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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2018 Serbia Team Selection Test
2018 Serbia Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(6)
4
1
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Dividing a rectangle into trapeziums.
An isosceles trapezium is called right if only one pair of its sides are parallel (i.e parallelograms are not right). A dissection of a rectangle into
n
n
n
(can be different shapes) right isosceles trapeziums is called strict if the union of any
i
,
(
2
≤
i
≤
n
)
i,(2\leq i \leq n)
i
,
(
2
≤
i
≤
n
)
trapeziums in the dissection do not form a right isosceles trapezium. Prove that for any
n
,
n
≥
9
n, n\geq 9
n
,
n
≥
9
there is a strict dissection of a
2017
×
2018
2017 \times 2018
2017
×
2018
rectangle into
n
n
n
right isosceles trapeziums.Proposed by Bojan Basic
3
1
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Weird game of geometry
Ana and Bob are playing the following game. [*] First, Bob draws triangle
A
B
C
ABC
A
BC
and a point
P
P
P
inside it. [*] Then Ana and Bob alternate, starting with Ana, choosing three different permutations
σ
1
\sigma_1
σ
1
,
σ
2
\sigma_2
σ
2
and
σ
3
\sigma_3
σ
3
of
{
A
,
B
,
C
}
\{A, B, C\}
{
A
,
B
,
C
}
. [*] Finally, Ana draw a triangle
V
1
V
2
V
3
V_1V_2V_3
V
1
V
2
V
3
.For
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
, let
ψ
i
\psi_i
ψ
i
be the similarity transformation which takes
σ
i
(
A
)
,
σ
i
(
B
)
\sigma_i(A), \sigma_i(B)
σ
i
(
A
)
,
σ
i
(
B
)
and
σ
i
(
C
)
\sigma_i(C)
σ
i
(
C
)
to
V
i
,
V
i
+
1
V_i, V_{i+1}
V
i
,
V
i
+
1
and
X
i
X_i
X
i
respectively (here
V
4
=
V
1
V_4=V_1
V
4
=
V
1
) where triangle
Δ
V
i
V
i
+
1
X
i
\Delta V_iV_{i+1}X_i
Δ
V
i
V
i
+
1
X
i
lies on the outside of triangle
V
1
V
2
V
3
V_1V_2V_3
V
1
V
2
V
3
. Finally, let
Q
i
=
ψ
i
(
P
)
Q_i=\psi_i(P)
Q
i
=
ψ
i
(
P
)
. Ana wins if triangles
Q
1
Q
2
Q
3
Q_1Q_2Q_3
Q
1
Q
2
Q
3
and
A
B
C
ABC
A
BC
are similar (in some order of vertices) and Bob wins otherwise. Determine who has the winning strategy.
6
1
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Is a linear combination of 2018th power bounded?
For any positive integer
n
n
n
, define
c
n
=
min
(
z
1
,
z
2
,
.
.
.
,
z
n
)
∈
{
−
1
,
1
}
n
∣
z
1
⋅
1
2018
+
z
2
⋅
2
2018
+
.
.
.
+
z
n
⋅
n
2018
∣
.
c_n=\min_{(z_1,z_2,...,z_n)\in\{-1,1\}^n} |z_1\cdot 1^{2018} + z_2\cdot 2^{2018} + ... + z_n\cdot n^{2018}|.
c
n
=
(
z
1
,
z
2
,
...
,
z
n
)
∈
{
−
1
,
1
}
n
min
∣
z
1
⋅
1
2018
+
z
2
⋅
2
2018
+
...
+
z
n
⋅
n
2018
∣.
Is the sequence
(
c
n
)
n
∈
Z
+
(c_n)_{n\in\mathbb{Z}^+}
(
c
n
)
n
∈
Z
+
bounded?
5
1
Hide problems
Tangency in nice configuration
Let
H
H
H
be the orthocenter of
A
B
C
ABC
A
BC
,
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
,and let
F
F
F
be a point on circumcircle of
A
B
C
ABC
A
BC
such that
∠
A
F
H
=
9
0
∘
\angle AFH=90^{\circ}
∠
A
F
H
=
9
0
∘
.
K
K
K
is the symmetric point of
H
H
H
wrt
B
B
B
.Let
P
P
P
be a point such that
∠
P
H
B
=
∠
P
B
C
=
9
0
∘
\angle PHB=\angle PBC=90^{\circ}
∠
P
H
B
=
∠
PBC
=
9
0
∘
,and
Q
Q
Q
is the foot of
B
B
B
to
C
P
CP
CP
.Prove that
H
Q
HQ
H
Q
is tangent to tge circumcircle of
F
H
K
FHK
F
HK
.
2
1
Hide problems
Inequality
Let
n
n
n
be a fixed positive integer and let
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
be positive real numbers. Prove that
x
1
(
1
−
x
1
2
)
+
x
2
(
1
−
(
x
1
+
x
2
)
2
)
+
⋯
+
x
n
(
1
−
(
x
1
+
.
.
.
+
x
n
)
2
)
<
2
3
.
x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.
x
1
(
1
−
x
1
2
)
+
x
2
(
1
−
(
x
1
+
x
2
)
2
)
+
⋯
+
x
n
(
1
−
(
x
1
+
...
+
x
n
)
2
)
<
3
2
.
1
1
Hide problems
Infinetly many composite numbers
Prove that there exists infinetly many natural number
n
n
n
such that at least one of the numbers
2
2
n
+
1
2^{2^n}+1
2
2
n
+
1
and
201
8
2
n
+
1
2018^{2^n}+1
201
8
2
n
+
1
is not a prime.