MathDB

1

Friends at a meeting

2021

people at a meeting. It is known that one person at the meeting doesn't have any friends there and another person has only one friend there. In addition, it is true that, given any

4

people, at least

2

of them are friends. Show that there are

2018

people at the meeting that are all friends with each other. Note. If

A

is friend of

B

then

B

is a friend of

A

.

6

1

More intersections and cyclic quadrilaterals

Let

ABC

be a triangle with

AB<AC

and let

M

be the midpoint of

AC

. A point

P

(other than

B

) is chosen on the segment

BC

in such a way that

AB=AP

. Let

D

be the intersection of

AC

with the circumcircle of

\bigtriangleup ABP

distinct from

A

, and

E

be the intersection of

PM

with the circumcircle of

\bigtriangleup ABP

distinct from

P

. Let

K

be the intersection of lines

AP

and

DE

. Let

F

be a point on

BC

(other than

P

) such that

KP=KF

. Show that

C,\ D,\ E

and

F

lie on the same circle.

5

1

Inequality with strange square condition

Let

n \geq 3

be an integer and

a_1,a_2,...,a_n

be positive real numbers such that

m

is the smallest and

M

is the largest of these numbers. It is known that for any distinct integers

1 \leq i,j,k \leq n

, if

a_i \leq a_j \leq a_k

then

a_ia_k \leq a_j^2

. Show that

a_1a_2 \cdots a_n \geq m^2M^{n-2}

and determine when equality holds

3

1

A table and a restless mouse

In a table consisting of

2021\times 2021

unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than the initial one if it is black). What is the maximum number of squares that can be colored black?

1

Parcera triples

An ordered triple

(p, q, r)

of prime numbers is called parcera if

p

divides

q^2-4

,

q

divides

r^2-4

and

r

divides

p^2-4

. Find all parcera triples.

2

1

Intersections and a vertex form a cyclic quadrilateral

Let

ABC

be a triangle and let

\Gamma

be its circumcircle. Let

D

be a point on

AB

such that

CD

is parallel to the line tangent to

\Gamma

at

A

. Let

E

be the intersection of

CD

with

\Gamma

distinct from

C

, and

F

the intersection of

BC

with the circumcircle of

\bigtriangleup ADC

distinct from

C

. Finally, let

G

be the intersection of the line

AB

and the internal bisector of

\angle DCF

. Show that

E,\ G,\ F

and

C

lie on the same circle.

2021 Centroamerican and Caribbean Math Olympiad

Subcontests

Friends at a meeting

More intersections and cyclic quadrilaterals

Inequality with strange square condition

A table and a restless mouse

Parcera triples

Intersections and a vertex form a cyclic quadrilateral