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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2006 Mediterranean Mathematics Olympiad
2006 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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Inequality for reals in [0,1]
Let
0
≤
x
i
,
j
≤
1
0\le x_{i,j} \le 1
0
≤
x
i
,
j
≤
1
, where
i
=
1
,
2
,
…
m
i=1,2, \ldots m
i
=
1
,
2
,
…
m
and
j
=
1
,
2
,
…
n
j=1,2, \ldots n
j
=
1
,
2
,
…
n
. Prove the inequality
∏
j
=
1
n
(
1
−
∏
i
=
1
m
x
i
,
j
)
+
∏
i
=
1
m
(
1
−
∏
j
=
1
n
(
1
−
x
i
,
j
)
)
≥
1
\prod_{j=1}^n\left(1-\prod_{i=1}^mx_{i,j} \right)+ \prod_{i=1}^m\left(1-\prod_{j=1}^n(1-x_{i,j}) \right) \ge 1
j
=
1
∏
n
(
1
−
i
=
1
∏
m
x
i
,
j
)
+
i
=
1
∏
m
(
1
−
j
=
1
∏
n
(
1
−
x
i
,
j
)
)
≥
1
3
1
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AB is a square of an integer
The side lengths
a
,
b
,
c
a,b,c
a
,
b
,
c
of a triangle
A
B
C
ABC
A
BC
are integers with
gcd
(
a
,
b
,
c
)
=
1
\gcd(a,b,c)=1
g
cd
(
a
,
b
,
c
)
=
1
. The bisector of angle
B
A
C
BAC
B
A
C
meets
B
C
BC
BC
at
D
D
D
.(a) show that if triangles
D
B
A
DBA
D
B
A
and
A
B
C
ABC
A
BC
are similar then
c
c
c
is a square.(b) If
c
=
n
2
c=n^2
c
=
n
2
is a square
(
n
≥
2
)
(n\ge 2)
(
n
≥
2
)
, find a triangle
A
B
C
ABC
A
BC
satisfying (a).
2
1
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Area Inequality
Let
P
P
P
be a point inside a triangle
A
B
C
ABC
A
BC
, and
A
1
B
2
,
B
1
C
2
,
C
1
A
2
A_1B_2,B_1C_2,C_1A_2
A
1
B
2
,
B
1
C
2
,
C
1
A
2
be segments passing through
P
P
P
and parallel to
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
respectively, where points
A
1
,
A
2
A_1, A_2
A
1
,
A
2
lie on
B
C
,
B
1
,
B
2
BC, B_1, B_2
BC
,
B
1
,
B
2
on
C
A
CA
C
A
, and
C
1
,
C
2
C_1,C_2
C
1
,
C
2
on
A
B
AB
A
B
. Prove that
Area
(
A
1
A
2
B
1
B
2
C
1
C
2
)
≥
1
2
Area
(
A
B
C
)
\text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)
Area
(
A
1
A
2
B
1
B
2
C
1
C
2
)
≥
2
1
Area
(
A
BC
)
1
1
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Every point is coloured red or blue
Every point of a plane is colored red or blue, not all with the same color. Can this be done in such a way that, on every circumference of radius 1,(a) there is exactly one blue point; (b) there are exactly two blue points?