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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2009 Mediterranean Mathematics Olympiad
2009 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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3-variable Inequality
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers. Prove that
∑
c
y
c
l
i
c
x
y
x
y
+
x
2
+
y
2
≤
∑
c
y
c
l
i
c
x
2
x
+
z
\sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z}
cyc
l
i
c
∑
x
y
+
x
2
+
y
2
x
y
≤
cyc
l
i
c
∑
2
x
+
z
x
(Proposed by Šefket Arslanagić, Bosnia and Herzegovina)
3
1
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Conditional Matrices
Decide whether the integers
1
,
2
,
…
,
100
1,2,\ldots,100
1
,
2
,
…
,
100
can be arranged in the cells
C
(
i
,
j
)
C(i, j)
C
(
i
,
j
)
of a
10
×
10
10\times10
10
×
10
matrix (where
1
≤
i
,
j
≤
10
1\le i,j\le 10
1
≤
i
,
j
≤
10
), such that the following conditions are fullfiled: i) In every row, the entries add up to the same sum
S
S
S
. ii) In every column, the entries also add up to this sum
S
S
S
. iii) For every
k
=
1
,
2
,
…
,
10
k = 1, 2, \ldots, 10
k
=
1
,
2
,
…
,
10
the ten entries
C
(
i
,
j
)
C(i, j)
C
(
i
,
j
)
with
i
−
j
≡
k
m
o
d
10
i-j\equiv k\bmod{10}
i
−
j
≡
k
mod
10
add up to
S
S
S
. (Proposed by Gerhard Woeginger, Austria)
2
1
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Prove that 4 Points are Collinear
Let
A
B
C
ABC
A
BC
be a triangle with
9
0
∘
≠
∠
A
≠
13
5
∘
90^\circ \ne \angle A \ne 135^\circ
9
0
∘
=
∠
A
=
13
5
∘
. Let
D
D
D
and
E
E
E
be external points to the triangle
A
B
C
ABC
A
BC
such that
D
A
B
DAB
D
A
B
and
E
A
C
EAC
E
A
C
are isoscele triangles with right angles at
D
D
D
and
E
E
E
. Let
F
=
B
E
∩
C
D
F = BE \cap CD
F
=
BE
∩
C
D
, and let
M
M
M
and
N
N
N
be the midpoints of
B
C
BC
BC
and
D
E
DE
D
E
, respectively.Prove that, if three of the points
A
A
A
,
F
F
F
,
M
M
M
,
N
N
N
are collinear, then all four are collinear.
1
1
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Fulfill Three Conditions
Determine all integers
n
≥
1
n\ge1
n
≥
1
for which there exists
n
n
n
real numbers
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
in the closed interval
[
−
4
,
2
]
[-4,2]
[
−
4
,
2
]
such that the following three conditions are fulfilled: - the sum of these real numbers is at least
n
n
n
. - the sum of their squares is at most
4
n
4n
4
n
. - the sum of their fourth powers is at least
34
n
34n
34
n
. (Proposed by Gerhard Woeginger, Austria)