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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2019 Greece Junior Math Olympiad
2019 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
1
1
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x^2+y^2+25z^2=6xz+8yz and 3x^2+2y^2+z^2=240 (Greece Juniors 2019 p1)
Find all triplets of real numbers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
that are solutions to the system of equations
x
2
+
y
2
+
25
z
2
=
6
x
z
+
8
y
z
x^2+y^2+25z^2=6xz+8yz
x
2
+
y
2
+
25
z
2
=
6
x
z
+
8
yz
3
x
2
+
2
y
2
+
z
2
=
240
3x^2+2y^2+z^2=240
3
x
2
+
2
y
2
+
z
2
=
240
4
1
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Combinatorics
In the table are written the positive integers
1
,
2
,
3
,
.
.
.
,
2018
1, 2,3,...,2018
1
,
2
,
3
,
...
,
2018
. John and Mary have the ability to make together the following move: They select two of the written numbers in the table, let
a
,
b
a,b
a
,
b
and they replace them with the numbers
5
a
−
2
b
5a-2b
5
a
−
2
b
and
3
a
−
4
b
3a-4b
3
a
−
4
b
. John claims that after a finite number of such moves, it is possible to triple all the numbers in the table, e.g. have the numbers:
3
,
6
,
9
,
.
.
.
,
6054
3, 6, 9,...,6054
3
,
6
,
9
,
...
,
6054
. Mary thinks a while and replies that this is not possible. Who of them is right?
3
1
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Number Theory
Determine all positive integers equal to 13 times the sum of their digits.
2
1
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concyclic lead to concyclic (Greece Junior 2019)
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in circle of center
O
O
O
. The perpendicular on the midpoint
E
E
E
of side
B
C
BC
BC
intersects line
A
B
AB
A
B
at point
Z
Z
Z
. The circumscribed circle of the triangle
C
E
Z
CEZ
CEZ
, intersects the side
A
B
AB
A
B
for the second time at point
H
H
H
and line
C
D
CD
C
D
at point
G
G
G
different than
D
D
D
. Line
E
G
EG
EG
intersects line
A
D
AD
A
D
at point
K
K
K
and line
C
H
CH
C
H
at point
L
L
L
. Prove that the points
A
,
H
,
L
,
K
A,H,L,K
A
,
H
,
L
,
K
are concyclic, e.g. lie on the same circle.