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National and Regional Contests
Germany Contests
Germany Team Selection Test
2003 Germany Team Selection Test
2003 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
3
1
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Least common multiple of any two of these n numbers
Let
N
N
N
be a natural number and
x
1
,
…
,
x
n
x_1, \ldots , x_n
x
1
,
…
,
x
n
further natural numbers less than
N
N
N
and such that the least common multiple of any two of these
n
n
n
numbers is greater than
N
N
N
. Prove that the sum of the reciprocals of these
n
n
n
numbers is always less than
2
2
2
:
∑
i
=
1
n
1
x
i
<
2.
\sum^n_{i=1} \frac{1}{x_i} < 2.
∑
i
=
1
n
x
i
1
<
2.
2
1
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MA,MB,MC intersect the lines BC,CA,AB
Given a triangle
A
B
C
ABC
A
BC
and a point
M
M
M
such that the lines
M
A
,
M
B
,
M
C
MA,MB,MC
M
A
,
MB
,
MC
intersect the lines
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
in this order in points
D
,
E
D,E
D
,
E
and
F
,
F,
F
,
respectively. Prove that there are numbers
ϵ
1
,
ϵ
2
,
ϵ
3
∈
{
−
1
,
1
}
\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 1\}
ϵ
1
,
ϵ
2
,
ϵ
3
∈
{
−
1
,
1
}
such that:
ϵ
1
⋅
M
D
A
D
+
ϵ
2
⋅
M
E
B
E
+
ϵ
3
⋅
M
F
C
F
=
1.
\epsilon_1 \cdot \frac{MD}{AD} + \epsilon_2 \cdot \frac{ME}{BE} + \epsilon_3 \cdot \frac{MF}{CF} = 1.
ϵ
1
⋅
A
D
M
D
+
ϵ
2
⋅
BE
ME
+
ϵ
3
⋅
CF
MF
=
1.
1
1
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At a chess tournament the winner gets 1 point
At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining
1
2
\frac{1}{2}
2
1
points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.