MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2000 Greece National Olympiad
2000 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
3
1
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Best constant
Find the maximum value of
k
k
k
such that
x
y
(
x
2
+
y
2
)
(
3
x
2
+
y
2
)
≤
1
k
\frac{xy}{\sqrt{(x^2 + y^2)(3x^2 + y^2)}}\leq \frac{1}{k}
(
x
2
+
y
2
)
(
3
x
2
+
y
2
)
x
y
≤
k
1
holds for all positive numbers
x
x
x
and
y
.
y.
y
.
2
1
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Prime number and perfect square
Find all prime numbers
p
p
p
such that
1
+
p
+
p
2
+
p
3
+
p
4
1 +p+p^2 +p^3 +p^4
1
+
p
+
p
2
+
p
3
+
p
4
is a perfect square.
1
1
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Rectangle and an arithmetic sequence
Consider a rectangle
A
B
C
D
ABCD
A
BC
D
with
A
B
=
a
AB = a
A
B
=
a
and
A
D
=
b
.
AD = b.
A
D
=
b
.
Let
l
l
l
be a line through
O
,
O,
O
,
the center of the rectangle, that cuts
A
D
AD
A
D
in
E
E
E
such that
A
E
/
E
D
=
1
/
2
AE/ED = 1/2
A
E
/
E
D
=
1/2
. Let
M
M
M
be any point on
l
,
l,
l
,
interior to the rectangle. Find the necessary and sufficient condition on
a
a
a
and
b
b
b
that the four distances from M to lines
A
D
,
A
B
,
D
C
,
B
C
AD, AB, DC, BC
A
D
,
A
B
,
D
C
,
BC
in this order form an arithmetic progression.
4
1
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Subsets.
The subsets
A
1
,
A
2
,
…
,
A
2000
A_1,A_2,\ldots ,A_{2000}
A
1
,
A
2
,
…
,
A
2000
of a finite set
M
M
M
satisfy
∣
A
i
∣
>
2
3
∣
M
∣
|A_i|>\frac{2}{3}|M|
∣
A
i
∣
>
3
2
∣
M
∣
for each
i
=
1
,
2
,
…
,
2000
i=1,2,\ldots ,2000
i
=
1
,
2
,
…
,
2000
. Prove that there exists
m
∈
M
m\in M
m
∈
M
which belongs to at least
1334
1334
1334
of the subsets
A
i
A_i
A
i
.