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Contests
International Contests
Mediterranean Mathematics Olympiad
2018 Mediterranean Mathematics OIympiad
2018 Mediterranean Mathematics OIympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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largest integer N, for which there exists a 6xN table, so that ...
Determine the largest integer
N
N
N
, for which there exists a
6
×
N
6\times N
6
×
N
table
T
T
T
that has the following properties:
∗
*
∗
Every column contains the numbers
1
,
2
,
…
,
6
1,2,\ldots,6
1
,
2
,
…
,
6
in some ordering.
∗
*
∗
For any two columns
i
≠
j
i\ne j
i
=
j
, there exists a row
r
r
r
such that
T
(
r
,
i
)
=
T
(
r
,
j
)
T(r,i)= T(r,j)
T
(
r
,
i
)
=
T
(
r
,
j
)
.
∗
*
∗
For any two columns
i
≠
j
i\ne j
i
=
j
, there exists a row
s
s
s
such that
T
(
s
,
i
)
≠
T
(
s
,
j
)
T(s,i)\ne T(s,j)
T
(
s
,
i
)
=
T
(
s
,
j
)
.(Proposed by Gerhard Woeginger, Austria)
3
1
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Aegean integer , none of the numbers a^{n+2}+3a^n+1 with n>=1 is prime.
An integer
a
≥
1
a\ge1
a
≥
1
is called Aegean, if none of the numbers
a
n
+
2
+
3
a
n
+
1
a^{n+2}+3a^n+1
a
n
+
2
+
3
a
n
+
1
with
n
≥
1
n\ge1
n
≥
1
is prime. Prove that there are at least 500 Aegean integers in the set
{
1
,
2
,
…
,
2018
}
\{1,2,\ldots,2018\}
{
1
,
2
,
…
,
2018
}
.(Proposed by Gerhard Woeginger, Austria)
2
1
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Geometrical equality
Let
A
B
C
ABC
A
BC
be acute triangle. Let
E
E
E
and
F
F
F
be points on
B
C
BC
BC
, such that angles
B
A
E
BAE
B
A
E
and
F
A
C
FAC
F
A
C
are equal. Lines
A
E
AE
A
E
and
A
F
AF
A
F
intersect cirumcircle of
A
B
C
ABC
A
BC
at points
M
M
M
and
N
N
N
. On rays
A
B
AB
A
B
and
A
C
AC
A
C
we have points
P
P
P
and
R
R
R
, such that angle
P
E
A
PEA
PE
A
is equal to angle
B
B
B
and angle
A
E
R
AER
A
ER
is equal to angle
C
C
C
. Let
L
L
L
be intersection of
A
E
AE
A
E
and
P
R
PR
PR
and
D
D
D
be intersection of
B
C
BC
BC
and
L
N
LN
L
N
. Prove that
1
∣
M
N
∣
+
1
∣
E
F
∣
=
1
∣
E
D
∣
.
\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.
∣
MN
∣
1
+
∣
EF
∣
1
=
∣
E
D
∣
1
.
1
1
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Inequality with sin
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
be more than one real numbers, such that
0
≤
a
i
≤
π
2
0\leq a_i\leq \frac{\pi}{2}
0
≤
a
i
≤
2
π
. Prove that
(
1
n
∑
i
=
1
n
1
1
+
sin
a
i
)
(
1
+
∏
i
=
1
n
(
sin
a
i
)
1
n
)
≤
1.
\Bigg(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{1+\sin a_i}\Bigg)\Bigg(1+\prod_{i=1}^{n}(\sin a_i)^{\frac{1}{n}}\Bigg)\leq1.
(
n
1
i
=
1
∑
n
1
+
sin
a
i
1
)
(
1
+
i
=
1
∏
n
(
sin
a
i
)
n
1
)
≤
1.