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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2004 Mediterranean Mathematics Olympiad
2004 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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Prove that the triangle is equilateral
Let
z
1
,
z
2
,
z
3
z_1, z_2, z_3
z
1
,
z
2
,
z
3
be pairwise distinct complex numbers satisfying
∣
z
1
∣
=
∣
z
2
∣
=
∣
z
3
∣
=
1
|z_1| = |z_2| = |z_3| = 1
∣
z
1
∣
=
∣
z
2
∣
=
∣
z
3
∣
=
1
and
1
2
+
∣
z
1
+
z
2
∣
+
1
2
+
∣
z
2
+
z
3
∣
+
1
2
+
∣
z
3
+
z
1
∣
=
1.
\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.
2
+
∣
z
1
+
z
2
∣
1
+
2
+
∣
z
2
+
z
3
∣
1
+
2
+
∣
z
3
+
z
1
∣
1
=
1.
If the points
A
(
z
1
)
,
B
(
z
2
)
,
C
(
z
3
)
A(z_1),B(z_2),C(z_3)
A
(
z
1
)
,
B
(
z
2
)
,
C
(
z
3
)
are vertices of an acute-angled triangle, prove that this triangle is equilateral.
2
1
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Prove that S{AQC}/S{CMT} = (sin B / cos C)^2
In a triangle
A
B
C
ABC
A
BC
, the altitude from
A
A
A
meets the circumcircle again at
T
T
T
. Let
O
O
O
be the circumcenter. The lines
O
A
OA
O
A
and
O
T
OT
OT
intersect the side
B
C
BC
BC
at
Q
Q
Q
and
M
M
M
, respectively. Prove that
S
A
Q
C
S
C
M
T
=
(
sin
B
cos
C
)
2
.
\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .
S
CMT
S
A
QC
=
(
cos
C
sin
B
)
2
.
1
1
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Factorial equation - Find all such m
Find all natural numbers
m
m
m
such that
1
!
⋅
3
!
⋅
5
!
⋯
(
2
m
−
1
)
!
=
(
m
(
m
+
1
)
2
)
!
.
1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.
1
!
⋅
3
!
⋅
5
!
⋯
(
2
m
−
1
)!
=
(
2
m
(
m
+
1
)
)
!
.
3
1
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Ab+bc+ca+2abc=1
Let
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
and
a
b
+
b
c
+
c
a
+
2
a
b
c
=
1
ab+bc+ca+2abc=1
ab
+
b
c
+
c
a
+
2
ab
c
=
1
then prove that
2
(
a
+
b
+
c
)
+
1
≥
32
a
b
c
2(a+b+c)+1\geq 32abc
2
(
a
+
b
+
c
)
+
1
≥
32
ab
c