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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2021 Greece Junior Math Olympiad
2021 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
1
1
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min A=x+1/x+y+1/y if 2(x+y)=1+xy for x,y>0 , 2020 ISL A3 for juniors
If positive reals
x
,
y
x,y
x
,
y
are such that
2
(
x
+
y
)
=
1
+
x
y
2(x+y)=1+xy
2
(
x
+
y
)
=
1
+
x
y
, find the minimum value of expression
A
=
x
+
1
x
+
y
+
1
y
A=x+\frac{1}{x}+y+\frac{1}{y}
A
=
x
+
x
1
+
y
+
y
1
2
1
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2 player game, n-> n+p, where p is prime divisor of n
Anna and Basilis play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer
n
n
n
, the player whose turn is chooses a prime divisor
p
p
p
of
n
n
n
and writes the numbers
n
+
p
n+p
n
+
p
. In the board, is written at the start number
2
2
2
and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to
31
31
31
. Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.
4
1
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equal circumcircles wanted, 3 more circles related
Given a triangle
A
B
C
ABC
A
BC
with
A
B
<
B
C
<
A
C
AB<BC<AC
A
B
<
BC
<
A
C
inscribed in circle
(
c
)
(c)
(
c
)
. The circle
c
(
A
,
A
B
)
c(A,AB)
c
(
A
,
A
B
)
(with center
A
A
A
and radius
A
B
AB
A
B
) interects the line
B
C
BC
BC
at point
D
D
D
and the circle
(
c
)
(c)
(
c
)
at point
H
H
H
. The circle
c
(
A
,
A
C
)
c(A,AC)
c
(
A
,
A
C
)
(with center
A
A
A
and radius
A
C
AC
A
C
) interects the line
B
C
BC
BC
at point
Z
Z
Z
and the circle
(
c
)
(c)
(
c
)
at point
E
E
E
. Lines
Z
H
ZH
Z
H
and
E
D
ED
E
D
intersect at point
T
T
T
. Prove that the circumscribed circles of triangles
T
D
Z
TDZ
T
D
Z
and
T
E
H
TEH
TE
H
are equal.
3
1
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8^n+47 is prime
Determine whether exists positive integer
n
n
n
such that the number
A
=
8
n
+
47
A=8^n+47
A
=
8
n
+
47
is prime.