2023 Junior Balkan Mathematical Olympiad

Part of Junior Balkan MO

Subcontests

(4)

1

Conditional geometry

ABC

be an acute triangle with circumcenter

O

D

be the foot of the altitude from

A

BC

M

be the midpoint of

OD

O_b

O_c

are the circumcenters of triangles

AOC

AOB

, respectively. If

AO=AD

, prove that points

A

O_b

M

O_c

are concyclic.
Marin Hristov and Bozhidar Dimitrov, Bulgaria

1

Alice and Bob strike again, and again, and again...

Alice and Bob play the following game on a

100\times 100

grid, taking turns, with Alice starting first. Initially the grid is empty. At their turn, they choose an integer from

1

100^2

that is not written yet in any of the cells and choose an empty cell, and place it in the chosen cell. When there is no empty cell left, Alice computes the sum of the numbers in each row, and her score is the maximum of these

100

numbers. Bob computes the sum of the numbers in each column, and his score is the maximum of these

100

numbers. Alice wins if her score is greater than Bob's score, Bob wins if his score is greater than Alice's score, otherwise no one wins.
Find if one of the players has a winning strategy, and if so which player has a winning strategy.
Théo Lenoir, France

1

Tricky inequality

Prove that for all non-negative real numbers

x,y,z

, not all equal to

0

, the following inequality holds

\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.

Determine all the triples

(x,y,z)

for which the equality holds.
Milan Mitreski, Serbia

1

Classical factorial number theory

(a,b)

of positive integers such that

a!+b

b!+a

are both powers of

5

.
Nikola Velov, North Macedonia