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National and Regional Contests
Germany Contests
Germany Team Selection Test
2023 Germany Team Selection Test
2023 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
2
1
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Orthocenter, Tangent points and circumcircles
Let
A
B
C
ABC
A
BC
be an acute angled triangle with orthocenter
H
H
H
and
A
B
<
A
C
AB<AC
A
B
<
A
C
. The point
T
T
T
lies on line
B
C
BC
BC
so that
A
T
AT
A
T
is a tangent to the circumcircle of
A
B
C
ABC
A
BC
. Let lines
A
H
AH
A
H
and
B
C
BC
BC
meet at point
D
D
D
and let
M
M
M
be the midpoint of
H
C
HC
H
C
. Let the circumcircle of
A
H
T
AHT
A
H
T
meets
C
H
CH
C
H
in
P
≠
H
P \not=H
P
=
H
and the circumcircle of
P
D
M
PDM
P
D
M
meet
B
C
BC
BC
in
Q
≠
D
Q \not=D
Q
=
D
. Prove that
Q
T
=
Q
A
QT=QA
QT
=
Q
A
.
3
2
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On elements of a finite set with sum 0
Let
A
A
A
be a non-empty set of integers with the following property: For each
a
∈
A
a \in A
a
∈
A
, there exist not necessarily distinct integers
b
,
c
∈
A
b,c \in A
b
,
c
∈
A
so that
a
=
b
+
c
a=b+c
a
=
b
+
c
. (a) Proof that there are examples of sets
A
A
A
fulfilling above property that do not contain
0
0
0
as element.(b) Proof that there exist
a
1
,
…
,
a
r
∈
A
a_1,\ldots,a_r \in A
a
1
,
…
,
a
r
∈
A
with
r
≥
1
r \ge 1
r
≥
1
and
a
1
+
⋯
+
a
r
=
0
a_1+\cdots+a_r=0
a
1
+
⋯
+
a
r
=
0
.(c) Proof that there exist pairwise distinct
a
1
,
…
,
a
r
a_1,\ldots,a_r
a
1
,
…
,
a
r
with
r
≥
1
r \ge 1
r
≥
1
and
a
1
+
⋯
+
a
r
=
0
a_1+\cdots+a_r=0
a
1
+
⋯
+
a
r
=
0
.
Control prime powers dividing product of polynomial values
Let
f
(
x
)
f(x)
f
(
x
)
be a monic polynomial of degree
2023
2023
2023
with positive integer coefficients. Show that for any sufficiently large integer
N
N
N
and any prime number
p
>
2023
N
p>2023N
p
>
2023
N
, the product
f
(
1
)
f
(
2
)
…
f
(
N
)
f(1)f(2)\dots f(N)
f
(
1
)
f
(
2
)
…
f
(
N
)
is at most
(
2023
2
)
\binom{2023}{2}
(
2
2023
)
times divisible by
p
p
p
. Proposed by Ashwin Sah
1
5
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