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National and Regional Contests
Germany Contests
Germany Team Selection Test
2007 Germany Team Selection Test
2007 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
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Prove that PA*BC = PB*AC = PC*AB
A point
P
P
P
in the interior of triangle
A
B
C
ABC
A
BC
satisfies \angle BPC \minus{} \angle BAC \equal{} \angle CPA \minus{} \angle CBA \equal{} \angle APB \minus{} \angle ACB. Prove that \bar{PA} \cdot \bar{BC} \equal{} \bar{PB} \cdot \bar{AC} \equal{} \bar{PC} \cdot \bar{AB}.
Sine inequality greater than twice triangle area
Let
A
B
C
ABC
A
BC
be a triangle and
P
P
P
an arbitrary point in the plane. Let
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
be interior angles of the triangle and its area is denoted by
F
.
F.
F
.
Prove: \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F When does equality occur?
Ratio of circumcircle radius to incircle diameter
In triangle
A
B
C
ABC
A
BC
we have
a
≥
b
a \geq b
a
≥
b
and
a
≥
c
.
a \geq c.
a
≥
c
.
Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis
s
a
s_a
s
a
to the altitude
a
a
.
a_a.
a
a
.
When do we have equality?
2
4
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3
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Weird if it contains exactly one of the distinct elements
Let
n
>
1
,
n
∈
Z
n > 1, n \in \mathbb{Z}
n
>
1
,
n
∈
Z
and B \equal{}\{1,2,\ldots, 2^n\}. A subset
A
A
A
of
B
B
B
is called weird if it contains exactly one of the distinct elements
x
,
y
∈
B
x,y \in B
x
,
y
∈
B
such that the sum of
x
x
x
and
y
y
y
is a power of two. How many weird subsets does
B
B
B
have?
Representation in ternary system
Let
k
∈
N
k \in \mathbb{N}
k
∈
N
. A polynomial is called
k
k
k
-valid if all its coefficients are integers between 0 and
k
k
k
inclusively. (Here we don't consider 0 to be a natural number.) a.) For
n
∈
N
n \in \mathbb{N}
n
∈
N
let
a
n
a_n
a
n
be the number of 5-valid polynomials
p
p
p
which satisfy
p
(
3
)
=
n
.
p(3) = n.
p
(
3
)
=
n
.
Prove that each natural number occurs in the sequence
(
a
n
)
n
(a_n)_n
(
a
n
)
n
at least once but only finitely often. b.) For
n
∈
N
n \in \mathbb{N}
n
∈
N
let
a
n
a_n
a
n
be the number of 4-valid polynomials
p
p
p
which satisfy
p
(
3
)
=
n
.
p(3) = n.
p
(
3
)
=
n
.
Prove that each natural number occurs infinitely often in the sequence
(
a
n
)
n
(a_n)_n
(
a
n
)
n
.
a%b + a%2b + a%3b + ... + a%nb = a + b
For a multiple of
k
b
kb
kb
of
b
b
b
let
a
%
k
b
a \% kb
a
%
kb
be the greatest number such that a \% kb \equal{} a \bmod b which is smaller than
k
b
kb
kb
and not greater than
a
a
a
itself. Let n \in \mathbb{Z}^ \plus{} . Determine all integer pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
with: a\%b \plus{} a\%2b \plus{} a\%3b \plus{} \ldots \plus{} a\%nb \equal{} a \plus{} b