MathDB
Problems
Contests
National and Regional Contests
Lithuania Contests
Lithuania National Olympiad
2006 Lithuania National Olympiad
2006 Lithuania National Olympiad
Part of
Lithuania National Olympiad
Subcontests
(2)
2
1
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Two tangent circles and an angle bisector
Two circles are tangent externaly at a point
B
B
B
. A line tangent to one of the circles at a point
A
A
A
intersects the other circle at points
C
C
C
and
D
D
D
. Show that
A
A
A
is equidistant to the lines
B
C
BC
BC
and
B
D
BD
B
D
.
4
1
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Maximal cardinality
Find the maximal cardinality
∣
S
∣
|S|
∣
S
∣
of the subset
S
⊂
A
=
{
1
,
2
,
3
,
…
,
9
}
S \subset A=\{1, 2, 3, \dots, 9\}
S
⊂
A
=
{
1
,
2
,
3
,
…
,
9
}
given that no two sums
a
+
b
∣
a
,
b
∈
S
,
a
≠
b
a+b | a, b \in S, a \neq b
a
+
b
∣
a
,
b
∈
S
,
a
=
b
are equal.