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Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2014 Serbia National Math Olympiad
2014 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
6
1
Hide problems
SMO 2014
In a triangle
A
B
C
ABC
A
BC
, let
D
D
D
and
E
E
E
be the feet of the angle bisectors of angles
A
A
A
and
B
B
B
, respectively. A rhombus is inscribed into the quadrilateral
A
E
D
B
AEDB
A
E
D
B
(all vertices of the rhombus lie on different sides of
A
E
D
B
AEDB
A
E
D
B
). Let
φ
\varphi
φ
be the non-obtuse angle of the rhombus. Prove that
φ
≤
max
{
∠
B
A
C
,
∠
A
B
C
}
\varphi \le \max \{ \angle BAC, \angle ABC \}
φ
≤
max
{
∠
B
A
C
,
∠
A
BC
}
Proposed by Dusan Djukic
I
M
O
S
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r
t
l
i
s
t
2013
IMO \ Shortlist \ 2013
I
MO
S
h
or
tl
i
s
t
2013
5
1
Hide problems
SMO 2014
Regular
n
n
n
-gon is divided to triangles using
n
−
3
n-3
n
−
3
diagonals of which none of them have common points with another inside polygon. How much among this triangles can there be the most not congruent?Proposed by Dusan Djukic
4
1
Hide problems
SMO 2014
We call natural number
n
n
n
c
r
a
z
y
crazy
cr
a
zy
iff there exist natural numbers
a
a
a
,
b
>
1
b >1
b
>
1
such that
n
=
a
b
+
b
n=a^b+b
n
=
a
b
+
b
. Whether there exist
2014
2014
2014
consecutive natural numbers among which are
2012
2012
2012
c
r
a
z
y
crazy
cr
a
zy
numbers?Proposed by Milos Milosavljevic
3
1
Hide problems
SMO 2014
Two players are playing game. Players alternately write down one natural number greater than
1
1
1
, but it is not allowed to write linear combination previously written numbers with nonnegative integer coefficients. Player lose a game if he can't write a new number. Does any of players can have wiining strategy, if yes, then which one of them?Journal "Kvant" / Aleksandar Ilic
2
1
Hide problems
SMO 2014
On sides
B
C
BC
BC
and
A
C
AC
A
C
of
△
A
B
C
\triangle ABC
△
A
BC
given are
D
D
D
and
E
E
E
, respectively. Let
F
F
F
(
F
≠
C
F \neq C
F
=
C
) be a point of intersection of circumcircle of
△
C
E
D
\triangle CED
△
CE
D
and line that is parallel to
A
B
AB
A
B
and passing through C. Let
G
G
G
be a point of intersection of line
F
D
FD
F
D
and side
A
B
AB
A
B
, and let
H
H
H
be on line
A
B
AB
A
B
such that
∠
H
D
A
=
∠
G
E
B
\angle HDA = \angle GEB
∠
HD
A
=
∠
GEB
and
H
−
A
−
B
H-A-B
H
−
A
−
B
. If
D
G
=
E
H
DG=EH
D
G
=
E
H
, prove that point of intersection of
A
D
AD
A
D
and
B
E
BE
BE
lie on angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
.Proposed by Milos Milosavljevic
1
1
Hide problems
SMO 2014
Determine all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that for all
x
x
x
,
y
∈
R
y \in \mathbb{R}
y
∈
R
hold:
f
(
x
f
(
y
)
−
y
f
(
x
)
)
=
f
(
x
y
)
−
x
y
f(xf(y)-yf(x))=f(xy)-xy
f
(
x
f
(
y
)
−
y
f
(
x
))
=
f
(
x
y
)
−
x
y
Proposed by Dusan Djukic