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Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2010 Serbia National Math Olympiad
2010 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(3)
3
2
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Find all n satisfying divisibility for all elements a of A
Let
A
A
A
be an infinite set of positive integers. Find all natural numbers
n
n
n
such that for each
a
∈
A
a \in A
a
∈
A
, a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots + a^{1!} + 1.Proposed by Milos Milosavljevic
There is a term of the sequence <sqrt{m}
Let
a
0
a_0
a
0
and
a
n
a_n
a
n
be different divisors of a natural number
m
m
m
, and
a
0
,
a
1
,
…
,
a
n
a_0, a_1, \ldots, a_n
a
0
,
a
1
,
…
,
a
n
be a sequence of natural numbers such that it satisfies
a
i
+
1
=
∣
a
i
±
a
i
−
1
∣
for
0
<
i
<
n
a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 < i < n
a
i
+
1
=
∣
a
i
±
a
i
−
1
∣
for
0
<
i
<
n
If
g
c
d
(
a
0
,
a
1
,
…
,
a
n
)
=
1
gcd(a_0,a_1,\ldots, a_n) = 1
g
c
d
(
a
0
,
a
1
,
…
,
a
n
)
=
1
, show that there exists a term of the sequence that is smaller than
m
\sqrt{m}
m
. Proposed by Dusan Djukic
2
2
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Product of n numbers in scattered cells gives same residue.
An n\times n table whose cells are numerated with numbers
1
,
2
,
⋯
,
n
2
1, 2,\cdots, n^2
1
,
2
,
⋯
,
n
2
in some order is called Naissus if all products of
n
n
n
numbers written in
n
n
n
scattered cells give the same residue when divided by
n
2
+
1
n^2+1
n
2
+
1
. Does there exist a Naissus table for
(
a
)
n
=
8
;
(a) n = 8;
(
a
)
n
=
8
;
(
b
)
n
=
10
?
(b) n = 10?
(
b
)
n
=
10
?
(
n
n
n
cells are scattered if no two are in the same row or column.)Proposed by Marko Djikic
Prove that <HMA=<GNS
In an acute-angled triangle
A
B
C
ABC
A
BC
,
M
M
M
is the midpoint of side
B
C
BC
BC
, and
D
,
E
D, E
D
,
E
and
F
F
F
the feet of the altitudes from
A
,
B
A, B
A
,
B
and
C
C
C
, respectively. Let
H
H
H
be the orthocenter of
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
S
S
S
the midpoint of
A
H
AH
A
H
, and
G
G
G
the intersection of
F
E
FE
FE
and
A
H
AH
A
H
. If
N
N
N
is the intersection of the median
A
M
AM
A
M
and the circumcircle of
Δ
B
C
H
\Delta BCH
Δ
BC
H
, prove that
∠
H
M
A
=
∠
G
N
S
\angle HMA = \angle GNS
∠
H
M
A
=
∠
GNS
. Proposed by Marko Djikic
1
2
Hide problems
Inequality with n towns and m two way airlines
Some of
n
n
n
towns are connected by two-way airlines. There are
m
m
m
airlines in total. For
i
=
1
,
2
,
⋯
,
n
i = 1, 2, \cdots, n
i
=
1
,
2
,
⋯
,
n
, let
d
i
d_i
d
i
be the number of airlines going from town
i
i
i
. If
1
≤
d
i
≤
2010
1\le d_i \le 2010
1
≤
d
i
≤
2010
for each
i
=
1
,
2
,
⋯
,
2010
i = 1, 2,\cdots, 2010
i
=
1
,
2
,
⋯
,
2010
, prove that \displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n Find all
n
n
n
for which equality can be attained. Proposed by Aleksandar Ilic
Prove that <APQ = 2<CAP if AQ/QB=DP/DE
Let
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
. A line through
O
O
O
intersects the sides
C
A
CA
C
A
and
C
B
CB
CB
at points
D
D
D
and
E
E
E
respectively, and meets the circumcircle of
A
B
O
ABO
A
BO
again at point
P
≠
O
P \neq O
P
=
O
inside the triangle. A point
Q
Q
Q
on side
A
B
AB
A
B
is such that
A
Q
Q
B
=
D
P
P
E
\frac{AQ}{QB}=\frac{DP}{PE}
QB
A
Q
=
PE
D
P
. Prove that
∠
A
P
Q
=
2
∠
C
A
P
\angle APQ = 2\angle CAP
∠
A
PQ
=
2∠
C
A
P
. Proposed by Dusan Djukic