is guadiana if the sequence

b_1, b_2, \dots, b_n

does not contain two consecutive equal terms. How many guadiana permutations exist?

1

Set closed under gcd

X

of positive integers is said to be iberic if

X

\{2, 3, \dots, 2018\}

m, n

X

\gcd(m, n)

X

. An iberic set is said to be olympic if it is not properly contained in any other iberic set. Find all olympic iberic sets that contain the number

33

1

Moving point in configuration of lines

In a plane we have

n

lines, no two of which are parallel or perpendicular, and no three of which are concurrent. A cartesian system of coordinates is chosen for the plane with one of the lines as the

x

P

is located at the origin of the coordinate system and starts moving along the positive

x

-axis with constant velocity. Whenever

P

reaches the intersection of two lines, it continues along the line it just reached in the direction that increases its

x

-coordinate. Show that it is possible to choose the system of coordinates in such a way that

P

visits points from all

n

1

Equal lengths in isosceles configuration

ABC

be a triangle such that

\angle BAC = 90^{\circ}

AB = AC

M

be the midpoint of

BC

D \neq A

is chosen on the semicircle with diameter

BC

A

. The circumcircle of triangle

DAM

DB

DC

E

F

respectively. Show that

BE = CF

1

Integer system of equations

For each integer

n \ge 2

, find all integer solutions of the following system of equations:

x_1 = (x_2 + x_3 + x_4 + ... + x_n)^{2018}

x_2 = (x_1 + x_3 + x_4 + ... + x_n)^{2018}

\vdots

x_n = (x_1 + x_2 + x_3 + ... + x_{n - 1})^{2018}