MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2003 Greece National Olympiad
2003 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
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An operation in plane
On the set
Σ
\Sigma
Σ
of points of the plane
Π
\Pi
Π
we define the operation
∗
*
∗
which maps each pair
(
X
,
Y
)
(X, Y )
(
X
,
Y
)
of points in
Σ
\Sigma
Σ
to the point
Z
=
X
∗
Y
Z = X * Y
Z
=
X
∗
Y
that is symmetric to
X
X
X
with respect to
Y
.
Y .
Y
.
Consider a square
A
B
C
D
ABCD
A
BC
D
in
Π
\Pi
Π
. Is it possible, using the points
A
,
B
,
C
A, B, C
A
,
B
,
C
and applying the operation
∗
*
∗
finitely many times, to construct the point
D
?
D?
D
?
3
1
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Unique point
Given are a circle
C
\mathcal{C}
C
with center
K
K
K
and radius
r
,
r,
r
,
point
A
A
A
on the circle and point
R
R
R
in its exterior. Consider a variable line
e
e
e
through
R
R
R
that intersects the circle at two points
B
B
B
and
C
.
C.
C
.
Let
H
H
H
be the orthocenter of triangle
A
B
C
.
ABC.
A
BC
.
Show that there is a unique point
T
T
T
in the plane of circle
C
\mathcal{C}
C
such that the sum
H
A
2
+
H
T
2
HA^2 + HT^2
H
A
2
+
H
T
2
remains constant (as
e
e
e
varies.)
2
1
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System of equations
Find all real solutions of the system
{
x
2
+
y
2
−
z
(
x
+
y
)
=
2
,
y
2
+
z
2
−
x
(
y
+
z
)
=
4
,
z
2
+
x
2
−
y
(
z
+
x
)
=
8.
\begin{cases}x^2 + y^2 - z(x + y) = 2, \\ y^2 + z^2 - x(y + z) = 4, \\ z^2 + x^2 - y(z + x) = 8.\end{cases}
⎩
⎨
⎧
x
2
+
y
2
−
z
(
x
+
y
)
=
2
,
y
2
+
z
2
−
x
(
y
+
z
)
=
4
,
z
2
+
x
2
−
y
(
z
+
x
)
=
8.
1
1
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Inequality
If
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are positive numbers satisfying
a
3
+
b
3
+
3
a
b
=
c
+
d
=
1
,
a^3 + b^3 +3ab = c + d = 1,
a
3
+
b
3
+
3
ab
=
c
+
d
=
1
,
prove that
(
a
+
1
a
)
3
+
(
b
+
1
b
)
3
+
(
c
+
1
c
)
3
+
(
d
+
1
d
)
3
≥
40.
\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.
(
a
+
a
1
)
3
+
(
b
+
b
1
)
3
+
(
c
+
c
1
)
3
+
(
d
+
d
1
)
3
≥
40.