2021 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

BMO 2021 problem 3

a, b

c

be positive integers satisfying the equation

(a, b) + [a, b]=2021^c

|a-b|

is a prime number, prove that the number

(a+b)^2+4

is composite.
Proposed by Serbia

1

BMO 2021 Problem 4

Problem 4. Angel has a warehouse, which initially contains

100

100

pieces of rubbish each. Each morning, Angel performs exactly one of the following moves: (a) He clears every piece of rubbish from a single pile. (b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves: (a) He adds one piece of rubbish to each non-empty pile. (b) He creates a new pile with one piece of rubbish.
What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?

1

BMO 2021 problem 2

Find all functions

f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}

f(x+f(x)+f(y))=2f(x)+y

for all positive reals

x,y

.
Proposed by Athanasios Kontogeorgis, Greece

1

BMO 2021 problem 1

ABC

be a triangle with

AB<AC

\omega

be a circle passing through

B, C

and assume that

A

\omega

X, Y

\omega

\angle BXA=\angle AYC

. Suppose also that

X

C

lie on opposite sides of the line

AB

Y

B

lie on opposite sides of the line

AC

. Show that, as

X, Y

\omega

XY

passes through a fixed point.
Proposed by Aaron Thomas, UK