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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2020 Serbia National Math Olympiad
2020 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
6
1
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Onedimensional board game.
We are given a natural number
k
k
k
. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute
n
n
n
coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves:
(
i
)
(i)
(
i
)
We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other.
(
i
i
)
(ii)
(
ii
)
We choose a field with at least
2
2
2
coins on it, and we transfer one coin from the chosen field to the
k
−
t
h
k-\mathrm{th}
k
−
th
field on the left , and one coin from the chosen field to the
k
−
t
h
k-\mathrm{th}
k
−
th
field on the right.
(
a
)
\mathbf{(a)}
(
a
)
If
n
≤
k
+
1
n\leq k+1
n
≤
k
+
1
, prove that we can play only finitely many moves.
(
b
)
\mathbf{(b)}
(
b
)
For which values of
k
k
k
we can choose a natural number
n
n
n
and distribute
n
n
n
coins on the given board such that we can play infinitely many moves.
5
1
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Number theory functional equation
For a natural number
n
n
n
, with
v
2
(
n
)
v_2(n)
v
2
(
n
)
we denote the largest integer
k
≥
0
k\geq0
k
≥
0
such that
2
k
∣
n
2^k|n
2
k
∣
n
. Let us assume that the function
f
:
N
→
N
f\colon\mathbb{N}\to\mathbb{N}
f
:
N
→
N
meets the conditions:
(
i
)
(i)
(
i
)
f
(
x
)
≤
3
x
f(x)\leq3x
f
(
x
)
≤
3
x
for all natural numbers
x
∈
N
x\in\mathbb{N}
x
∈
N
.
(
i
i
)
(ii)
(
ii
)
v
2
(
f
(
x
)
+
f
(
y
)
)
=
v
2
(
x
+
y
)
v_2(f(x)+f(y))=v_2(x+y)
v
2
(
f
(
x
)
+
f
(
y
))
=
v
2
(
x
+
y
)
for all natural numbers
x
,
y
∈
N
x,y\in\mathbb{N}
x
,
y
∈
N
.Prove that for every natural number
a
a
a
there exists exactly one natural number
x
x
x
such that
f
(
x
)
=
3
a
f(x)=3a
f
(
x
)
=
3
a
.
4
1
Hide problems
Trapezoid circumscribed circle
In a trapezoid
A
B
C
D
ABCD
A
BC
D
such that the internal angles are not equal to
9
0
∘
90^{\circ}
9
0
∘
, the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at the point
E
E
E
. Let
P
P
P
and
Q
Q
Q
be the feet of the altitudes from
A
A
A
and
B
B
B
to the sides
B
C
BC
BC
and
A
D
AD
A
D
respectively. Circumscribed circles of the triangles
C
E
Q
CEQ
CEQ
and
D
E
P
DEP
D
EP
intersect at the point
F
≠
E
F\neq E
F
=
E
. Prove that the lines
A
P
AP
A
P
,
B
Q
BQ
BQ
and
E
F
EF
EF
are either parallel to each other, or they meet at exactly one point.
3
1
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Circumscribed circles
We are given a triangle
A
B
C
ABC
A
BC
. Points
D
D
D
and
E
E
E
on the line
A
B
AB
A
B
are such that
A
D
=
A
C
AD=AC
A
D
=
A
C
and
B
E
=
B
C
BE=BC
BE
=
BC
, with the arrangment of points
D
−
A
−
B
−
E
D - A - B - E
D
−
A
−
B
−
E
. The circumscribed circles of the triangles
D
B
C
DBC
D
BC
and
E
A
C
EAC
E
A
C
meet again at the point
X
≠
C
X\neq C
X
=
C
, and the circumscribed circles of the triangles
D
E
C
DEC
D
EC
and
A
B
C
ABC
A
BC
meet again at the point
Y
≠
C
Y\neq C
Y
=
C
. Find the measure of
∠
A
C
B
\angle ACB
∠
A
CB
given the condition
D
Y
+
E
Y
=
2
X
Y
DY+EY=2XY
D
Y
+
E
Y
=
2
X
Y
.
2
1
Hide problems
Combinatorial geometry with polyhedra
We are given a polyhedron with at least
5
5
5
vertices, such that exactly
3
3
3
edges meet in each of the vertices. Prove that we can assign a rational number to every vertex of the given polyhedron such that the following conditions are met:
(
i
)
(i)
(
i
)
At least one of the numbers assigned to the vertices is equal to
2020
2020
2020
.
(
i
i
)
(ii)
(
ii
)
For every polygonal face, the product of the numbers assigned to the vertices of that face is equal to
1
1
1
.
1
1
Hide problems
Monic Polynomial Divisibility
Find all monic polynomials
P
(
x
)
P(x)
P
(
x
)
such that the polynomial
P
(
x
)
2
−
1
P(x)^2-1
P
(
x
)
2
−
1
is divisible by the polynomial
P
(
x
+
1
)
P(x+1)
P
(
x
+
1
)
.