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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2017 Greece Junior Math Olympiad
2017 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
3
1
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1\p=1/a^2+1/b^2 diophantine with prime p (Greece Juniors 2017 p3)
Find all triplets
(
a
,
b
,
p
)
(a,b,p)
(
a
,
b
,
p
)
where
a
,
b
a,b
a
,
b
are positive integers and
p
p
p
is a prime number satisfying:
1
p
=
1
a
2
+
1
b
2
\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}
p
1
=
a
2
1
+
b
2
1
4
1
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largest possible value of n people in board game (Greece Juniors 2017 p4)
A group of
n
n
n
people play a board game with the following rules: 1) In each round of the game exactly
3
3
3
people play 2) The game ends after exactly
n
n
n
rounds 3) Every pair of players has played together at least at one round Find the largest possible value of
n
n
n
1
1
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area computational with square (Greece Junior 2017)
Let
A
B
C
D
ABCD
A
BC
D
be a square of side
a
a
a
. On side
A
D
AD
A
D
consider points
E
E
E
and
Z
Z
Z
such that
D
E
=
a
/
3
DE=a/3
D
E
=
a
/3
and
A
Z
=
a
/
4
AZ=a/4
A
Z
=
a
/4
. If the lines
B
Z
BZ
BZ
and
C
E
CE
CE
intersect at point
H
H
H
, calculate the area of the triangle
B
C
H
BCH
BC
H
in terms of
a
a
a
.
2
1
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Solve the system
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
is positive. Solve:
{
x
(
6
−
y
)
=
9
y
(
6
−
z
)
=
9
z
(
6
−
x
)
=
9
\begin{cases}{x\left( {6 - y} \right) = 9}\\ {y\left( {6 - z} \right) = 9}\\ {z\left( {6 - x} \right) = 9}\end{cases}
⎩
⎨
⎧
x
(
6
−
y
)
=
9
y
(
6
−
z
)
=
9
z
(
6
−
x
)
=
9