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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2016 Greece Junior Math Olympiad
2016 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
2
1
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Algebra equations
Given is that
x
,
y
,
z
x, y, z
x
,
y
,
z
are real numbers, different from 0,
x
x
x
and
z
z
z
are different, such that
(
x
+
y
)
2
+
(
2
−
x
y
)
=
9
(x+y) ^2+(2-xy)=9
(
x
+
y
)
2
+
(
2
−
x
y
)
=
9
and
(
y
+
z
)
2
−
(
3
+
y
z
)
=
4
(y+z) ^2-(3+yz)=4
(
y
+
z
)
2
−
(
3
+
yz
)
=
4
Find the value of
A
=
(
x
/
y
+
y
2
/
x
2
+
z
3
/
x
2
y
)
(
y
/
z
+
z
2
/
y
2
+
x
3
/
y
2
z
)
(
z
/
x
+
x
2
/
z
2
+
y
3
/
z
2
x
)
=
?
A=(x/y+y^2/x^2+z^3/x^2y)(y/z+z^2/y^2+x^3/y^2z)(z/x+x^2/z^2+y^3/z^2x)=?
A
=
(
x
/
y
+
y
2
/
x
2
+
z
3
/
x
2
y
)
(
y
/
z
+
z
2
/
y
2
+
x
3
/
y
2
z
)
(
z
/
x
+
x
2
/
z
2
+
y
3
/
z
2
x
)
=
?
4
1
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Easy Combinatorics
Find the number ot 6-tuples
(
x
1
,
x
2
,
.
.
.
,
x
6
)
(x_1, x_2,...,x_6)
(
x
1
,
x
2
,
...
,
x
6
)
, where
x
i
=
0
,
1
o
r
2
x_i=0,1 or 2
x
i
=
0
,
1
or
2
and
x
1
+
x
2
+
.
.
.
+
x
6
x_1+x_2+...+x_6
x
1
+
x
2
+
...
+
x
6
is even
1
1
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Easy Number Theory
If
n
n
n
is positive integer and
p
,
q
,
r
p, q, r
p
,
q
,
r
are primes solve the system:
p
q
r
=
n
pqr=n
pq
r
=
n
and
(
p
+
1
)
(
q
+
1
)
r
=
n
+
138
(p+1)(q+1)r=n+138
(
p
+
1
)
(
q
+
1
)
r
=
n
+
138
3
1
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perpencularities on trapezoid with right angles (Greece Junior 2016)
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid (
A
D
/
/
B
C
AD//BC
A
D
//
BC
) with
∠
A
=
∠
B
=
9
0
o
\angle A=\angle B= 90^o
∠
A
=
∠
B
=
9
0
o
and
A
D
<
B
C
AD<BC
A
D
<
BC
. Let
E
E
E
be the intersection point of the non parallel sides
A
B
AB
A
B
and
C
D
CD
C
D
,
Z
Z
Z
be the symmetric point of
A
A
A
wrt line
B
C
BC
BC
and
M
M
M
be the midpoint of
E
Z
EZ
EZ
. If it is given than line
C
M
CM
CM
is perpendicular on line
D
Z
DZ
D
Z
, then prove that line
Z
C
ZC
ZC
is perpendicular on line
E
C
EC
EC
.