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Contests
National and Regional Contests
Germany Contests
Germany Team Selection Test
2013 Germany Team Selection Test
2013 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
3
1
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Circles Through Fixed Point
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with circumcircle
ω
\omega
ω
. Prove that there exists a point
J
J
J
such that for any point
X
X
X
inside
A
B
C
ABC
A
BC
if
A
X
,
B
X
,
C
X
AX,BX,CX
A
X
,
BX
,
CX
intersect
ω
\omega
ω
in
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
and
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
be reflections of
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
in midpoints of
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
respectively then
A
2
,
B
2
,
C
2
,
J
A_2,B_2,C_2,J
A
2
,
B
2
,
C
2
,
J
lie on a circle.
2
1
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Formula on Grid Polygon
Given a
m
×
n
m\times n
m
×
n
grid rectangle with
m
,
n
≥
4
m,n \ge 4
m
,
n
≥
4
and a closed path
P
P
P
that is not self intersecting from inner points of the grid, let
A
A
A
be the number of points on
P
P
P
such that
P
P
P
does not turn in them and let
B
B
B
be the number of squares that
P
P
P
goes through two non-adjacent sides of them furthermore let
C
C
C
be the number of squares with no side in
P
P
P
. Prove that
A
=
B
−
C
+
m
+
n
−
1.
A=B-C+m+n-1.
A
=
B
−
C
+
m
+
n
−
1.
1
2
Hide problems
Injectivity on Ratinoals
n
n
n
is an odd positive integer and
x
,
y
x,y
x
,
y
are two rational numbers satisfying
x
n
+
2
y
=
y
n
+
2
x
.
x^n+2y=y^n+2x.
x
n
+
2
y
=
y
n
+
2
x
.
Prove that
x
=
y
x=y
x
=
y
.
A TST Problem That Makes You Think Is This Really a TST Problem
Two concentric circles
ω
,
Ω
\omega, \Omega
ω
,
Ω
with radii
8
,
13
8,13
8
,
13
are given.
A
B
AB
A
B
is a diameter of
Ω
\Omega
Ω
and the tangent from
B
B
B
to
ω
\omega
ω
touches
ω
\omega
ω
at
D
D
D
. What is the length of
A
D
AD
A
D
.