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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2010 Mediterranean Mathematics Olympiad
2010 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
3
1
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Mediterranean M.C. 2010 Problem 3
Let
A
′
∈
(
B
C
)
,
A'\in(BC),
A
′
∈
(
BC
)
,
B
′
∈
(
C
A
)
,
C
′
∈
(
A
B
)
B'\in(CA),C'\in(AB)
B
′
∈
(
C
A
)
,
C
′
∈
(
A
B
)
be the points of tangency of the excribed circles of triangle
△
A
B
C
\triangle ABC
△
A
BC
with the sides of
△
A
B
C
.
\triangle ABC.
△
A
BC
.
Let
R
′
R'
R
′
be the circumradius of triangle
△
A
′
B
′
C
′
.
\triangle A'B'C'.
△
A
′
B
′
C
′
.
Show that
R
′
=
1
2
r
2
R
(
2
R
−
h
a
)
(
2
R
−
h
b
)
(
2
R
−
h
c
)
R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}
R
′
=
2
r
1
2
R
(
2
R
−
h
a
)
(
2
R
−
h
b
)
(
2
R
−
h
c
)
where as usual,
R
R
R
is the circumradius of
△
A
B
C
,
\triangle ABC,
△
A
BC
,
r is the inradius of
△
A
B
C
,
\triangle ABC,
△
A
BC
,
and
h
a
,
h
b
,
h
c
h_{a},h_{b},h_{c}
h
a
,
h
b
,
h
c
are the lengths of altitudes of
△
A
B
C
.
\triangle ABC.
△
A
BC
.
4
1
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Mediterranean M.C. 2010 Problem 4
Let
p
p
p
be a positive integer,
p
>
1.
p>1.
p
>
1.
Find the number of
m
×
n
m\times n
m
×
n
matrices with entries in the set
{
1
,
2
,
…
,
p
}
\left\{ 1,2,\dots,p\right\}
{
1
,
2
,
…
,
p
}
and such that the sum of elements on each row and each column is not divisible by
p
.
p.
p
.
1
1
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Mediterranean M.C. 2010 Problem 1
Real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are given. Solve the system of equations (unknowns
x
,
y
,
z
,
u
)
x,y,z,u)
x
,
y
,
z
,
u
)
x
2
−
y
z
−
z
u
−
y
u
=
a
x^{2}-yz-zu-yu=a
x
2
−
yz
−
z
u
−
y
u
=
a
y
2
−
z
u
−
u
x
−
x
z
=
b
y^{2}-zu-ux-xz=b
y
2
−
z
u
−
ux
−
x
z
=
b
z
2
−
u
x
−
x
y
−
y
u
=
c
z^{2}-ux-xy-yu=c
z
2
−
ux
−
x
y
−
y
u
=
c
u
2
−
x
y
−
y
z
−
z
x
=
d
u^{2}-xy-yz-zx=d
u
2
−
x
y
−
yz
−
z
x
=
d
2
1
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Mediterranean M.C. 2010 Problem 2
Given the positive real numbers
a
1
,
a
2
,
…
,
a
n
,
a_{1},a_{2},\dots,a_{n},
a
1
,
a
2
,
…
,
a
n
,
such that
n
>
2
n>2
n
>
2
and
a
1
+
a
2
+
⋯
+
a
n
=
1
,
a_{1}+a_{2}+\dots+a_{n}=1,
a
1
+
a
2
+
⋯
+
a
n
=
1
,
prove that the inequality
a
2
⋅
a
3
⋅
⋯
⋅
a
n
a
1
+
n
−
2
+
a
1
⋅
a
3
⋅
⋯
⋅
a
n
a
2
+
n
−
2
+
⋯
+
a
1
⋅
a
2
⋅
⋯
⋅
a
n
−
1
a
n
+
n
−
2
≤
1
(
n
−
1
)
2
\frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}
a
1
+
n
−
2
a
2
⋅
a
3
⋅
⋯
⋅
a
n
+
a
2
+
n
−
2
a
1
⋅
a
3
⋅
⋯
⋅
a
n
+
⋯
+
a
n
+
n
−
2
a
1
⋅
a
2
⋅
⋯
⋅
a
n
−
1
≤
(
n
−
1
)
2
1
does holds.