2024 Chile TST Ibero.

Part of Chile TST Ibero

Subcontests

(5)

1

Solve by titus??

a

b

c

are positive real numbers, then the following inequality holds:

\frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6.

1

Coloring again

Find all natural numbers

n

for which it is possible to construct an

n \times n

square using only tetrominoes like the one below:

1

Navas' Classic again???

A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.

1

CMAT in TST????

Determine all integers

x

for which the expression

x^2 + 10x + 160

is a perfect square.

1

You know how to draw?

\triangle ABC

be an acute-angled triangle. Let

P

be the midpoint of

BC

K

the foot of the altitude from

A

BC

D

be a point on segment

AP

\angle BDC = 90^\circ

E

be the second point of intersection of line

BC

with the circumcircle of

\triangle ADK

F

be the second point of intersection of line

AE

with the circumcircle of

\triangle ABC

\angle AFD = 90^\circ