MathDB

1

Viktor and Natalia play a game on a cube

Viktor and Natalia play a colouring game with a 3-dimensional cube taking turns alternatingly. Viktor goes first, and on each of his turns, he selects an unpainted edge, and paints it violet. On each of Natalia's turns, she selects an unpainted edge, or at most once during the game a face diagonal, and paints it neon green. If the player on turn cannot make a legal move, then the turn switches to the other player. The game ends when nobody can make any more legal moves.
Natalia wins if at the end of the game every vertex of the cube can be reached from every other vertex by traveling only along neon green segments (edges or diagonal), otherwise Viktor wins.
Who has a winning strategy? (Prove your answer.)
Proposed by Viktor Simjanoski

Problem 3

1

Upper bound for sum of k-th roots

Suppose that

a_1, a_2, \dots a_{2021}

are non-negative numbers such that

\sum_{k=1}^{2021} a_k=1

. Prove that

\sum_{k=1}^{2021}\sqrt[k]{a_1 a_2\dots a_k} \leq 3.

Problem 2

1

Sequence generating all pairs of coprime positive integers

Define a sequence:

x_0=1

and for all

n\ge 0

,

x_{2n+1}=x_{n}

and

x_{2n+2}=x_{n}+x_{n+1}

. Prove that for any relatively prime positive integers

a

and

b

, there is a non-negative integer

n

such that

a=x_n

and

b=x_{n+1}

.

Problem 1

1

Arbitary point on nine-point circle and cyclic quadrilaterals

Let

ABC

be an acute triangle. Let

D

,

E

and

F

be the feet of the altitudes from

A

,

B

and

C

respectively and let

H

be the orthocenter of

\triangle ABC

. Let

X

be an arbitrary point on the circumcircle of

\triangle DEF

and let the circumcircles of

\triangle EHX

and

\triangle FHX

intersect the second time the lines

CF

and

BE

second at

Y

and

Z

, respectively. Prove that the line

YZ

passes through the midpoint of

BC

.

2021 Macedonian Balkan MO TST

Subcontests

Viktor and Natalia play a game on a cube

Upper bound for sum of k-th roots

Sequence generating all pairs of coprime positive integers

Arbitary point on nine-point circle and cyclic quadrilaterals