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International Contests
Mediterranean Mathematics Olympiad
2015 Mediterranean Mathematical Olympiad
2015 Mediterranean Mathematical Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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Friends in a mathematical contest
In a mathematical contest, some of the competitors are friends and friendship is mutual. Prove that there is a subset
M
M
M
of the competitors such that each element of
M
M
M
has at most three friends in
M
M
M
and such that each competitor who is not in
M
,
M,
M
,
has at least four friends in
M
.
M.
M
.
3
1
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Mediterranean points
In the Cartesian plane
R
2
,
\mathbb{R}^2,
R
2
,
each triangle contains a Mediterranean point on its sides or in its interior, even if the triangle is degenerated into a segment or a point. The Mediterranean points have the following properties: (i) If a triangle is symmetric with respect to a line which passes through the origin
(
0
,
0
)
(0,0)
(
0
,
0
)
, then the Mediterranean point lies on this line. (ii) If the triangle
D
E
F
DEF
D
EF
contains the triangle
A
B
C
ABC
A
BC
and if the triangle
A
B
C
ABC
A
BC
contains the Mediterranean points
M
M
M
of
D
E
F
,
DEF,
D
EF
,
then
M
M
M
is the Mediterranean point of the triangle
A
B
C
.
ABC.
A
BC
.
Find all possible positions for the Mediterranean point of the triangle with vertices
(
−
3
,
5
)
,
(
12
,
5
)
,
(
3
,
11
)
.
(-3,5),\ (12,5),\ (3,11).
(
−
3
,
5
)
,
(
12
,
5
)
,
(
3
,
11
)
.
2
1
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Construction of a triangle: two sides and a cevian
Prove that for each triangle, there exists a vertex, such that with the two sides starting from that vertex and each cevian starting from that vertex, is possible to construct a triangle.
1
1
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Infinite values of n, P(3^n) is not prime
Let
P
(
x
)
=
x
4
−
x
3
−
3
x
2
−
x
+
1.
P(x)=x^4-x^3-3x^2-x+1.
P
(
x
)
=
x
4
−
x
3
−
3
x
2
−
x
+
1.
Prove that there are infinitely many positive integers
n
n
n
such that
P
(
3
n
)
P(3^n)
P
(
3
n
)
is not a prime.