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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2009 Bundeswettbewerb Mathematik
2009 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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decimal palindrome
A positive integer is called decimal palindrome if its decimal representation
z
n
.
.
.
z
0
z_n...z_0
z
n
...
z
0
with
z
n
≠
0
z_n\ne 0
z
n
=
0
is mirror symmetric, i.e. if
z
k
=
z
n
−
k
z_k = z_{n-k}
z
k
=
z
n
−
k
applies to all
k
=
0
,
.
.
.
,
n
k= 0, ..., n
k
=
0
,
...
,
n
. Show that each integer that is not divisible by
10
10
10
has a positive multiple, which is a decimal palindrome.
no of diagonals inside a 2009-gon, every diagonals cuts at most another
How many diagonals can you draw in a convex
2009
2009
2009
-gon if in the finished drawing, every drawn diagonal inside the
2009
2009
2009
-gon may cut at most another drawn diagonal?
3
2
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concurrency related to symmetric wrt side midpoints
Let
P
P
P
be a point inside the triangle
A
B
C
ABC
A
BC
and
P
a
,
P
b
,
P
c
P_a, P_b ,P_c
P
a
,
P
b
,
P
c
be the symmetric points wrt the midpoints of the sides
B
C
,
C
A
,
A
B
BC, CA,AB
BC
,
C
A
,
A
B
respectively. Prove that that the lines
A
P
a
,
B
P
b
AP_a, BP_b
A
P
a
,
B
P
b
and
C
P
c
CP_c
C
P
c
are concurrent.
PQ/CQ<= [AB/(AC+CB)]^2 , angle bisector and circumcircle related
Given a triangle
A
B
C
ABC
A
BC
and a point
P
P
P
on the side
A
B
AB
A
B
. Let
Q
Q
Q
be the intersection of the straight line
C
P
CP
CP
(different from
C
C
C
) with the circumcicle of the triangle. Prove the inequality
P
Q
‾
C
Q
‾
≤
(
A
B
‾
A
C
‾
+
C
B
‾
)
2
\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2
CQ
PQ
≤
(
A
C
+
CB
A
B
)
2
and that equality holds if and only if the
C
P
CP
CP
is bisector of the angle
A
C
B
ACB
A
CB
. https://cdn.artofproblemsolving.com/attachments/b/1/068fafd5564e77930160115a1cd409c4fdbf61.png
1
2
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Find the missing digit
Determine all possible digits
z
z
z
for which
9...9
⏟
100
z
0...0
⏟
100
9
\underbrace{9...9}_{100}z\underbrace{0...0}_{100}9
100
9...9
z
100
0...0
9
is a square number.
game with 3 boxes and 2008, 2009,2010 game pieces, winning strategy
At the start of a game there are three boxes with
2008
,
2009
2008, 2009
2008
,
2009
and
2010
2010
2010
game pieces Anja and Bernd play in turns according to the following rule: When it is your turn, select two boxes, empty them and then distribute the pieces from the third box to the three boxes, such that no box may remain empty.If you can no longer complete a turn, you have lost. Who has a winning strategy when Anja starts?
2
2
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Maximum of m(a,b)
Let
a
,
b
a,b
a
,
b
be positive real numbers. Define
m
(
a
,
b
)
m(a,b)
m
(
a
,
b
)
as the minimum of $ a,\frac{1}{b} \text{ and } \frac{1}{a}+b.
F
i
n
d
t
h
e
m
a
x
i
m
u
m
o
f
Find the maximum of
F
in
d
t
h
e
ma
x
im
u
m
o
f
m(a,b).$
trinomial has 2 real roots such |x_2-x_1|><=1/n <=> n has 2 prime divisors
Let
n
n
n
be an integer that is greater than
1
1
1
. Prove that the following two statements are equivalent: (A) There are positive integers
a
,
b
a, b
a
,
b
and
c
c
c
that are not greater than
n
n
n
and for which that polynomial
a
x
2
+
b
x
+
c
ax^2 + bx + c
a
x
2
+
b
x
+
c
has two different real roots
x
1
x_1
x
1
and
x
2
x_2
x
2
with
∣
x
2
−
x
1
∣
≤
1
n
| x_2- x_1 | \le \frac{1}{n}
∣
x
2
−
x
1
∣
≤
n
1
(B) The number
n
n
n
has at least two different prime divisors.