2019 Regional Olympiad of Mexico Southeast

Part of Regional Olympiad of Mexico Southeast

Subcontests

(6)

1

divisibility with primes

p\geq 3

a prime number,

a

b

integers such that

\gcd(a, b)=1

n

a natural number such that

p

a^{2^n}+b^{2^n}

2^{n+1}

p-1

1

inequality

n

a natural number and

A=\{1, 2, 3, \cdots, 2^{n+1}-1\}

. Prove that if we choose

2n+1

elements differents of the set

A

, then among them are three distinct number

a,b

c

bc<2a^2<4bc

1

Prove two lines are parallel

\Gamma

a circumference.

T

\Gamma

P

A

two points outside

\Gamma

PT

\Gamma

PA=PT

C

\Gamma (C\neq T)

AC

PC

intersect again

\Gamma

D

B

, respectively.

AB

\Gamma

E

DE

it´s parallel to

AP

1

Probablity problem

Eight teams are competing in a tournament all against all (every pair of team play exactly one time among them). There are not ties and both results of every game are equally probable. What is the probability that in the tournament every team had lose at least one game and won at least one game?

1

prove two lines are perpendicular

ABCD

a convex quadrilateral. Suppose that the circumference with center

B

BC

AD

F

and the circumference with center

A

AD

BC

E

DE

CF

are perpendicular.

1

found a multiple of 2019

Found the smaller multiple of

2019

abcabc\dots abc

a,b

c