MathDB

1

Junior Balkan Mathematical Olympiad 2021- P4

M

be a subset of the set of

2021

integers

\{1, 2, 3, ..., 2021\}

such that for any three elements (not necessarily distinct)

a, b, c

of

M

we have

|a + b - c | > 10

. Determine the largest possible number of elements of

M

.

3

1

Junior Balkan Mathematical Olympiad 2021- P3

Let

ABC

be an acute scalene triangle with circumcenter

O

. Let

D

be the foot of the altitude from

A

to the side

BC

. The lines

BC

and

AO

intersect at

E

. Let

s

be the line through

E

perpendicular to

AO

. The line

s

intersects

AB

and

AC

at

K

and

L

, respectively. Denote by

\omega

the circumcircle of triangle

AKL

. Line

AD

intersects

\omega

again at

X

. Prove that

\omega

and the circumcircles of triangles

ABC

and

DEX

have a common point.

2

1

Junior Balkan Mathematical Olympiad 2021- P2

For any set

A = \{x_1, x_2, x_3, x_4, x_5\}

of five distinct positive integers denote by

S_A

the sum of its elements, and denote by

T_A

the number of triples

(i, j, k)

with

1 \le i < j < k \le 5

for which

x_i + x_j + x_k

divides

S_A

. Find the largest possible value of

T_A

.

1

Junior Balkan Mathematical Olympiad 2021- P1

Let

n

(

n \ge 1

) be an integer. Consider the equation

2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)

,
where

x

is the unknown real variable.
(a) Solve the equation for

n = 8

. (b) Prove that there exists an integer

n

for which the equation has at least

2021

solutions. (For any real number

y

by

\lfloor{y} \rfloor

we denote the largest integer

m

such that

m \le y

.)

2021 Junior Balkаn Mathematical Olympiad

Subcontests

Junior Balkan Mathematical Olympiad 2021- P4

Junior Balkan Mathematical Olympiad 2021- P3

Junior Balkan Mathematical Olympiad 2021- P2

Junior Balkan Mathematical Olympiad 2021- P1