MathDB

1

Wim and his challenging little brother in the Centro

n

\geq

2

and

k

\geq

2

be positive integers. A cat and a mouse are playing Wim, which is a stone removal game. The game starts with

n

stones and they take turns removing stones, with the cat going first. On each turn they are allowed to remove

1

,

2

,

\dotsb

, or

k

stones, and the player who cannot remove any stones on their turn loses. \\\\ A raccoon finds Wim very boring and creates Wim 2, which is Wim but with the following additional rule: You cannot remove the same number of stones that your opponent removed on the previous turn. \\\\Find all values of

k

such that for every

n

, the cat has a winning strategy in Wim if and only if it has a winning strategy in Wim 2.

5

1

Tricky system of equations in the Centro

Let

x

and

y

be positive real numbers satisfying the following system of equations:

\begin{cases} \sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\ \sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2 \end{cases}

Find the maximum value of

x + y

.

4

1

Nice triangular similarity problem in the Centro

Let

ABC

be a triangle,

I

its incenter, and

\Gamma

its circumcircle. Let

D

be the second point of intersection of

AI

with

\Gamma

. The line parallel to

BC

through

I

intersects

AB

and

AC

at

P

and

Q

, respectively. The lines

PD

and

QD

intersect

BC

at

E

and

F

, respectively. Prove that triangles

IEF

and

ABC

are similar.

3

1

Collinearity with altitudes and circumcircles in the Centro

Let

ABC

be a triangle,

H

its orthocenter, and

\Gamma

its circumcircle. Let

J

be the point diametrically opposite to

A

on

\Gamma

. The points

D

,

E

and

F

are the feet of the altitudes from

A

,

B

and

C

, respectively. The line

AD

intersects

\Gamma

again at

P

. The circumcircle of

EFP

intersects

\Gamma

again at

Q

. Let

K

be the second point of intersection of

JH

with

\Gamma

. Prove that

K

,

D

and

Q

are collinear.

2

1

Winning strategy for a 2024 cell-row in the Centro

There is a row with

2024

cells. Ana and Beto take turns playing, with Ana going first. On each turn, the player selects an empty cell and places a digit in that space. Once all

2024

cells are filled, the number obtained from reading left to right is considered, ignoring any leading zeros. Beto wins if the resulting number is a multiple of

99

, otherwise Ana wins. Determine which of the two players has a winning strategy and describe it.

1

Alero Numbers in the Centro

Let

n

be a positive integer with

k

digits. A number

m

is called an

alero

of

n

if there exist distinct digits

a_1

,

a_2

,

\dotsb

,

a_k

, all different from each other and from zero, such that

m

is obtained by adding the digit

a_i

to the

i

-th digit of

n

, and no sum exceeds 9. For example, if

n

=

2024

and we choose

a_1

=

2

,

a_2

=

1

,

a_3

=

5

,

a_4

=

3

, then

m

=

4177

is an alero of

n

, but if we choose the digits

a_1

=

2

,

a_2

=

1

,

a_3

=

5

,

a_4

=

6

, then we don't obtain an alero of

n

, because

4

+

6

exceeds

9

. Find the smallest

n

which is a multiple of

2024

that has an alero which is also a multiple of

2024

.

2024 Centroamerican and Caribbean Math Olympiad

Subcontests

Wim and his challenging little brother in the Centro

Tricky system of equations in the Centro

Nice triangular similarity problem in the Centro

Collinearity with altitudes and circumcircles in the Centro

Winning strategy for a 2024 cell-row in the Centro

Alero Numbers in the Centro