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Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2009 Serbia Team Selection Test
2009 Serbia Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
2
2
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Sum of digits is 2009 and divisibility by 2009
Find the least number which is divisible by 2009 and its sum of digits is 2009.
Inequality with xy + yz + zx = x + y + z
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers such that xy \plus{} yz \plus{} zx \equal{} x \plus{} y \plus{} z. Prove the inequality \frac1{x^2 \plus{} y \plus{} 1} \plus{} \frac1{y^2 \plus{} z \plus{} 1} \plus{} \frac1{z^2 \plus{} x \plus{} 1}\le1 When does the equality hold?
3
2
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Biggest number of sets
Find the largest natural number
n
n
n
for which there exist different sets
S
1
,
S
2
,
…
,
S
n
S_1,S_2,\ldots,S_n
S
1
,
S
2
,
…
,
S
n
such that:
1
∘
1^\circ
1
∘
∣
S
i
∪
S
j
∣
≤
2004
|S_i\cup S_j|\leq 2004
∣
S
i
∪
S
j
∣
≤
2004
for each two
1
≤
i
,
j
≤
n
1\leq i,j\le n
1
≤
i
,
j
≤
n
and
2
∘
2^\circ
2
∘
S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\} for each three integers
1
≤
i
<
j
<
k
≤
n
1\le i<j<k\le n
1
≤
i
<
j
<
k
≤
n
.
Circles and collinearity
Let
k
k
k
be the inscribed circle of non-isosceles triangle
△
A
B
C
\triangle ABC
△
A
BC
, which center is
S
S
S
. Circle
k
k
k
touches sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
in points
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
respectively. Line
Q
R
QR
QR
intersects
B
C
BC
BC
in point
M
M
M
. Let a circle which contains points
B
B
B
and
C
C
C
touch
k
k
k
in point
N
N
N
. Circumscribed circle of
△
M
N
P
\triangle MNP
△
MNP
intersects line
A
P
AP
A
P
in point
L
L
L
, different from
P
P
P
. Prove that points
S
,
L
S,L
S
,
L
and
M
M
M
are collinear.
1
1
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Geometric angle inequality
Let
α
\alpha
α
and
β
\beta
β
be the angles of a non-isosceles triangle
A
B
C
ABC
A
BC
at points
A
A
A
and
B
B
B
, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in
D
D
D
and
E
E
E
, respectively. Prove that the acute angle between the lines
D
E
DE
D
E
and
A
B
AB
A
B
isn't greater than \frac{|\alpha\minus{}\beta|}3.