MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2004 Greece National Olympiad
2004 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
3
1
Hide problems
Some fixed point
Consider a circle
K
(
O
,
r
)
K(O,r)
K
(
O
,
r
)
and a point
A
A
A
outside
K
.
K.
K
.
A line
ϵ
\epsilon
ϵ
different from
A
O
AO
A
O
cuts
K
K
K
at
B
B
B
and
C
,
C,
C
,
where
B
B
B
lies between
A
A
A
and
C
.
C.
C
.
Now the symmetric line of
ϵ
\epsilon
ϵ
with respect to axis of symmetry the line
A
O
AO
A
O
cuts
K
K
K
at
E
E
E
and
D
,
D,
D
,
where
E
E
E
lies between
A
A
A
and
D
.
D.
D
.
Show that the diagonals of the quadrilateral
B
C
D
E
BCDE
BC
D
E
intersect in a fixed point.
4
1
Hide problems
Minimum of maximum element
Let
M
⊂
N
∗
M\subset \Bbb{N}^*
M
⊂
N
∗
such that
∣
M
∣
=
2004.
|M|=2004.
∣
M
∣
=
2004.
If no element of
M
M
M
is equal to the sum of any two elements of
M
,
M,
M
,
find the least value that the greatest element of
M
M
M
can take.
2
1
Hide problems
There doesnt exist integers such that...
If
m
≥
2
m\geq 2
m
≥
2
show that there does not exist positive integers
x
1
,
x
2
,
.
.
.
,
x
m
,
x_1, x_2, ..., x_m,
x
1
,
x
2
,
...
,
x
m
,
such that
x
1
<
x
2
<
.
.
.
<
x
m
and
1
x
1
3
+
1
x
2
3
+
.
.
.
+
1
x
m
3
=
1.
x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.
x
1
<
x
2
<
...
<
x
m
and
x
1
3
1
+
x
2
3
1
+
...
+
x
m
3
1
=
1.
1
1
Hide problems
Best constant
Find the greatest value of
M
M
M
∈
R
\in \mathbb{R}
∈
R
such that the following inequality is true
∀
\forall
∀
x
,
y
,
z
x, y, z
x
,
y
,
z
∈
R
\in \mathbb{R}
∈
R
x
4
+
y
4
+
z
4
+
x
y
z
(
x
+
y
+
z
)
≥
M
(
x
y
+
y
z
+
z
x
)
2
x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2
x
4
+
y
4
+
z
4
+
x
yz
(
x
+
y
+
z
)
≥
M
(
x
y
+
yz
+
z
x
)
2
.