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National and Regional Contests
Greece Contests
Greece National Olympiad
2019 Greece National Olympiad
2019 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
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2019 Greece National Olympiad Q4
Given a
n
×
m
n\times m
n
×
m
grid we play the following game . Initially we place
M
M
M
tokens in each of
M
M
M
empty cells and at the end of the game we need to fill the whole grid with tokens.For that purpose we are allowed to make the following move:If an empty cell shares a common side with at least two other cells that contain a token then we can place a token in this cell.Find the minimum value of
M
M
M
in terms of
m
,
n
m,n
m
,
n
that enables us to win the game.
3
1
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2019 Greece National Olympiad Q3
Find all positive rational
(
x
,
y
)
(x,y)
(
x
,
y
)
that satisfy the equation :
y
x
y
=
y
+
1
yx^y=y+1
y
x
y
=
y
+
1
2
1
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2019 Greece National Olympiad Q2
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
.Let
O
O
O
be the center of it's circumcircle and
D
D
D
be the center of minor arc \overarc{AB}.Line
A
D
AD
A
D
intersects
B
C
BC
BC
at
E
E
E
and the circumcircle of
B
D
E
BDE
B
D
E
intersects
A
B
AB
A
B
at
Z
Z
Z
,(
Z
≠
B
Z\not=B
Z
=
B
).The circumcircle of
A
D
Z
ADZ
A
D
Z
intersects
A
C
AC
A
C
at
H
H
H
,(
H
≠
A
H\not=A
H
=
A
),prove that
B
E
=
A
H
BE=AH
BE
=
A
H
.
1
1
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2019 Greece National Olympiad Q1
Define the sequnce
(
a
n
)
n
≥
1
{(a_n)}_{n\ge1}
(
a
n
)
n
≥
1
by
a
1
=
1
a_1=1
a
1
=
1
and
a
n
=
5
a
n
−
1
+
3
n
−
1
a_n=5a_{n-1}+3^{n-1}
a
n
=
5
a
n
−
1
+
3
n
−
1
for
n
≥
2
n\ge2
n
≥
2
. Find the greatest power of
2
2
2
that divides
a
2
2019
a_{2^{2019}}
a
2
2019
.