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Contests
National and Regional Contests
Germany Contests
Germany Team Selection Test
2016 Germany Team Selection Test
2016 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
3
1
Hide problems
Game involving Divisibility by 4
In the beginning there are
100
100
100
integers in a row on the blackboard. Kain and Abel then play the following game: A move consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers
1
,
2
,
…
,
100
1,2,\dots,100
1
,
2
,
…
,
100
and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add
1
1
1
to each of the chosen numbers or instead subtract
1
1
1
from of the numbers. After that the next move begins, and so on.If there are at least
98
98
98
numbers on the blackboard that are divisible by
4
4
4
after a move, then Kain has won.Prove that Kain can force a win in a finite number of moves.
2
1
Hide problems
Integer inequality with integers on a circle
The positive integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
are aligned clockwise in a circular line with
n
≥
5
n \geq 5
n
≥
5
. Let
a
0
=
a
n
a_0=a_n
a
0
=
a
n
and
a
n
+
1
=
a
1
a_{n+1}=a_1
a
n
+
1
=
a
1
. For each
i
∈
{
1
,
2
,
…
,
n
}
i \in \{1,2,\dots,n \}
i
∈
{
1
,
2
,
…
,
n
}
the quotient
q
i
=
a
i
−
1
+
a
i
+
1
a
i
q_i=\frac{a_{i-1}+a_{i+1}}{a_i}
q
i
=
a
i
a
i
−
1
+
a
i
+
1
is an integer. Prove
2
n
≤
q
1
+
q
2
+
⋯
+
q
n
<
3
n
.
2n \leq q_1+q_2+\dots+q_n < 3n.
2
n
≤
q
1
+
q
2
+
⋯
+
q
n
<
3
n
.
1
1
Hide problems
Easy Geo involving Concyclicity
The two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
with the midpoints
O
1
O_1
O
1
resp.
O
2
O_2
O
2
intersect in the two distinct points
A
A
A
and
B
B
B
. A line through
A
A
A
meets
Γ
1
\Gamma_1
Γ
1
in
C
≠
A
C \neq A
C
=
A
and
Γ
2
\Gamma_2
Γ
2
in
D
≠
A
D \neq A
D
=
A
. The lines
C
O
1
CO_1
C
O
1
and
D
O
2
DO_2
D
O
2
intersect in
X
X
X
. Prove that the four points
O
1
,
O
2
,
B
O_1,O_2,B
O
1
,
O
2
,
B
and
X
X
X
are concyclic.