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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2016 Serbia Additional Team Selection Test
2016 Serbia Additional Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
1
1
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Polynomial induction, Serbia additional TST 2016
Let
P
0
(
x
)
=
x
3
−
4
x
P_0(x)=x^3-4x
P
0
(
x
)
=
x
3
−
4
x
. Sequence of polynomials is defined as following:\\
P
n
+
1
=
P
n
(
1
+
x
)
P
n
(
1
−
x
)
−
1
P_{n+1}=P_n(1+x)P_n(1-x)-1
P
n
+
1
=
P
n
(
1
+
x
)
P
n
(
1
−
x
)
−
1
.\\ Prove that
x
2016
∣
P
2016
(
x
)
x^{2016}|P_{2016}(x)
x
2016
∣
P
2016
(
x
)
.
2
1
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Combinatorial geometry
Let
A
B
C
D
ABCD
A
BC
D
be a square with side
4
4
4
. Find, with proof, the biggest
k
k
k
such that no matter how we place
k
k
k
points into
A
B
C
D
ABCD
A
BC
D
, such that they are on the interior but not on the sides, we always have a square with sidr length
1
1
1
, which is inside the square
A
B
C
D
ABCD
A
BC
D
, such that it contains no points in its interior(they can be on the sides).
3
1
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Number theory, biggest odd divisor, additional TST,Serbia
Let
w
(
x
)
w(x)
w
(
x
)
be largest odd divisor of
x
x
x
. Let
a
,
b
a,b
a
,
b
be natural numbers such that
(
a
,
b
)
=
1
(a,b)=1
(
a
,
b
)
=
1
and \\
a
+
w
(
b
+
1
)
a+w(b+1)
a
+
w
(
b
+
1
)
and
b
+
w
(
a
+
1
)
b+w(a+1)
b
+
w
(
a
+
1
)
are powers of two. Prove that
a
+
1
a+1
a
+
1
and
b
+
1
b+1
b
+
1
are powers of two.