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Contests
International Contests
Mediterranean Mathematics Olympiad
2011 Mediterranean Mathematics Olympiad
2011 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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Show Two Segments Meet on AD
Let
D
D
D
be the foot of the internal bisector of the angle
∠
A
\angle A
∠
A
of the triangle
A
B
C
ABC
A
BC
. The straight line which joins the incenters of the triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
cut
A
B
AB
A
B
and
A
C
AC
A
C
at
M
M
M
and
N
N
N
, respectively. Show that
B
N
BN
BN
and
C
M
CM
CM
meet on the bisector
A
D
AD
A
D
.
3
1
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Tipping a Tetrahedron
A regular tetrahedron of height
h
h
h
has a tetrahedron of height
x
h
xh
x
h
cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.) Show that
x
x
x
is a root of the equation
x
3
+
x
2
+
x
=
2
x^3 + x^2 + x = 2
x
3
+
x
2
+
x
=
2
.
2
1
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Sets of Positive Reals
Let
A
A
A
be a finite set of positive reals, let
B
=
{
x
/
y
∣
x
,
y
∈
A
}
B = \{x/y\mid x,y\in A\}
B
=
{
x
/
y
∣
x
,
y
∈
A
}
and let
C
=
{
x
y
∣
x
,
y
∈
A
}
C = \{xy\mid x,y\in A\}
C
=
{
x
y
∣
x
,
y
∈
A
}
. Show that
∣
A
∣
⋅
∣
B
∣
≤
∣
C
∣
2
|A|\cdot|B|\le|C|^2
∣
A
∣
⋅
∣
B
∣
≤
∣
C
∣
2
. (Proposed by Gerhard Woeginger, Austria)
1
1
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Mediterranean Polynomials
A Mediterranean polynomial has only real roots and it is of the form
P
(
x
)
=
x
10
−
20
x
9
+
135
x
8
+
a
7
x
7
+
a
6
x
6
+
a
5
x
5
+
a
4
x
4
+
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0
P
(
x
)
=
x
10
−
20
x
9
+
135
x
8
+
a
7
x
7
+
a
6
x
6
+
a
5
x
5
+
a
4
x
4
+
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
with real coefficients
a
0
…
,
a
7
a_0\ldots,a_7
a
0
…
,
a
7
. Determine the largest real number that occurs as a root of some Mediterranean polynomial. (Proposed by Gerhard Woeginger, Austria)