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National and Regional Contests
Greece Contests
Greece JBMO TST
2005 Greece JBMO TST
2005 Greece JBMO TST
Part of
Greece JBMO TST
Subcontests
(4)
4
1
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find n >=3 such that n | (n-2)!
Find all the positive integers
n
,
n
≥
3
n , n\ge 3
n
,
n
≥
3
such that
n
∣
(
n
−
2
)
!
n\mid (n-2)!
n
∣
(
n
−
2
)!
1
1
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can 9 convex 6-angled polygons construct a convex 39-angled polygon?
Examine if we can place
9
9
9
convex
6
6
6
-angled polygons the one next to the other (with common only one side or part of her) to construct a convex
39
39
39
-angled polygon.
2
1
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(x^2-y^2)/ (2x^2+1) + (y^2-z^2)/(2y^2+1)+(z^2-x^2)/ (2z^2+1)<=0
Prove that for each
x
,
y
,
z
∈
R
x,y,z \in R
x
,
y
,
z
∈
R
it holds that
x
2
−
y
2
2
x
2
+
1
+
y
2
−
z
2
2
y
2
+
1
+
z
2
−
x
2
2
z
2
+
1
≤
0
\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0
2
x
2
+
1
x
2
−
y
2
+
2
y
2
+
1
y
2
−
z
2
+
2
z
2
+
1
z
2
−
x
2
≤
0
3
1
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Clever and good
I have a very good solution of this but I want to see others.Let the midpoint
M
M
M
of the side
A
B
AB
A
B
of an inscribed quardiletar,
A
B
C
D
ABCD
A
BC
D
.Let
P
P
P
the point of intersection of
M
C
MC
MC
with
B
D
BD
B
D
. Let the parallel from the point
C
C
C
to the
A
P
AP
A
P
which intersects the
B
D
BD
B
D
at
S
S
S
. If
C
A
D
CAD
C
A
D
angle=
P
A
B
PAB
P
A
B
angle=
B
M
C
2
\frac{BMC}{2}
2
BMC
angle, prove that
B
P
=
S
D
BP=SD
BP
=
S
D
.