MathDB
Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2013 Mediterranean Mathematics Olympiad
2013 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
3
1
Hide problems
\sum (xy)^{2}=6xyz
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive reals for which:
∑
(
x
y
)
2
=
6
x
y
z
\sum (xy)^{2}=6xyz
∑
(
x
y
)
2
=
6
x
yz
Prove that:
∑
x
x
+
y
z
≥
3
\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}
∑
x
+
yz
x
≥
3
.
2
1
Hide problems
Chess tournament
Determine the least integer
k
k
k
for which the following story could hold true: In a chess tournament with
24
24
24
players, every pair of players plays at least
2
2
2
and at most
k
k
k
games against each other. At the end of the tournament, it turns out that every player has played a different number of games.
1
1
Hide problems
Polynomial with nine distinct nonegative integer roots
Do there exist two real monic polynomials
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
of degree 3,such that the roots of
P
(
Q
(
X
)
)
P(Q(X))
P
(
Q
(
X
))
are nine pairwise distinct nonnegative integers that add up to
72
72
72
? (In a monic polynomial of degree 3, the coefficient of
x
3
x^{3}
x
3
is
1
1
1
.)
4
1
Hide problems
Two orthogonal circles
A
B
C
D
ABCD
A
BC
D
is quadrilateral inscribed in a circle
Γ
\Gamma
Γ
.Lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect at
E
E
E
and lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at
F
F
F
. Prove that the circle with diameter
E
F
EF
EF
and circle
Γ
\Gamma
Γ
are orthogonal.