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Contests
International Contests
Mediterranean Mathematics Olympiad
2002 Mediterranean Mathematics Olympiad
2002 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
3
1
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CD contains the circumcenter O of ABC
In an acute-angled triangle
A
B
C
ABC
A
BC
,
M
M
M
and
N
N
N
are points on the sides
A
C
AC
A
C
and
B
C
BC
BC
respectively, and
K
K
K
the midpoint of
M
N
MN
MN
. The circumcircles of triangles
A
C
N
ACN
A
CN
and
B
C
M
BCM
BCM
meet again at a point
D
D
D
. Prove that the line
C
D
CD
C
D
contains the circumcenter
O
O
O
of
△
A
B
C
\triangle ABC
△
A
BC
if and only if
K
K
K
is on the perpendicular bisector of
A
B
.
AB.
A
B
.
2
1
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Find all possible values for a
Suppose
x
,
y
,
a
x, y, a
x
,
y
,
a
are real numbers such that
x
+
y
=
x
3
+
y
3
=
x
5
+
y
5
=
a
x+y = x^3 +y^3 = x^5 +y^5 = a
x
+
y
=
x
3
+
y
3
=
x
5
+
y
5
=
a
. Find all possible values of
a
.
a.
a
.
4
1
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a^2+b^2+c^2=1
If
a
,
b
,
c
a, b, c
a
,
b
,
c
are non-negative real numbers with a^2 \plus{} b^2 \plus{} c^2 \equal{} 1, prove that: \frac {a}{b^2 \plus{} 1} \plus{} \frac {b}{c^2 \plus{} 1} \plus{} \frac {c}{a^2 \plus{} 1} \geq \frac {3}{4}(a\sqrt {a} \plus{} b\sqrt {b} \plus{} c\sqrt {c})^2
1
1
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Mediterranean mathematical contest 2002 (1)
Find all natural numbers
x
,
y
x,y
x
,
y
such that
y
∣
(
x
2
+
1
)
y| (x^{2}+1)
y
∣
(
x
2
+
1
)
and
x
2
∣
(
y
3
+
1
)
x^{2}| (y^{3}+1)
x
2
∣
(
y
3
+
1
)
.