2022 Junior Balkan Mathematical Olympiad

Part of Junior Balkan MO

Subcontests

(4)

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2022 Junior Balkan MO, Problem 4

We call an even positive integer

n

nice if the set

\{1, 2, \dots, n\}

can be partitioned into

\frac{n}{2}

two-element subsets, such that the sum of the elements in each subset is a power of

3

6

is nice, because the set

\{1, 2, 3, 4, 5, 6\}

can be partitioned into subsets

\{1, 2\}

\{3, 6\}

\{4, 5\}

. Find the number of nice positive integers which are smaller than

3^{2022}

1

2022 Junior Balkan MO, Problem 3

Find all quadruples of positive integers

(p, q, a, b)

p

q

are prime numbers and

a > 1

p^a = 1 + 5q^b.

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2022 Junior Balkan MO, Problem 2

ABC

be an acute triangle such that

AH = HD

H

is the orthocenter of

ABC

D \in BC

is the foot of the altitude from the vertex

A

\ell

denote the line through

H

which is tangent to the circumcircle of the triangle

BHC

S

T

be the intersection points of

\ell

AB

AC

, respectively. Denote the midpoints of

BH

CH

M

N

, respectively. Prove that the lines

SM

TN

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2022 Junior Balkan MO, Problem 1

Find all pairs of positive integers

(a, b)

11ab \le a^3 - b^3 \le 12ab.