MathDB

1

Esteban the alchemist

8088

copper pieces,

6066

bronze pieces,

4044

silver pieces and

2022

gold pieces. He can take two pieces of different metals and use a magic hammer to turn them into two pieces of different metals that he take and different each other. Find the largest number of gold pieces that Esteban can obtain after using the magic hammer a finite number of times.

<span class='latex-bold'>Note:</span>

If Esteban takes a copper and bronze pieces, then he turn them into a silver and a gold pieces.

2

1

numbers in a circle

Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers

a,b,c,d,e,f

respectively. At the end of each minute, all the people simultaneously replace the number on their paper by the sum of three numbers; the number that was at the beginning of the minute on his paper and on the papers of his two neighbors. At the end of the minute

2022, 2022

replacements have been made and each person have in his paper it´s initial number. Find all the posible values of

abc+def

.

<span class='latex-bold'>Note:</span>

If at the beginning of the minute

N

Ana, Beto, Carlos have the numbers

x,y,z

, respectively, then at the end of the minute

N

, Beto is going to have the number

x+y+z

.

6

1

sum and product equal to $n$

A positive integer

n

is

inverosimil

if there exists

n

integers not necessarily distinct such that the sum and the product of this integers are equal to

n

. How many positive integers less than or equal to

2022

are

inverosimils

?

4

1

prove a rectangle it´s a square

Let

A_1A_2A_3A_4

be a rectangle and let

S_1,S_2,S_3,S_4

four circumferences inside of the rectangle such that

S_k

and

S_{k+1}

are tangent to each other and tangent to the side

A_kA_{k+1}

for

k=1,2,3,4

, where

A_5=A_1

and

S_5=S_1

. Prove that

A_1A_2A_3A_4

is a square.

3

1

cyclic quadrilateral with orthocenter and circumcenter

Let

ABC

an acutangle triangle with orthocenter

H

and circumcenter

O

. Let

D

the intersection of

AO

and

BH

. Let

P

be the point on

AB

such that

PH=PD

. Prove that the points

B, D, O

and

P

lie on a circle.

1

game with the sum of the digits

There is a pile with 2022 rocks. Ana y Beto play by turns to the following game, starting with Ana: in each turn, if there are

n

rocks in the pile, the player can remove

S(n)

rocks or

n-S(n)

rocks, where

S(n)

is the sum of the the digits of

n

. The person who removes the last rock wins. Determine which of the two players has a winning strategy and describe it.

2022 Centroamerican and Caribbean Math Olympiad

Subcontests

Esteban the alchemist

numbers in a circle

sum and product equal to $n$

prove a rectangle it´s a square

cyclic quadrilateral with orthocenter and circumcenter

game with the sum of the digits

2022 Centroamerican and Caribbean Math Olympiad

Subcontests

Esteban the alchemist

numbers in a circle

sum and product equal to $n$

prove a rectangle it&acute;s a square

cyclic quadrilateral with orthocenter and circumcenter

game with the sum of the digits

prove a rectangle it´s a square