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Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2023 Junior Macedonian Mathematical Olympiad
2023 Junior Macedonian Mathematical Olympiad
Part of
JBMO TST - Macedonia
Subcontests
(5)
5
1
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Consider a $2023\times2023$ board split into unit squares.
Consider a
2023
×
2023
2023\times2023
2023
×
2023
board split into unit squares. Two unit squares are called adjacent is they share at least one vertex. Mahler and Srecko play a game on this board. Initially, Mahler has one piece placed on the square marked M, and Srecko has a piece placed on the square marked by S (see the attachment). The players alternate moving their piece, following three rules: 1. A piece can only be moved to a unit square adjacent to the one it is placed on. 2. A piece cannot be placed on a unit square on which a piece has been placed before (once used, a unit square can never be used again). 3. A piece cannot be moved to a unit square adjacent to the square occupied by the opponent’s piece. A player wins the game if his piece gets to the corner diagonally opposite to its starting position (i.e. Srecko moves to
s
p
s_p
s
p
, Mahler moves to
m
p
m_p
m
p
) or if the opponent has to move but has no legal move. Mahler moves first. Which player has a winning strategy?
4
1
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Perpendicular bisector meets the circumcircle of another triangle
We are given an acute
△
A
B
C
\triangle ABC
△
A
BC
with circumcenter
O
O
O
such that
B
C
<
A
B
BC<AB
BC
<
A
B
. The bisector of
∠
A
C
B
\angle ACB
∠
A
CB
meets the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at a second point
D
D
D
. The perpendicular bisector of
A
C
AC
A
C
meets the circumcircle of
△
B
O
D
\triangle BOD
△
BO
D
for the second time at
E
E
E
. The line
D
E
DE
D
E
meets the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
for the second time at
F
F
F
. Prove that the lines
C
F
CF
CF
,
O
E
OE
OE
and
A
B
AB
A
B
are concurrent.Proposed by Petar Filipovski
3
1
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Cyclic inequality with product of fractions
Let
a
a
a
,
b
b
b
and
c
c
c
be positive real numbers such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove the inequality
(
1
+
a
b
+
2
)
(
1
+
b
c
+
2
)
(
1
+
c
a
+
2
)
≥
216.
\left ( \frac{1+a}{b}+2 \right ) \left ( \frac{1+b}{c}+2 \right ) \left ( \frac{1+c}{a}+2 \right )\geq 216.
(
b
1
+
a
+
2
)
(
c
1
+
b
+
2
)
(
a
1
+
c
+
2
)
≥
216.
When does equality hold?Proposed by Anastasija Trajanova
2
1
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Superprime numbers
A positive integer is called superprime if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers.Proposed by Nikola Velov
1
1
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Group of kids forms bipartite graph
In a group of kids there are
2022
2022
2022
boys and
2023
2023
2023
girls. Every girl is a friend with exactly
2021
2021
2021
boys. Friendship is a symmetric relation: if A is a friend of B, then B is also a friend of A. Prove that it is not possible that all boys have the same number of girl friends. Proposed by the JMMO Problem Selection Committee