MathDB

1

Centroamerican 2018, problem 6

A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled

0, 1,\ldots, 2017

are marked in a circumference and each couple is placed on a different point. For

i\geq1

, let

s_i=i\ (\textrm{mod}\ 2018)

and

r_i=2i\ (\textrm{mod}\ 2018)

. The dance begins at minute

0

. On the

i

-th minute, the couple at point

s_i

(if there's any) moves to point

r_i

, the couple on point

r_i

(if there's any) drops out, and the dance continues with the remaining couples. The dance ends after

2018^2

minutes. Determine how many couples remain at the end.
Note: If

r_i=s_i

, the couple on

s_i

stays there and does not drop out.

5

1

Centroamerican 2018, problem 5

Let

n

be a positive integer,

1<n<2018

. For each

i=1, 2, \ldots ,n

we define the polynomial

S_i(x)=x^2-2018x+l_i

, where

l_1, l_2, \ldots, l_n

are distinct positive integers. If the polynomial

S_1(x)+S_2(x)+\cdots+S_n(x)

has at least an integer root, prove that at least one of the

l_i

is greater or equal than

2018

.

4

1

Centroamerican 2018, problem 4

Determine all triples

(p, q, r)

of positive integers, where

p, q

are also primes, such that

\frac{r^2-5q^2}{p^2-1}=2

.

3

1

Centroamerican 2018, problem 3

Let

x, y

be real numbers such that

x-y, x^2-y^2, x^3-y^3

are all prime numbers. Prove that

x-y=3

.
EDIT: Problem submitted by Leonel Castillo, Panama.

2

1

Centroamerican 2018, problem 2

Let

\Delta ABC

be a triangle inscribed in the circumference

\omega

of center

O

. Let

T

be the symmetric of

C

respect to

O

and

T'

be the reflection of

T

respect to line

AB

. Line

BT'

intersects

\omega

again at

R

. The perpendicular to

CT

through

O

intersects line

AC

at

L

. Let

N

be the intersection of lines

TR

and

AC

. Prove that

\overline{CN}=2\overline{AL}

.

1

Centroamerican 2018, Problem 1

There are 2018 cards numbered from 1 to 2018. The numbers of the cards are visible at all times. Tito and Pepe play a game. Starting with Tito, they take turns picking cards until they're finished. Then each player sums the numbers on his cards and whoever has an even sum wins. Determine which player has a winning strategy and describe it.

P.S. Proposed by yours truly :-D

2018 Centroamerican and Caribbean Math Olympiad

Subcontests

Centroamerican 2018, problem 6

Centroamerican 2018, problem 5

Centroamerican 2018, problem 4

Centroamerican 2018, problem 3

Centroamerican 2018, problem 2

Centroamerican 2018, Problem 1