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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2000 Mediterranean Mathematics Olympiad
2000 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
2
1
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Equilateral triangles on the exterior of a quadrilater ABCD
Suppose that in the exterior of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
equilateral triangles
X
A
B
,
Y
B
C
,
Z
C
D
,
W
D
A
XAB,YBC,ZCD,WDA
X
A
B
,
Y
BC
,
ZC
D
,
W
D
A
with centroids
S
1
,
S
2
,
S
3
,
S
4
S_1,S_2,S_3,S_4
S
1
,
S
2
,
S
3
,
S
4
respectively are constructed. Prove that
S
1
S
3
⊥
S
2
S
4
S_1S_3\perp S_2S_4
S
1
S
3
⊥
S
2
S
4
if and only if
A
C
=
B
D
AC=BD
A
C
=
B
D
.
3
1
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Equation.
Let
c
1
,
c
2
,
…
,
c
n
,
b
1
,
b
2
,
…
,
b
n
c_1,c_2,\ldots ,c_n,b_1,b_2,\ldots ,b_n
c
1
,
c
2
,
…
,
c
n
,
b
1
,
b
2
,
…
,
b
n
(
n
≥
2
)
(n\geq 2)
(
n
≥
2
)
be positive real numbers. Prove that the equation
∑
i
=
1
n
c
i
x
i
−
b
i
=
1
2
∑
i
=
1
n
x
i
\sum_{i=1}^nc_i\sqrt{x_i-b_i}=\frac{1}{2}\sum_{i=1}^nx_i
i
=
1
∑
n
c
i
x
i
−
b
i
=
2
1
i
=
1
∑
n
x
i
has a unique solution
(
x
1
,
…
,
x
n
)
(x_1,\ldots ,x_n)
(
x
1
,
…
,
x
n
)
if and only if
∑
i
=
1
n
c
i
2
=
∑
i
=
1
n
b
i
\sum_{i=1}^nc_i^2=\sum_{i=1}^nb_i
∑
i
=
1
n
c
i
2
=
∑
i
=
1
n
b
i
.
1
1
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Elements.
Let
F
=
{
1
,
2
,
.
.
.
,
100
}
F=\{1,2,...,100\}
F
=
{
1
,
2
,
...
,
100
}
and let
G
G
G
be any
10
10
10
-element subset of
F
F
F
. Prove that there exist two disjoint nonempty subsets
S
S
S
and
T
T
T
of
G
G
G
with the same sum of elements.
4
1
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Ineq
Let
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
be the midpoints of the sides
B
C
,
C
D
,
D
A
,
A
B
BC,CD,DA,AB
BC
,
C
D
,
D
A
,
A
B
of a convex quadrilateral, respectively. Prove that
4
(
A
P
2
+
B
Q
2
+
C
R
2
+
D
S
2
)
≤
5
(
A
B
2
+
B
C
2
+
C
D
2
+
D
A
2
)
4(AP^2+BQ^2+CR^2+DS^2)\le 5(AB^2+BC^2+CD^2+DA^2)
4
(
A
P
2
+
B
Q
2
+
C
R
2
+
D
S
2
)
≤
5
(
A
B
2
+
B
C
2
+
C
D
2
+
D
A
2
)