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National and Regional Contests
Kosovo Contests
Kosovo Team Selection Test
2016 Kosovo Team Selection Test
2016 Kosovo Team Selection Test
Part of
Kosovo Team Selection Test
Subcontests
(5)
5
1
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Kosovo Mathematical Olympiad 2016 TST , Problem 5
Let be
A
B
C
ABC
A
BC
an acute triangle with
∣
A
B
∣
>
∣
A
C
∣
|AB|>|AC|
∣
A
B
∣
>
∣
A
C
∣
. Let be
D
D
D
point in side
A
B
AB
A
B
such that
∠
A
C
D
=
∠
C
B
D
\angle ACD=\angle CBD
∠
A
C
D
=
∠
CB
D
. Let be
E
E
E
the midpoint of segment
B
D
BD
B
D
and
S
S
S
let be the circumcenter of triangle
B
C
D
BCD
BC
D
. Show that points
A
,
E
,
S
A,E,S
A
,
E
,
S
and
C
C
C
lie on a circle .
4
1
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Kosovo Mathematical Olympiad 2016 TST , Problem 4
It is given the function
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
fow which
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
and for all
x
∈
R
x\in\mathbb{R}
x
∈
R
satisfied
f
(
x
+
5
)
≥
f
(
x
)
+
5
f(x+5)\geq f(x)+5
f
(
x
+
5
)
≥
f
(
x
)
+
5
and
f
(
x
+
1
)
≤
f
(
x
)
+
1
f(x+1)\leq f(x)+1
f
(
x
+
1
)
≤
f
(
x
)
+
1
If
g
(
x
)
=
f
(
x
)
−
x
+
1
g(x)=f(x)-x+1
g
(
x
)
=
f
(
x
)
−
x
+
1
then find
g
(
2016
)
g(2016)
g
(
2016
)
.
3
1
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Kosovo Mathematical Olympiad 2016 TST , Problem 3
If quadratic equations
x
2
+
a
x
+
b
=
0
x^2+ax+b=0
x
2
+
a
x
+
b
=
0
and
x
2
+
p
x
+
q
=
0
x^2+px+q=0
x
2
+
p
x
+
q
=
0
share one similar root then find quadratic equation for which has roots of other roots of both quadratic equations .
2
1
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Kosovo Mathematical Olympiad 2016 TST , Problem 2
Show that for all positive integers
n
≥
2
n\geq 2
n
≥
2
the last digit of the number
2
2
n
+
1
2^{2^n}+1
2
2
n
+
1
is
7
7
7
.
1
1
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Kosovo Mathematical Olympiad 2016 TST , Problem 1
Solve equation in real numbers
x
+
4
x
+
16
x
+
…
+
4
n
x
+
3
−
x
=
1
\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1
x
+
4
x
+
16
x
+
…
+
4
n
x
+
3
−
x
=
1