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National and Regional Contests
Greece Contests
Greece JBMO TST
2015 Greece JBMO TST
2015 Greece JBMO TST
Part of
Greece JBMO TST
Subcontests
(4)
2
1
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two circumcenters and one orthocenter, vertices of parallelogram
Let
A
B
C
ABC
A
BC
be an acute triangle inscribed in a circle of center
O
O
O
. If the altitudes
B
D
,
C
E
BD,CE
B
D
,
CE
intersect at
H
H
H
and the circumcenter of
△
B
H
C
\triangle BHC
△
B
H
C
is
O
1
O_1
O
1
, prove that
A
H
O
1
O
AHO_1O
A
H
O
1
O
is a parallelogram.
4
1
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112 groups of 11 members each with one common in every two groups
Pupils of a school are divided into
112
112
112
groups, of
11
11
11
members each. Any two groups have exactly one common pupil. Prove that: a) there is a pupil that belongs to at least
12
12
12
groups. b) there is a pupil that belongs to all the groups.
3
1
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(n+1)2^n, (n+3)2^{n+2} not perfect squares for the same n
Prove that there is not a positive integer
n
n
n
such that numbers
(
n
+
1
)
2
n
,
(
n
+
3
)
2
n
+
2
(n+1)2^n, (n+3)2^{n+2}
(
n
+
1
)
2
n
,
(
n
+
3
)
2
n
+
2
are both perfect squares.
1
1
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(3x+y)(3y+z)(3z+x) \ge 64xyz if x,y,z>0
If
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
, prove that
(
3
x
+
y
)
(
3
y
+
z
)
(
3
z
+
x
)
≥
64
x
y
z
(3x+y)(3y+z)(3z+x) \ge 64xyz
(
3
x
+
y
)
(
3
y
+
z
)
(
3
z
+
x
)
≥
64
x
yz
. When we have equality;