2023 Centroamerican and Caribbean Math Olympiad

Part of CentroAmerican

Subcontests

(6)

1

Peaches in El Salvador

In a pond there are

n \geq 3

stones arranged in a circle. A princess wants to label the stones with the numbers

1, 2, \dots, n

in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone labeled with the number

k

k

minutes and then jumps to the adjacent stone.
What is the greatest number of toads for which the princess can label the stones and place the toads in such a way that at no time are two toads occupying a stone at the same time?
Note: A stone is considered occupied by two toads at the same time only if there are two toads that are on the stone for at least one minute.

1

Guanaco numbers

A four-digit number

n=\overline{a b c d}

a, b, c

d

are digits, with

a \neq 0

, is said to be guanaco if the product

\overline{a b} \times \overline{c d}

is a positive divisor of

n

. Find all guanaco numbers.

1

Two points lying on a triangle's circumcircle

ABC

be an acute-angled triangle with

AB < AC

\Gamma

the circumference that passes through

A,\ B

C

D

be the point diametrically opposite

A

\Gamma

\ell

the tangent through

D

\Gamma

P, Q

R

be the intersection points of

B C

\ell

A P

\Gamma

Q \neq A

Q D

A

-altitude of the triangle

ABC

, respectively. Define

S

to be the intersection of

AB

\ell

T

to be the intersection of

A C

\ell

S

T

lie on the circumference that passes through

A, Q

R

1

Stepping numbers

Octavio writes an integer

n \geq 1

on a blackboard and then he starts a process in which, at each step he erases the integer

k

written on the blackboard and replaces it with one of the following numbers: 3k-1, 2k+1, \frac{k}{2}. provided that the result is an integer.
Show that for any integer

n \geq 1

, Octavio can write on the blackboard the number

3^{2023}

after a finite number of steps.

1

Coloration of Integers

A coloring of the set of integers greater than or equal to

1

, must be done according to the following rule: Each number is colored blue or red, so that the sum of any two numbers (not necessarily different) of the same color is blue. Determine all the possible colorings of the set of integers greater than or equal to

1

that follow this rule.

1

Mount inequality erupts in El Salvador

a,\ b

c

be positive real numbers such that

a b+b c+c a=1

\frac{a^3}{a^2+3 b^2+3 a b+2 b c}+\frac{b^3}{b^2+3 c^2+3 b c+2 c a}+\frac{c^3}{c^2+3 a^2+3 c a+2 a b}>\frac{1}{6\left(a^2+b^2+c^2\right)^2} .