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Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2008 Serbia National Math Olympiad
2008 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
5
1
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Nice sequence
The sequence
(
a
n
)
n
≥
1
(a_n)_{n\ge 1}
(
a
n
)
n
≥
1
is defined by a_1 \equal{} 3, a_2 \equal{} 11 and a_n \equal{} 4a_{n\minus{}1}\minus{}a_{n\minus{}2}, for
n
≥
3
n \ge 3
n
≥
3
. Prove that each term of this sequence is of the form a^2 \plus{} 2b^2 for some natural numbers
a
a
a
and
b
b
b
.
4
1
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Colors of point
Each point of a plane is painted in one of three colors. Show that there exists a triangle such that:
(
i
)
(i)
(
i
)
all three vertices of the triangle are of the same color;
(
i
i
)
(ii)
(
ii
)
the radius of the circumcircle of the triangle is
2008
2008
2008
;
(
i
i
i
)
(iii)
(
iii
)
one angle of the triangle is either two or three times greater than one of the other two angles.
6
1
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pentagon area
In a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
, let \angle EAB \equal{} \angle ABC \equal{} 120^{\circ}, \angle ADB \equal{} 30^{\circ} and \angle CDE \equal{} 60^{\circ}. Let AB \equal{} 1. Prove that the area of the pentagon is less than
3
\sqrt {3}
3
.
1
1
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exponential diofantine with 2008
Find all nonegative integers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that 12^x\plus{}y^4\equal{}2008^z
2
1
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Hard geometry
Triangle
△
A
B
C
\triangle ABC
△
A
BC
is given. Points
D
D
D
i
E
E
E
are on line
A
B
AB
A
B
such that D \minus{} A \minus{} B \minus{} E, AD \equal{} AC and BE \equal{} BC. Bisector of internal angles at
A
A
A
and
B
B
B
intersect
B
C
,
A
C
BC,AC
BC
,
A
C
at
P
P
P
and
Q
Q
Q
, and circumcircle of
A
B
C
ABC
A
BC
at
M
M
M
and
N
N
N
. Line which connects
A
A
A
with center of circumcircle of
B
M
E
BME
BME
and line which connects
B
B
B
and center of circumcircle of
A
N
D
AND
A
N
D
intersect at
X
X
X
. Prove that
C
X
⊥
P
Q
CX \perp PQ
CX
⊥
PQ
.
3
1
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Hard inequality
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers such that a \plus{} b \plus{} c \equal{} 1. Prove inequality: \frac{1}{bc \plus{} a \plus{} \frac{1}{a}} \plus{} \frac{1}{ac \plus{} b \plus{} \frac{1}{b}} \plus{} \frac{1}{ab \plus{} c \plus{} \frac{1}{c}} \leqslant \frac{27}{31}.