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Problems
Contests
International Contests
Balkan MO
1987 Balkan MO
1987 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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two circles - classic and nice problem of Bulgaria
Two circles
K
1
K_{1}
K
1
and
K
2
K_{2}
K
2
, centered at
O
1
O_{1}
O
1
and
O
2
O_{2}
O
2
with radii
1
1
1
and
2
\sqrt{2}
2
respectively, intersect at
A
A
A
and
B
B
B
. Let
C
C
C
be a point on
K
2
K_{2}
K
2
such that the midpoint of
A
C
AC
A
C
lies on
K
1
K_{1}
K
1
. Find the length of the segment
A
C
AC
A
C
if
O
1
O
2
=
2
O_{1}O_{2}=2
O
1
O
2
=
2
3
1
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easy trigonometry, but strange
In the triangle
A
B
C
ABC
A
BC
the following equality holds:
sin
23
A
2
cos
48
B
2
=
sin
23
B
2
cos
48
A
2
\sin^{23}{\frac{A}{2}}\cos^{48}{\frac{B}{2}}=\sin^{23}{\frac{B}{2}}\cos^{48}{\frac{A}{2}}
sin
23
2
A
cos
48
2
B
=
sin
23
2
B
cos
48
2
A
Determine the value of
A
C
B
C
\frac{AC}{BC}
BC
A
C
.
2
1
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find x,y \geq 1
Find all real numbers
x
,
y
x,y
x
,
y
greater than
1
1
1
, satisfying the condition that the numbers
x
−
1
+
y
−
1
\sqrt{x-1}+\sqrt{y-1}
x
−
1
+
y
−
1
and
x
+
1
+
y
+
1
\sqrt{x+1}+\sqrt{y+1}
x
+
1
+
y
+
1
are nonconsecutive integers.
1
1
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functional equation f(x+y)=f(x)f(a-y)+f(y)f(a-x) - show f is constant
Let
a
a
a
be a real number and let
f
:
R
→
R
f : \mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
be a function satisfying
f
(
0
)
=
1
2
f(0)=\frac{1}{2}
f
(
0
)
=
2
1
and f(x+y)=f(x)f(a-y)+f(y)f(a-x), \forall x,y \in \mathbb{R}. Prove that
f
f
f
is constant.