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Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2006 Greece National Olympiad
2006 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(3)
1
1
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5-digit numbers with digits in ascending order
How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit?
4
1
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Greek salad : functions and ... a question !!!
Does there exist a function
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
, which satisfies both conditions : a)
f
(
x
+
y
+
z
)
≤
3
(
x
y
+
y
z
+
z
x
)
f( x + y + z) \leq 3(xy + yz + zx)
f
(
x
+
y
+
z
)
≤
3
(
x
y
+
yz
+
z
x
)
for all real numbers
x
,
y
,
z
x , y , z
x
,
y
,
z
and b) there exist function
g
g
g
and natural number
n
n
n
, such that
g
(
g
(
x
)
)
=
x
2
n
+
1
g(g(x)) = x ^ {2n + 1}
g
(
g
(
x
))
=
x
2
n
+
1
and
f
(
g
(
x
)
)
=
(
g
(
x
)
)
2
f(g(x)) = (g(x)) ^2
f
(
g
(
x
))
=
(
g
(
x
)
)
2
for every real number
x
x
x
?
3
1
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It seems difficult ! harmonic ratios?
Let a triangle
A
B
C
ABC
A
BC
and the cevians
A
L
,
B
N
,
C
M
AL, BN , CM
A
L
,
BN
,
CM
such that
A
L
AL
A
L
is the bisector of angle
A
A
A
. If
∠
A
L
B
=
∠
A
N
M
\angle ALB = \angle ANM
∠
A
L
B
=
∠
A
NM
, prove that
∠
M
N
L
=
90
\angle MNL = 90
∠
MN
L
=
90
.