MathDB
Problems
Contests
National and Regional Contests
Kosovo Contests
Kosovo Team Selection Test
2017 Kosovo Team Selection Test
2017 Kosovo Team Selection Test
Part of
Kosovo Team Selection Test
Subcontests
(5)
5
1
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2017 Kosovo TST Problem 5
Given triangle
A
B
C
ABC
A
BC
. Let
P
P
P
,
Q
Q
Q
,
R
R
R
, be the tangency points of inscribed circle of
△
A
B
C
\triangle ABC
△
A
BC
on sides
A
B
AB
A
B
,
B
C
BC
BC
,
A
C
AC
A
C
respectively. We take the reflection of these points with respect to midpoints of the sides they lie on, and denote them as
P
′
,
Q
′
P',Q'
P
′
,
Q
′
and
R
′
R'
R
′
. Prove that
A
P
′
AP'
A
P
′
,
B
Q
′
BQ'
B
Q
′
, and
C
R
′
CR'
C
R
′
are concurrent.
4
1
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2017 Kosovo TST Problem 4
For every
n
∈
N
0
n \in \mathbb{N}_{0}
n
∈
N
0
, prove that
∑
k
=
0
[
n
2
]
2
n
−
2
k
(
n
2
k
)
=
3
n
+
1
2
\sum_{k=0}^{\left[\frac{n}{2} \right]}{2}^{n-2k} \binom{n}{2k}=\frac{3^{n}+1}{2}
∑
k
=
0
[
2
n
]
2
n
−
2
k
(
2
k
n
)
=
2
3
n
+
1
3
1
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2017 Kosovo TST Problem 3
If
a
a
a
and
b
b
b
are positive real numbers with sum
3
3
3
, and
x
,
y
,
z
x, y, z
x
,
y
,
z
positive real numbers with product
1
1
1
, prove that :
(
a
x
+
b
)
(
a
y
+
b
)
(
a
z
+
b
)
≥
27
(ax+b)(ay+b)(az+b)\geq 27
(
a
x
+
b
)
(
a
y
+
b
)
(
a
z
+
b
)
≥
27
2
1
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2017 Kosovo TST Problem 2
Prove that there doesn't exist any function
f
:
N
→
N
f:\mathbb{N}\rightarrow \mathbb{N}
f
:
N
→
N
such that :
f
(
f
(
n
−
1
)
=
f
(
n
+
1
)
−
f
(
n
)
f(f(n-1)=f(n+1)-f(n)
f
(
f
(
n
−
1
)
=
f
(
n
+
1
)
−
f
(
n
)
, for every natural
n
≥
2
n\geq2
n
≥
2
1
1
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2017 Kosovo TST Problem 1
Find all positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
, such that
a
2
2
a
b
2
−
b
3
+
1
\frac{a^2}{2ab^2-b^3+1}
2
a
b
2
−
b
3
+
1
a
2
is also a positive integer.