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Problems
Contests
International Contests
Balkan MO
2003 Balkan MO
2003 Balkan MO
Part of
Balkan MO
Subcontests
(4)
1
1
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Can one find 4004 positive integers such that the sum ...
Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?
3
1
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rational function with f(1)+1>0
Find all functions
f
:
Q
→
R
f: \mathbb{Q}\to\mathbb{R}
f
:
Q
→
R
which fulfill the following conditions: a)
f
(
1
)
+
1
>
0
f(1)+1>0
f
(
1
)
+
1
>
0
; b)
f
(
x
+
y
)
−
x
f
(
y
)
−
y
f
(
x
)
=
f
(
x
)
f
(
y
)
−
x
−
y
+
x
y
f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy
f
(
x
+
y
)
−
x
f
(
y
)
−
y
f
(
x
)
=
f
(
x
)
f
(
y
)
−
x
−
y
+
x
y
, for all
x
,
y
∈
Q
x,y\in\mathbb{Q}
x
,
y
∈
Q
; c)
f
(
x
)
=
2
f
(
x
+
1
)
+
x
+
2
f(x) = 2f(x+1) +x+2
f
(
x
)
=
2
f
(
x
+
1
)
+
x
+
2
, for every
x
∈
Q
x\in\mathbb{Q}
x
∈
Q
.
4
1
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balkan problem
A rectangle
A
B
C
D
ABCD
A
BC
D
has side lengths
A
B
=
m
AB = m
A
B
=
m
,
A
D
=
n
AD = n
A
D
=
n
, with
m
m
m
and
n
n
n
relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points
A
1
=
A
,
A
2
,
A
3
,
…
,
A
k
=
C
A_1 = A, A_2, A_3, \ldots , A_k = C
A
1
=
A
,
A
2
,
A
3
,
…
,
A
k
=
C
. Show that
A
1
A
2
−
A
2
A
3
+
A
3
A
4
−
⋯
+
A
k
−
1
A
k
=
m
2
+
n
2
m
n
.
A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}.
A
1
A
2
−
A
2
A
3
+
A
3
A
4
−
⋯
+
A
k
−
1
A
k
=
mn
m
2
+
n
2
.
2
1
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balkan geo 2003
Let
A
B
C
ABC
A
BC
be a triangle, and let the tangent to the circumcircle of the triangle
A
B
C
ABC
A
BC
at
A
A
A
meet the line
B
C
BC
BC
at
D
D
D
. The perpendicular to
B
C
BC
BC
at
B
B
B
meets the perpendicular bisector of
A
B
AB
A
B
at
E
E
E
. The perpendicular to
B
C
BC
BC
at
C
C
C
meets the perpendicular bisector of
A
C
AC
A
C
at
F
F
F
. Prove that the points
D
D
D
,
E
E
E
and
F
F
F
are collinear. Valentin Vornicu