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National and Regional Contests
Kosovo Contests
Kosovo National Mathematical Olympiad
2017 Kosovo National Mathematical Olympiad
2017 Kosovo National Mathematical Olympiad
Part of
Kosovo National Mathematical Olympiad
Subcontests
(5)
5
3
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2017 Kosovo MO 9th grade
5. Given the point T in rectangle ABCD, the distances from T to A,B,C is 15,20,25. Find the distance from T to D.
Kosovo National MO Grade 11 P5
A sphere with ray
R
R
R
is cut by two parallel planes. such that the center of the sphere is outside the region determined by these planes. Let
S
1
S_{1}
S
1
and
S
2
S_{2}
S
2
be the areas of the intersections, and
d
d
d
the distance between these planes. Find the area of the intersection of the sphere with the plane parallel with these two planes, with equal distance from them.
Kosovo National MO Grade 12 P5
Lines determined by sides
A
B
AB
A
B
and
C
D
CD
C
D
of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at point
P
P
P
. Prove that
α
+
γ
=
β
+
δ
\alpha +\gamma =\beta +\delta
α
+
γ
=
β
+
δ
if and only if
P
A
⋅
P
B
=
P
C
⋅
P
D
PA\cdot PB=PC\cdot PD
P
A
⋅
PB
=
PC
⋅
P
D
, where
α
,
β
,
γ
,
δ
\alpha ,\beta ,\gamma ,\delta
α
,
β
,
γ
,
δ
are the measures of the internal angles of vertices
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
respectively.
4
3
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2017 Kosovo MO 9th grade
4. Find all triples of consecutive numbers ,whose sum of squares is equal to some fourdigit number with all four digits being equal.
Trigonometry equality
Prove that :
cos
36
−
sin
18
=
1
2
\cos36-\sin18=\frac{1}{2}
cos
36
−
sin
18
=
2
1
Kosovo National MO Grade 12 P4
Prove the identity
∑
k
=
2
n
k
(
k
−
1
)
(
n
k
)
=
(
n
2
)
2
n
−
1
\sum_{k=2}^{n} k(k-1)\binom{n}{k} =\binom{n}{2} 2^{n-1}
∑
k
=
2
n
k
(
k
−
1
)
(
k
n
)
=
(
2
n
)
2
n
−
1
for all
n
=
2
,
3
,
4
,
.
.
.
n=2,3,4,...
n
=
2
,
3
,
4
,
...
3
3
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Kosovo National MO Grade 11 P3
n
n
n
teams participated in a basketball tournament. Each team has played with each team exactly one game. There was no tie. If in the end of the tournament the
i
i
i
-th team has
x
i
x_{i}
x
i
wins and
y
i
y_{i}
y
i
loses
(
1
≤
i
≤
n
)
(1\leq i \leq n)
(
1
≤
i
≤
n
)
prove that:
∑
i
=
1
n
x
i
2
=
∑
i
=
1
n
y
i
2
\sum_{i=1}^{n} {x_{i}}^2=\sum_{i=1}^{n} {y_{i}}^2
∑
i
=
1
n
x
i
2
=
∑
i
=
1
n
y
i
2
2017 Kosovo MO 9th grade
3. 3 red birds for 4 days eat 36 grams of seed, 5 blue birds for 3 days eat 60 gram of seed. For how many days could be feed 2 red birds and 4 blue birds with 88 gr seed?
Kosovo National MO Grade 12 P3
Let
a
≥
2
a\geq 2
a
≥
2
a fixed natural number, and let
a
n
a_{n}
a
n
be the sequence
a
n
=
a
a
.
.
a
a_{n}=a^{a^{.^{.^{a}}}}
a
n
=
a
a
.
.
a
(e.g
a
1
=
a
a_{1}=a
a
1
=
a
,
a
2
=
a
a
a_{2}=a^a
a
2
=
a
a
, etc.). Prove that
(
a
n
+
1
−
a
n
)
∣
(
a
n
+
2
−
a
n
+
1
)
(a_{n+1}-a_{n})|(a_{n+2}-a_{n+1})
(
a
n
+
1
−
a
n
)
∣
(
a
n
+
2
−
a
n
+
1
)
for every natural number
n
n
n
.
2
3
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2017 Kosovo MO
2 .Solve the inequality
∣
x
−
1
∣
−
2
∣
x
−
4
∣
>
3
+
2
x
|x-1|-2|x-4|>3+2x
∣
x
−
1∣
−
2∣
x
−
4∣
>
3
+
2
x
Kosovo National MO Grade 11 P2
Prove that for every positive real
a
,
b
,
c
a,b,c
a
,
b
,
c
the inequality holds :
a
b
+
b
c
+
c
a
+
1
≥
2
2
3
(
a
+
b
c
+
b
+
c
a
+
c
+
a
b
)
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1 \geq \frac{2\sqrt2}{3} (\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}})
b
a
+
c
b
+
a
c
+
1
≥
3
2
2
(
c
a
+
b
+
a
b
+
c
+
b
c
+
a
)
When does the equality hold?
Kosovo National MO Grade 12 P2
Solve the system of equations
x
+
y
+
z
=
π
x+y+z=\pi
x
+
y
+
z
=
π
tan
x
tan
z
=
2
\tan x\tan z=2
tan
x
tan
z
=
2
tan
y
tan
z
=
18
\tan y\tan z=18
tan
y
tan
z
=
18
1
3
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2017 Kosovo MO
1. Find all primes of the form
n
3
−
1
n^3-1
n
3
−
1
.
Kosovo National MO Grade 11.
Find all ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
, of natural numbers, where
1
<
a
,
b
≤
100
1<a,b\leq 100
1
<
a
,
b
≤
100
, such that
1
log
a
10
+
1
log
b
10
\frac{1}{\log_{a}{10}}+\frac{1}{\log_{b}{10}}
l
o
g
a
10
1
+
l
o
g
b
10
1
is a natural number.
Kosovo National MO Grade 12 P1
The sequence
a
n
{a_{n}}
a
n
n
∈
N
n\in \mathbb{N}
n
∈
N
is given in a recursive way with
a
1
=
1
a_{1}=1
a
1
=
1
,
a
n
=
∏
i
=
1
n
−
1
a
i
+
1
a_{n}=\prod_{i=1}^{n-1} a_{i}+1
a
n
=
∏
i
=
1
n
−
1
a
i
+
1
, for all
n
≥
2
n\geq 2
n
≥
2
. Determine the least number
M
M
M
, such that
∑
n
=
1
m
1
a
n
<
M
\sum_{n=1}^{m} \frac{1}{a_{n}} <M
∑
n
=
1
m
a
n
1
<
M
for all
m
∈
N
m\in \mathbb{N}
m
∈
N