2024 Iberoamerican

Part of IberoAmerican

Subcontests

(6)

1

Infinite set with a curious propety

Determine all infinite sets

A

of positive integers with the following propety: If

a,b \in A

a \ge b

\left\lfloor \frac{a}{b} \right\rfloor \in A

1

Board numbers can be eventually all equal

n \ge 2

be an integer and let

a_1, a_2, \cdots a_n

be fixed positive integers (not necessarily all distinct) in such a way that

\gcd(a_1, a_2 \cdots a_n)=1

. In a board the numbers

a_1, a_2 \cdots a_n

are all written along with a positive integer

x

. A move consists of choosing two numbers

a>b

n+1

numbers in the board and replace them with

a-b,2b

. Find all possible values of

x

, with respect of the values of

a_1, a_2 \cdots a_n

, for which it is possible to achieve a finite sequence of moves (possibly none) such that eventually all numbers written in the board are equal.

1

Red geometry!?

We color some points in the plane with red, in such way that if

P,Q

X

is a point such that triangle

\triangle PQX

30º, 60º, 90º

in some order, then

X

is also red. If we have vertices

A, B, C

all red, prove that the barycenter of triangle

\triangle ABC

1

Bolivian triangles can get messy sometimes

O

be a fixed point in the plane. We have

2024

2024

yellow points and

2024

green points in the plane, where there isn't any three colinear points and all points are distinct from

O

. It is known that for any two colors, the convex hull of the points that are from any of those two colors contains

O

(it maybe contain it in it's interior or in a side of it). We say that a red point, a yellow point and a green point make a bolivian triangle if said triangle contains

O

in its interior or in one of its sides. Determine the greatest positive integer

k

such that, no matter how such points are located, there is always at least

k

bolivian triangles.

1

American point shows in Ibero geo!

\triangle ABC

be an acute triangle and let

M, N

be the midpoints of

AB, AC

respectively. Given a point

D

in the interior of segment

BC

DB<DC

P, Q

the intersections of

DM, DN

AC, AB

respectively. Let

R \ne A

be the intersection of circumcircles of triangles

\triangle PAQ

\triangle AMN

K

AR

\angle MKN=2\angle BAC

1

Inequality with number of divisors

For each positive integer

n

d(n)

be the number of positive integer divisors of

n

. Prove that for all pairs of positive integers

(a,b)

d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b))

and determine all pairs of positive integers

(a,b)

where we have equality case.