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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2007 Mediterranean Mathematics Olympiad
2007 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
Hide problems
Inequality on integer part and decimal part of x
Let
x
>
1
x > 1
x
>
1
be a non-integer number. Prove that
(
x
+
{
x
}
[
x
]
−
[
x
]
x
+
{
x
}
)
+
(
x
+
[
x
]
{
x
}
−
{
x
}
x
+
[
x
]
)
>
9
2
\biggl( \frac{x+\{x\}}{[x]} - \frac{[x]}{x+\{x\}} \biggr) + \biggl( \frac{x+[x]}{ \{x \} } - \frac{ \{ x \}}{x+[x]} \biggr) > \frac 92
(
[
x
]
x
+
{
x
}
−
x
+
{
x
}
[
x
]
)
+
(
{
x
}
x
+
[
x
]
−
x
+
[
x
]
{
x
}
)
>
2
9
3
1
Hide problems
Compute all possible lengths of sides AB and AC
In the triangle
A
B
C
ABC
A
BC
, the angle
α
=
∠
B
A
C
\alpha = \angle BAC
α
=
∠
B
A
C
and the side
a
=
B
C
a = BC
a
=
BC
are given. Assume that
a
=
r
R
a = \sqrt{rR}
a
=
r
R
, where
r
r
r
is the inradius and
R
R
R
the circumradius. Compute all possible lengths of sides
A
B
AB
A
B
and
A
C
.
AC.
A
C
.
2
1
Hide problems
Determine the length of CD
The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of a convex cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at point
E
E
E
. Given that
A
B
=
39
,
A
E
=
45
,
A
D
=
60
AB = 39, AE = 45, AD = 60
A
B
=
39
,
A
E
=
45
,
A
D
=
60
and
B
C
=
56
BC = 56
BC
=
56
, determine the length of
C
D
.
CD.
C
D
.
1
1
Hide problems
Prove that xz< 1/2
Let
x
≥
y
≥
z
x \geq y \geq z
x
≥
y
≥
z
be real numbers such that
x
y
+
y
z
+
z
x
=
1
xy + yz + zx = 1
x
y
+
yz
+
z
x
=
1
. Prove that
x
z
<
1
2
.
xz < \frac 12.
x
z
<
2
1
.
Is it possible to improve the value of constant
1
2
?
\frac 12 \ ?
2
1
?