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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2016 Mediterranean Mathematics Olympiad
2016 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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Prime n^8 + n^6 + n^4 + 4
Determine all integers
n
≥
1
n\ge1
n
≥
1
for which the number
n
8
+
n
6
+
n
4
+
4
n^8+n^6+n^4+4
n
8
+
n
6
+
n
4
+
4
is prime.(Proposed by Gerhard Woeginger, Austria)
3
1
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Coloring 25 x 25 chessboard
Consider a
25
×
25
25\times25
25
×
25
chessboard with cells
C
(
i
,
j
)
C(i,j)
C
(
i
,
j
)
for
1
≤
i
,
j
≤
25
1\le i,j\le25
1
≤
i
,
j
≤
25
. Find the smallest possible number
n
n
n
of colors with which these cells can be colored subject to the following condition: For
1
≤
i
<
j
≤
25
1\le i<j\le25
1
≤
i
<
j
≤
25
and for
1
≤
s
<
t
≤
25
1\le s<t\le25
1
≤
s
<
t
≤
25
, the three cells
C
(
i
,
s
)
C(i,s)
C
(
i
,
s
)
,
C
(
j
,
s
)
C(j,s)
C
(
j
,
s
)
,
C
(
j
,
t
)
C(j,t)
C
(
j
,
t
)
carry at least two different colors.(Proposed by Gerhard Woeginger, Austria)
2
1
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Inequality: a + b + c = 3; sums of square roots
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers with
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
. Prove that
b
a
2
+
3
+
c
b
2
+
3
+
a
c
2
+
3
≤
3
2
1
a
b
c
4
\sqrt{\frac{b}{a^2+3}}+ \sqrt{\frac{c}{b^2+3}}+ \sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}
a
2
+
3
b
+
b
2
+
3
c
+
c
2
+
3
a
≤
2
3
4
ab
c
1
1
1
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Orthogonal circumradius
Let
A
B
C
ABC
A
BC
be a triangle. Let
D
D
D
be the intersection point of the angle bisector at
A
A
A
with
B
C
BC
BC
. Let
T
T
T
be the intersection point of the tangent line to the circumcircle of triangle
A
B
C
ABC
A
BC
at point
A
A
A
with the line through
B
B
B
and
C
C
C
. Let
I
I
I
be the intersection point of the orthogonal line to
A
T
AT
A
T
through point
D
D
D
with the altitude
h
a
h_a
h
a
of the triangle at point
A
A
A
. Let
P
P
P
be the midpoint of
A
B
AB
A
B
, and let
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
. Let
M
M
M
be the intersection point of
A
B
AB
A
B
and
T
I
TI
T
I
, and let
F
F
F
be the intersection point of
P
T
PT
PT
and
A
D
AD
A
D
. Prove:
M
F
MF
MF
and
A
O
AO
A
O
are orthogonal to each other.