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National and Regional Contests
Kosovo Contests
Kosovo National Mathematical Olympiad
2011 Kosovo National Mathematical Olympiad
2011 Kosovo National Mathematical Olympiad
Part of
Kosovo National Mathematical Olympiad
Subcontests
(5)
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4
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Find the length of AB [KMO #4, Grade 9]
In triangle
A
B
C
ABC
A
BC
medians of triangle
B
E
BE
BE
and
A
D
AD
A
D
are perpendicular to each other. Find the length of
A
B
‾
\overline{AB}
A
B
, if
B
C
‾
=
6
\overline{BC}=6
BC
=
6
and
A
C
‾
=
8
\overline{AC}=8
A
C
=
8
KMO #4 (Grade 11) [Area of the square]
A point
P
P
P
is given in the square
A
B
C
D
ABCD
A
BC
D
such that
P
A
‾
=
3
\overline{PA}=3
P
A
=
3
,
P
B
‾
=
7
\overline{PB}=7
PB
=
7
and
P
D
‾
=
5
\overline{PD}=5
P
D
=
5
. Find the area of the square.
Kosovo Mathematical Olympiad, #4. (Grade 12)
It is given a convex hexagon
A
1
A
2
⋯
A
6
A_1A_2 \cdots A_6
A
1
A
2
⋯
A
6
such that all its interior angles are same valued (congruent). Denote by
a
1
=
A
1
A
2
‾
,
a
2
=
A
2
A
3
‾
,
⋯
,
a
6
=
A
6
A
1
‾
.
a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.
a
1
=
A
1
A
2
,
a
2
=
A
2
A
3
,
⋯
,
a
6
=
A
6
A
1
.
a
)
a)
a
)
Prove that holds:
a
1
−
a
4
=
a
2
−
a
5
=
a
3
−
a
6
a_1-a_4=a_2-a_5=a_3-a_6
a
1
−
a
4
=
a
2
−
a
5
=
a
3
−
a
6
b
)
b)
b
)
Prove that if
a
1
,
a
2
,
a
3
,
.
.
.
,
a
6
a_1,a_2,a_3,...,a_6
a
1
,
a
2
,
a
3
,
...
,
a
6
satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent) interior angles.
3
4
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5
1
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Kosovo Mathematical Olympiad, #5. (Grade 12) [Permutations]
Let
n
>
1
n>1
n
>
1
be an integer and
S
n
S_n
S
n
the set of all permutations
π
:
{
1
,
2
,
⋯
,
n
}
→
{
1
,
2
,
⋯
,
n
}
\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}
π
:
{
1
,
2
,
⋯
,
n
}
→
{
1
,
2
,
⋯
,
n
}
where
π
\pi
π
is bijective function. For every permutation
π
∈
S
n
\pi \in S_n
π
∈
S
n
we define:
F
(
π
)
=
∑
k
=
1
n
∣
k
−
π
(
k
)
∣
and
M
n
=
1
n
!
∑
π
∈
S
n
F
(
π
)
F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi)
F
(
π
)
=
k
=
1
∑
n
∣
k
−
π
(
k
)
∣
and
M
n
=
n
!
1
π
∈
S
n
∑
F
(
π
)
where
M
n
M_n
M
n
is taken with all permutations
π
∈
S
n
\pi \in S_n
π
∈
S
n
. Calculate the sum
M
n
M_n
M
n
.
1
4
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