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International Contests
Mediterranean Mathematics Olympiad
2012 Mediterranean Mathematics Olympiad
2012 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
3
1
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Inequality in 0-1 matrix
Consider a binary matrix
M
M
M
(all entries are
0
0
0
or
1
1
1
) on
r
r
r
rows and
c
c
c
columns, where every row and every column contain at least one entry equal to
1
1
1
. Prove that there exists an entry
M
(
i
,
j
)
=
1
M(i,j) = 1
M
(
i
,
j
)
=
1
, such that the corresponding row-sum
R
(
i
)
R(i)
R
(
i
)
and column-sum
C
(
j
)
C(j)
C
(
j
)
satisfy
r
R
(
i
)
≥
c
C
(
j
)
r R(i)\ge c C(j)
r
R
(
i
)
≥
c
C
(
j
)
. (Proposed by Gerhard Woeginger, Austria)
1
1
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Least alpha for which this sequence is positive
For a real number
α
>
0
\alpha>0
α
>
0
, consider the infinite real sequence defined by
x
1
=
1
x_1=1
x
1
=
1
and \alpha x_n = x_1+x_2+\cdots+x_{n+1} \mbox{\qquad for } n\ge1. Determine the smallest
α
\alpha
α
for which all terms of this sequence are positive reals. (Proposed by Gerhard Woeginger, Austria)
4
1
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MMC 2012 problem 4
Let
O
O
O
be the circumcenter,
R
R
R
be the circumradius, and
k
k
k
be the circumcircle of a triangle
A
B
C
ABC
A
BC
. Let
k
1
k_1
k
1
be a circle tangent to the rays
A
B
AB
A
B
and
A
C
AC
A
C
, and also internally tangent to
k
k
k
. Let
k
2
k_2
k
2
be a circle tangent to the rays
A
B
AB
A
B
and
A
C
AC
A
C
, and also externally tangent to
k
k
k
. Let
A
1
A_1
A
1
and
A
2
A_2
A
2
denote the respective centers of
k
1
k_1
k
1
and
k
2
k_2
k
2
. Prove that:
(
O
A
1
+
O
A
2
)
2
−
A
1
A
2
2
=
4
R
2
.
(OA_1+OA_2)^2-A_1A_2^2 = 4R^2.
(
O
A
1
+
O
A
2
)
2
−
A
1
A
2
2
=
4
R
2
.
2
1
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Triangle inequality (48)
In an acute
△
A
B
C
\triangle ABC
△
A
BC
, prove that \begin{align*}\frac{1}{3}\left(\frac{\tan^2A}{\tan B\tan C}+\frac{\tan^2 B}{\tan C\tan A}+\frac{\tan^2 C}{\tan A\tan B}\right) \\ +3\left(\frac{1}{\tan A+\tan B+\tan C}\right)^{\frac{2}{3}}\ge 2.\end{align*}