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Problems
Contests
National and Regional Contests
Germany Contests
Germany Team Selection Test
2022 Germany Team Selection Test
2022 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
3
1
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Another perpendicular to the Euler line
Let
A
B
C
ABC
A
BC
be a triangle with orthocenter
H
H
H
and circumcenter
O
O
O
. Let
P
P
P
be a point in the plane such that
A
P
⊥
B
C
AP \perp BC
A
P
⊥
BC
. Let
Q
Q
Q
and
R
R
R
be the reflections of
P
P
P
in the lines
C
A
CA
C
A
and
A
B
AB
A
B
, respectively. Let
Y
Y
Y
be the orthogonal projection of
R
R
R
onto
C
A
CA
C
A
. Let
Z
Z
Z
be the orthogonal projection of
Q
Q
Q
onto
A
B
AB
A
B
. Assume that
H
≠
O
H \neq O
H
=
O
and
Y
≠
Z
Y \neq Z
Y
=
Z
. Prove that
Y
Z
⊥
H
O
YZ \perp HO
Y
Z
⊥
H
O
.[asy] import olympiad; unitsize(30); pair A,B,C,H,O,P,Q,R,Y,Z,Q2,R2,P2; A = (-14.8, -6.6); B = (-10.9, 0.3); C = (-3.1, -7.1); O = circumcenter(A,B,C); H = orthocenter(A,B,C); P = 1.2 * H - 0.2 * A; Q = reflect(A, C) * P; R = reflect(A, B) * P; Y = foot(R, C, A); Z = foot(Q, A, B); P2 = foot(A, B, C); Q2 = foot(P, C, A); R2 = foot(P, A, B); draw(B--(1.6*A-0.6*B)); draw(B--C--A); draw(P--R, blue); draw(R--Y, red); draw(P--Q, blue); draw(Q--Z, red); draw(A--P2, blue); draw(O--H, darkgreen+linewidth(1.2)); draw((1.4*Z-0.4*Y)--(4.6*Y-3.6*Z), red+linewidth(1.2)); draw(rightanglemark(R,Y,A,10), red); draw(rightanglemark(Q,Z,B,10), red); draw(rightanglemark(C,Q2,P,10), blue); draw(rightanglemark(A,R2,P,10), blue); draw(rightanglemark(B,P2,H,10), blue); label("
H
\textcolor{blue}{H}
H
",H,NW); label("
P
\textcolor{blue}{P}
P
",P,N); label("
A
A
A
",A,W); label("
B
B
B
",B,N); label("
C
C
C
",C,S); label("
O
O
O
",O,S); label("
Q
\textcolor{blue}{Q}
Q
",Q,E); label("
R
\textcolor{blue}{R}
R
",R,W); label("
Y
\textcolor{red}{Y}
Y
",Y,S); label("
Z
\textcolor{red}{Z}
Z
",Z,NW); dot(A, filltype=FillDraw(black)); dot(B, filltype=FillDraw(black)); dot(C, filltype=FillDraw(black)); dot(H, filltype=FillDraw(blue)); dot(P, filltype=FillDraw(blue)); dot(Q, filltype=FillDraw(blue)); dot(R, filltype=FillDraw(blue)); dot(Y, filltype=FillDraw(red)); dot(Z, filltype=FillDraw(red)); dot(O, filltype=FillDraw(black)); [/asy]
2
1
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Equally many L-tetrominos and ┐-tetrominos are monochromatic
Given two positive integers
n
n
n
and
m
m
m
and a function
f
:
Z
×
Z
→
{
0
,
1
}
f : \mathbb{Z} \times \mathbb{Z} \to \left\{0,1\right\}
f
:
Z
×
Z
→
{
0
,
1
}
with the property that \begin{align*} f\left(i, j\right) = f\left(i+n, j\right) = f\left(i, j+m\right) \qquad \text{for all } \left(i, j\right) \in \mathbb{Z} \times \mathbb{Z} . \end{align*} Let
[
k
]
=
{
1
,
2
,
…
,
k
}
\left[k\right] = \left\{1,2,\ldots,k\right\}
[
k
]
=
{
1
,
2
,
…
,
k
}
for each positive integer
k
k
k
. Let
a
a
a
be the number of all
(
i
,
j
)
∈
[
n
]
×
[
m
]
\left(i, j\right) \in \left[n\right] \times \left[m\right]
(
i
,
j
)
∈
[
n
]
×
[
m
]
satisfying \begin{align*} f\left(i, j\right) = f\left(i+1, j\right) = f\left(i, j+1\right) . \end{align*} Let
b
b
b
be the number of all
(
i
,
j
)
∈
[
n
]
×
[
m
]
\left(i, j\right) \in \left[n\right] \times \left[m\right]
(
i
,
j
)
∈
[
n
]
×
[
m
]
satisfying \begin{align*} f\left(i, j\right) = f\left(i-1, j\right) = f\left(i, j-1\right) . \end{align*} Prove that
a
=
b
a = b
a
=
b
.
1
2
Hide problems
Common refinement of two integer factorizations
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be
n
n
n
positive integers, and let
b
1
,
b
2
,
…
,
b
m
b_1, b_2, \ldots, b_m
b
1
,
b
2
,
…
,
b
m
be
m
m
m
positive integers such that
a
1
a
2
⋯
a
n
=
b
1
b
2
⋯
b
m
a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m
a
1
a
2
⋯
a
n
=
b
1
b
2
⋯
b
m
. Prove that a rectangular table with
n
n
n
rows and
m
m
m
columns can be filled with positive integer entries in such a way that* the product of the entries in the
i
i
i
-th row is
a
i
a_i
a
i
(for each
i
∈
{
1
,
2
,
…
,
n
}
i \in \left\{1,2,\ldots,n\right\}
i
∈
{
1
,
2
,
…
,
n
}
);* the product of the entries in the
j
j
j
-th row is
b
j
b_j
b
j
(for each
i
∈
{
1
,
2
,
…
,
m
}
i \in \left\{1,2,\ldots,m\right\}
i
∈
{
1
,
2
,
…
,
m
}
).
Concurrent perpendicular bisectors from three circles in a triangle
Given a triangle
A
B
C
ABC
A
BC
and three circles
x
x
x
,
y
y
y
and
z
z
z
such that
A
∈
y
∩
z
A \in y \cap z
A
∈
y
∩
z
,
B
∈
z
∩
x
B \in z \cap x
B
∈
z
∩
x
and
C
∈
x
∩
y
C \in x \cap y
C
∈
x
∩
y
.The circle
x
x
x
intersects the line
A
C
AC
A
C
at the points
X
b
X_b
X
b
and
C
C
C
, and intersects the line
A
B
AB
A
B
at the points
X
c
X_c
X
c
and
B
B
B
. The circle
y
y
y
intersects the line
B
A
BA
B
A
at the points
Y
c
Y_c
Y
c
and
A
A
A
, and intersects the line
B
C
BC
BC
at the points
Y
a
Y_a
Y
a
and
C
C
C
. The circle
z
z
z
intersects the line
C
B
CB
CB
at the points
Z
a
Z_a
Z
a
and
B
B
B
, and intersects the line
C
A
CA
C
A
at the points
Z
b
Z_b
Z
b
and
A
A
A
. (Yes, these definitions have the symmetries you would expect.)Prove that the perpendicular bisectors of the segments
Y
a
Z
a
Y_a Z_a
Y
a
Z
a
,
Z
b
X
b
Z_b X_b
Z
b
X
b
and
X
c
Y
c
X_c Y_c
X
c
Y
c
concur.