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Contests
National and Regional Contests
Kosovo Contests
Kosovo National Mathematical Olympiad
2009 Kosovo National Mathematical Olympiad
2009 Kosovo National Mathematical Olympiad
Part of
Kosovo National Mathematical Olympiad
Subcontests
(5)
5
1
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Kosovo MO 2009 Grade 10, Problem 5
In a circle four distinct points are fixed and each of them is assigned with a real number. Let those numbers be
x
1
,
x
2
,
x
3
,
x
4
x_1,x_2,x_3,x_4
x
1
,
x
2
,
x
3
,
x
4
such that
x
1
+
x
2
+
x
3
+
x
4
>
0
x_1+x_2+x_3+x_4>0
x
1
+
x
2
+
x
3
+
x
4
>
0
. Now we define a game with these numbers: If one of them, i.e.
x
i
x_i
x
i
, is a negative number, the player makes a move by adding the number
x
i
x_i
x
i
to his neighbors and changes the sign of the chosen number. The game ends when all the numbers are negative. Prove that this game ends in a finite number of steps.
4
3
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Kosovo MO 2009 Grade 10, Problem 4
Prove that if in the product of four consequtive natural numbers we add
1
1
1
, we get a perfect square.
Kosovo MO 2009 Grade 11, Problem 4
Prove that
n
11
−
n
n^{11}-n
n
11
−
n
is divisible by
11
11
11
.
Kosovo MO 2009 Grade 12, Problem 4
(
a
)
(a)
(
a
)
Let
a
1
,
a
2
,
a
3
a_1,a_2,a_3
a
1
,
a
2
,
a
3
be three real numbers. Prove that
(
a
1
−
a
2
)
(
a
1
−
a
3
)
+
(
a
2
−
a
1
)
(
a
2
−
a
3
)
+
(
a
3
−
a
1
)
(
a
2
−
a
2
)
≥
0
(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0
(
a
1
−
a
2
)
(
a
1
−
a
3
)
+
(
a
2
−
a
1
)
(
a
2
−
a
3
)
+
(
a
3
−
a
1
)
(
a
2
−
a
2
)
≥
0
.
(
b
)
(b)
(
b
)
Prove that the inequality above doesn't hold if we use four number instead of three.
3
3
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$\sqrt 2$
Prove that
2
\sqrt 2
2
is irrational.
Kosovo MO 2009 Grade 11, Problem 3
Let
n
≥
2
n\geq2
n
≥
2
be an integer.
n
n
n
is a prime if it is only divisible by
1
1
1
and
n
n
n
. Prove that there are infinitely many prime numbers.
Kosovo MO 2009 Grade 12, Problem 3
Let
a
,
b
a,b
a
,
b
and
c
c
c
be the sides of a triangle, prove that
a
b
+
c
+
b
c
+
a
+
c
a
+
b
<
2
\frac {a}{b+c}+\frac {b}{c+a}+\frac {c}{a+b}<2
b
+
c
a
+
c
+
a
b
+
a
+
b
c
<
2
.
2
3
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Roots and inequalities
If
x
1
x_1
x
1
and
x
2
x_2
x
2
are the solutions of the equation
x
2
−
(
m
+
3
)
x
+
m
+
2
=
0
x^2-(m+3)x+m+2=0
x
2
−
(
m
+
3
)
x
+
m
+
2
=
0
Find all real values of
m
m
m
such that the following inequations are valid
1
x
1
+
1
x
2
>
1
2
\frac {1}{x_1}+\frac {1}{x_2}>\frac{1}{2}
x
1
1
+
x
2
1
>
2
1
and
x
1
2
+
x
2
2
<
5
x_1^2+x_2^2<5
x
1
2
+
x
2
2
<
5
Kosovo MO 2009 Grade 11, Problem 2
Solve the equation:
x
2
+
2
x
c
o
s
(
x
−
y
)
+
1
=
0
x^2+2xcos(x-y)+1=0
x
2
+
2
x
cos
(
x
−
y
)
+
1
=
0
Kosovo MO 2009 Grade 12, Problem 2
Let
p
p
p
be a prime number and
n
n
n
a natural one. How many natural numbers are between
1
1
1
and
p
n
p^n
p
n
that are relatively prime with
p
n
p^n
p
n
?
1
3
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Kosovo MO 2009 Grade 10, Problem 1
Find the graph of the function
y
=
x
−
∣
x
+
x
2
∣
y=x-|x+x^2|
y
=
x
−
∣
x
+
x
2
∣
Kosovo MO 2009 Grade 11, Problem 1
Find the graph of the function
y
=
1
−
∣
1
−
s
i
n
x
∣
y=1-|1-sin x|
y
=
1
−
∣1
−
s
in
x
∣
.
Kosovo MO 2009 Grade 12, Problem 1
Find the graph of the function
y
=
x
+
∣
1
−
x
3
∣
y=x+|1-x^3|
y
=
x
+
∣1
−
x
3
∣
.