MathDB

1

Junior Balkan Mathematical Olympiad 2020- P4

p

and

q

such that

1 + \frac{p^q - q^p}{p + q}

is a prime number.
Proposed by Dorlir Ahmeti, Albania

3

1

Junior Balkan Mathematical Olympiad 2020- P3

Alice and Bob play the following game: Alice picks a set

A = \{1, 2, ..., n \}

for some natural number

n \ge 2

. Then, starting from Bob, they alternatively choose one number from the set

A

, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by

1

from an already chosen number. The game ends when all numbers from the set

A

are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy.
Proposed by Demetres Christofides, Cyprus

2

1

Junior Balkan Mathematical Olympiad 2020- P2

Let

\triangle ABC

be a right-angled triangle with

\angle BAC = 90^{\circ}

and let

E

be the foot of the perpendicular from

A

to

BC

. Let

Z \ne A

be a point on the line

AB

with

AB = BZ

. Let

(c)

be the circumcircle of the triangle

\triangle AEZ

. Let

D

be the second point of intersection of

(c)

with

ZC

and let

F

be the antidiametric point of

D

with respect to

(c)

. Let

P

be the point of intersection of the lines

FE

and

CZ

. If the tangent to

(c)

at

Z

meets

PA

at

T

, prove that the points

T

,

E

,

B

,

Z

are concyclic.
Proposed by Theoklitos Parayiou, Cyprus

1

Junior Ballkan Mathematical Olympiad 2020, Problem 1

Find all triples

(a,b,c)

of real numbers such that the following system holds:

\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}

Proposed by Dorlir Ahmeti, Albania

2020 Junior Balkаn MO

Subcontests

Junior Balkan Mathematical Olympiad 2020- P4

Junior Balkan Mathematical Olympiad 2020- P3

Junior Balkan Mathematical Olympiad 2020- P2

Junior Ballkan Mathematical Olympiad 2020, Problem 1