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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2022 Serbia Team Selection Test
2022 Serbia Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(5)
P1
1
Hide problems
Polynomials
For a non-constant polynomial
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
…
+
a
1
x
+
a
0
∈
R
[
x
]
,
a
n
≠
0
,
n
∈
N
P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} \in \mathbb{R}[x], a_{n} \neq 0, n \in \mathbb{N}
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
…
+
a
1
x
+
a
0
∈
R
[
x
]
,
a
n
=
0
,
n
∈
N
, we say that
P
P
P
is symmetric if
a
k
=
a
n
−
k
a_{k}=a_{n-k}
a
k
=
a
n
−
k
for every
k
=
0
,
1
,
…
,
⌈
n
2
⌉
k=0,1, \ldots,\left\lceil\frac{n}{2}\right\rceil
k
=
0
,
1
,
…
,
⌈
2
n
⌉
. We define the weight of a non-constant polynomial
P
∈
R
[
x
]
P \in \mathbb{R}[x]
P
∈
R
[
x
]
, denoted by
t
(
P
)
t(P)
t
(
P
)
, as the multiplicity of its zero with the highest multiplicity.a) Prove that there exist non-constant, monic, pairwise distinct polynomials
P
1
,
P
2
,
…
,
P
2021
∈
R
[
x
]
P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]
P
1
,
P
2
,
…
,
P
2021
∈
R
[
x
]
, none of which is symmetric, such that the product of any two (distinct) polynomials is symmetric.b) What is the smallest possible value of
t
(
P
1
⋅
P
2
⋅
…
⋅
P
2021
)
t\left(P_{1} \cdot P_{2} \cdot \ldots \cdot P_{2021}\right)
t
(
P
1
⋅
P
2
⋅
…
⋅
P
2021
)
, if
P
1
,
P
2
,
…
,
P
2021
∈
R
[
x
]
P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]
P
1
,
P
2
,
…
,
P
2021
∈
R
[
x
]
are non-constant, monic, pairwise distinct polynomials, none of which is symmetric, and the product of any two (distinct) polynomials is symmetric?
P6
1
Hide problems
Weird geometry
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with bases
A
B
,
C
D
AB,CD
A
B
,
C
D
such that
C
D
=
k
⋅
A
B
CD=k \cdot AB
C
D
=
k
⋅
A
B
(
0
<
k
<
1
0<k<1
0
<
k
<
1
). Point
P
P
P
is such that
∠
P
A
B
=
∠
C
A
D
\angle PAB=\angle CAD
∠
P
A
B
=
∠
C
A
D
and
∠
P
B
A
=
∠
D
B
C
\angle PBA=\angle DBC
∠
PB
A
=
∠
D
BC
. Prove that
P
A
+
P
B
≤
1
1
−
k
2
⋅
A
B
PA+PB \leq \dfrac{1}{\sqrt{1-k^2}} \cdot AB
P
A
+
PB
≤
1
−
k
2
1
⋅
A
B
.
P2
1
Hide problems
Extension of a classic config
Given is a triangle
A
B
C
ABC
A
BC
with circumcircle
γ
\gamma
γ
. Points
E
,
F
E, F
E
,
F
lie on
A
B
,
A
C
AB, AC
A
B
,
A
C
such that
B
E
=
C
F
BE=CF
BE
=
CF
. Let
(
A
E
F
)
(AEF)
(
A
EF
)
meet
γ
\gamma
γ
at
D
D
D
. The perpendicular from
D
D
D
to
E
F
EF
EF
meets
γ
\gamma
γ
at
G
G
G
and
A
D
AD
A
D
meets
E
F
EF
EF
at
P
P
P
. If
P
G
PG
PG
meets
γ
\gamma
γ
at
J
J
J
, prove that
J
E
J
F
=
A
E
A
F
\frac {JE} {JF}=\frac{AE} {AF}
J
F
J
E
=
A
F
A
E
.
P5
1
Hide problems
Polynomial being product of the first few consecutive primes at a point $x>n$
Given is a positive integer
n
n
n
divisible by
3
3
3
and such that
2
n
−
1
2n-1
2
n
−
1
is a prime. Does there exist a positive integer
x
>
n
x>n
x
>
n
such that
n
x
n
+
1
+
(
2
n
+
1
)
x
n
−
3
(
n
−
1
)
x
n
−
1
−
x
−
3
nx^{n+1}+(2n+1)x^n-3(n-1)x^{n-1}-x-3
n
x
n
+
1
+
(
2
n
+
1
)
x
n
−
3
(
n
−
1
)
x
n
−
1
−
x
−
3
is a product of the first few odd primes?
P3
1
Hide problems
Why not combine NT and combo in one problem?
Let
n
n
n
be an odd positive integer. Given are
n
n
n
balls - black and white, placed on a circle. For a integer
1
≤
k
≤
n
−
1
1\leq k \leq n-1
1
≤
k
≤
n
−
1
, call
f
(
k
)
f(k)
f
(
k
)
the number of balls, such that after shifting them with
k
k
k
positions clockwise, their color doesn't change. a) Prove that for all
n
n
n
, there is a
k
k
k
with
f
(
k
)
≥
n
−
1
2
f(k) \geq \frac{n-1}{2}
f
(
k
)
≥
2
n
−
1
. b) Prove that there are infinitely many
n
n
n
(and corresponding colorings for them) such that
f
(
k
)
≤
n
−
1
2
f(k)\leq \frac{n-1}{2}
f
(
k
)
≤
2
n
−
1
for all
k
k
k
.