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National and Regional Contests
Kosovo Contests
Kosovo National Mathematical Olympiad
2010 Kosovo National Mathematical Olympiad
2010 Kosovo National Mathematical Olympiad
Part of
Kosovo National Mathematical Olympiad
Subcontests
(5)
5
1
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Kosovo MO 2010 Problem 5
Let
x
,
y
x,y
x
,
y
be positive real numbers such that
x
+
y
=
1
x+y=1
x
+
y
=
1
. Prove that
(
1
+
1
x
)
(
1
+
1
y
)
≥
9
\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9
(
1
+
x
1
)
(
1
+
y
1
)
≥
9
.
4
4
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3
3
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Kosovo MO 2010 Grade 9, Problem 3
Prove that in any polygon, there exist two sides whose radio is less than
2
2
2
.(Essentialy if
a
1
≥
a
2
≥
.
.
.
≥
a
n
a_1\geq a_2\geq...\geq a_n
a
1
≥
a
2
≥
...
≥
a
n
are the sides of a polygon prove that there exist
i
,
j
∈
{
1
,
2
,
.
.
,
n
}
i,j\in\{1,2,..,n\}
i
,
j
∈
{
1
,
2
,
..
,
n
}
so that
i
<
j
i<j
i
<
j
and
a
i
a
j
<
2
\frac {a_i}{a_j}<2
a
j
a
i
<
2
).
Kosovo MO 2010 Grade 11, Problem 3
Arrange the numbers
cos
2
,
cos
4
,
cos
6
\cos 2, \cos 4, \cos 6
cos
2
,
cos
4
,
cos
6
and
cos
8
\cos 8
cos
8
from the biggest to the smallest where
2
,
4
,
6
,
8
2,4,6,8
2
,
4
,
6
,
8
are given as radians.
Kosovo MO 2010 Grade 12, Problem 3
Let
n
∈
N
n\in \mathbb{N}
n
∈
N
. Prove that the polynom
p
(
x
)
=
x
2
n
−
2
x
2
n
−
1
+
3
x
2
n
−
2
−
.
.
.
−
2
n
x
+
2
n
+
1
p(x)=x^{2n}-2x^{2n-1}+3x^{2n-2}-...-2nx+2n+1
p
(
x
)
=
x
2
n
−
2
x
2
n
−
1
+
3
x
2
n
−
2
−
...
−
2
n
x
+
2
n
+
1
doesn't have real roots.
2
4
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1
4
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