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Contests
National and Regional Contests
Germany Contests
Germany Team Selection Test
2015 Germany Team Selection Test
2015 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
3
1
Hide problems
Circumcircles and intersections madness
Let
A
B
C
ABC
A
BC
be an acute triangle with
∣
A
B
∣
≠
∣
A
C
∣
|AB| \neq |AC|
∣
A
B
∣
=
∣
A
C
∣
and the midpoints of segments
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
be
D
D
D
resp.
E
E
E
. The circumcircles of the triangles
B
C
D
BCD
BC
D
and
B
C
E
BCE
BCE
intersect the circumcircle of triangle
A
D
E
ADE
A
D
E
in
P
P
P
resp.
Q
Q
Q
with
P
≠
D
P \neq D
P
=
D
and
Q
≠
E
Q \neq E
Q
=
E
. Prove
∣
A
P
∣
=
∣
A
Q
∣
|AP|=|AQ|
∣
A
P
∣
=
∣
A
Q
∣
.(Notation:
∣
⋅
∣
|\cdot|
∣
⋅
∣
denotes the length of a segment and
[
⋅
]
[\cdot]
[
⋅
]
denotes the line segment.)
2
2
Hide problems
Naughty consecutive integers - a^b+b
A positive integer
n
n
n
is called naughty if it can be written in the form
n
=
a
b
+
b
n=a^b+b
n
=
a
b
+
b
with integers
a
,
b
≥
2
a,b \geq 2
a
,
b
≥
2
. Is there a sequence of
102
102
102
consecutive positive integers such that exactly
100
100
100
of those numbers are naughty?
Perpendicularity implies special intersection
Let
A
B
C
ABC
A
BC
be an acute triangle with the circumcircle
k
k
k
and incenter
I
I
I
. The perpendicular through
I
I
I
in
C
I
CI
C
I
intersects segment
[
B
C
]
[BC]
[
BC
]
in
U
U
U
and
k
k
k
in
V
V
V
. In particular
V
V
V
and
A
A
A
are on different sides of
B
C
BC
BC
. The parallel line through
U
U
U
to
A
I
AI
A
I
intersects
A
V
AV
A
V
in
X
X
X
. Prove: If
X
I
XI
X
I
and
A
I
AI
A
I
are perpendicular to each other, then
X
I
XI
X
I
intersects segment
[
A
C
]
[AC]
[
A
C
]
in its midpoint
M
M
M
.(Notation:
[
⋅
]
[\cdot]
[
⋅
]
denotes the line segment.)
1
1
Hide problems
Large real polynomial with real roots
Find the least positive integer
n
n
n
, such that there is a polynomial
P
(
x
)
=
a
2
n
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
⋯
+
a
1
x
+
a
0
P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0
P
(
x
)
=
a
2
n
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
⋯
+
a
1
x
+
a
0
with real coefficients that satisfies both of the following properties: - For
i
=
0
,
1
,
…
,
2
n
i=0,1,\dots,2n
i
=
0
,
1
,
…
,
2
n
it is
2014
≤
a
i
≤
2015
2014 \leq a_i \leq 2015
2014
≤
a
i
≤
2015
. - There is a real number
ξ
\xi
ξ
with
P
(
ξ
)
=
0
P(\xi)=0
P
(
ξ
)
=
0
.