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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2010 Bundeswettbewerb Mathematik
2010 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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2010 different ways as sum of powers of 2
Find all numbers that can be expressed in exactly
2010
2010
2010
different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand.
f (f (f (n))) = P (n), for all n \in N_0, all coefficients of P(x) are in N_0
In the following, let
N
0
N_0
N
0
denotes the set of non-negative integers. Find all polynomials
P
(
x
)
P(x)
P
(
x
)
that fulfill the following two properties: (1) All coefficients of
P
(
x
)
P(x)
P
(
x
)
are from
N
0
N_0
N
0
. (2) Exists a function
f
:
N
0
→
N
0
f : N_0 \to N_0
f
:
N
0
→
N
0
such as
f
(
f
(
f
(
n
)
)
)
=
P
(
n
)
f (f (f (n))) = P (n)
f
(
f
(
f
(
n
)))
=
P
(
n
)
for all
n
∈
N
0
n \in N_0
n
∈
N
0
.
3
2
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P, Q, R, S lie either on a circle or on one straight line
Given an acute-angled triangle
A
B
C
ABC
A
BC
. Let
C
B
CB
CB
be the altitude and
E
E
E
a random point on the line
C
D
CD
C
D
. Finally, let
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
and
S
S
S
are the projections of
D
D
D
on the straight lines
A
C
,
A
E
,
B
E
AC, AE, BE
A
C
,
A
E
,
BE
and
B
C
BC
BC
. Prove that the points
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
and
S
S
S
lie either on a circle or on one straight line.
similar triangles externally to a triangle lead to a similar of circumcenters
On the sides of a triangle
X
Y
Z
XYZ
X
Y
Z
to the outside construct similar triangles
Y
D
Z
,
E
X
Z
,
Y
X
F
YDZ, EXZ ,YXF
Y
D
Z
,
EXZ
,
Y
XF
with circumcenters
K
,
L
,
M
K, L ,M
K
,
L
,
M
respectively. Here are
∠
Z
D
Y
=
∠
Z
X
E
=
∠
F
X
Y
\angle ZDY = \angle ZXE = \angle FXY
∠
Z
D
Y
=
∠
ZXE
=
∠
FX
Y
and
∠
Y
Z
D
=
∠
E
Z
X
=
∠
Y
F
X
\angle YZD = \angle EZX = \angle YFX
∠
Y
Z
D
=
∠
EZX
=
∠
Y
FX
. Show that the triangle
K
L
M
KLM
K
L
M
is similar to the triangles . https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png
2
2
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3 number game from 1 to 9999, 3 left to construct triangle, winning strategy
There are
9999
9999
9999
rods with lengths
1
,
2
,
.
.
.
,
9998
,
9999
1, 2, ..., 9998, 9999
1
,
2
,
...
,
9998
,
9999
. The players Anja and Bernd alternately remove one of the sticks, with Anja starting. The game ends when there are only three bars left. If from those three bars, a not degenerate triangle can be constructed then Anja wins, otherwise Bernd. Who has a winning strategy?
a_1 = 1, a_{n + 1} = [\sqrt{a_1+a_2+...+a_n} ]
The sequence of numbers
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
is defined recursively by
a
1
=
1
,
a
n
+
1
=
⌊
a
1
+
a
2
+
.
.
.
+
a
n
⌋
a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor
a
1
=
1
,
a
n
+
1
=
⌊
a
1
+
a
2
+
...
+
a
n
⌋
for
n
≥
1
n \ge 1
n
≥
1
. Find all numbers that appear more than twice at this sequence.
1
2
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t (a, b, c)= min{b/a,c/b} where a <= b <= c , sidelengths
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side lengths of an non-degenerate triangle with
a
≤
b
≤
c
a \le b \le c
a
≤
b
≤
c
. With
t
(
a
,
b
,
c
)
t (a, b, c)
t
(
a
,
b
,
c
)
denote the minimum of the quotients
b
a
\frac{b}{a}
a
b
and
c
b
\frac{c}{b}
b
c
. Find all values that
t
(
a
,
b
,
c
)
t (a, b, c)
t
(
a
,
b
,
c
)
can take.
1...1 2 1...1 is a prime
Exists a positive integer
n
n
n
such that the number
1...1
⏟
n
o
n
e
s
2
1...1
⏟
n
o
n
e
s
\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}
n
o
n
es
1...1
2
n
o
n
es
1...1
is a prime number?