MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2014 German National Olympiad
2014 German National Olympiad
Part of
German National Olympiad
Subcontests
(5)
6
1
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Angles in a circumscribed quadrilateral.
Let
A
B
C
D
ABCD
A
BC
D
be a circumscribed quadrilateral and
M
M
M
the centre of the incircle. There are points
P
P
P
and
Q
Q
Q
on the lines
M
A
MA
M
A
and
M
C
MC
MC
such that
∠
C
B
A
=
2
∠
Q
B
P
.
\angle CBA= 2\angle QBP.
∠
CB
A
=
2∠
QBP
.
Prove that
∠
A
D
C
=
2
∠
P
D
Q
.
\angle ADC = 2 \angle PDQ.
∠
A
D
C
=
2∠
P
D
Q
.
5
1
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Fake coins again, but now the scale is broken, too
There are
9
9
9
visually indistinguishable coins, and one of them is fake and thus lighter. We are given
3
3
3
indistinguishable balance scales to find the fake coin; however, one of the scales is defective and shows a random result each time. Show that the fake coin can still be found with
4
4
4
weighings.
3
1
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Sum of integers of the same colour has the same colour again
Given two positive integers
n
n
n
and
k
k
k
, we say that
k
k
k
is
n
n
n
-ergetic if: However the elements of
M
=
{
1
,
2
,
…
,
k
}
M=\{1,2,\ldots, k\}
M
=
{
1
,
2
,
…
,
k
}
are coloured in red and green, there exist
n
n
n
not necessarily distinct integers of the same colour whose sum is again an element of
M
M
M
of the same colour. For each positive integer
n
n
n
, determine the least
n
n
n
-ergetic integer, if it exists.
2
1
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Divisibility of a certain sequence
For a positive integer
n
n
n
, let
y
n
y_n
y
n
be the number of
n
n
n
-digit positive integers containing only the digits
2
,
3
,
5
,
7
2,3,5, 7
2
,
3
,
5
,
7
and which do not have a
5
5
5
directly to the right of a
2.
2.
2.
If
r
≥
1
r\geq 1
r
≥
1
and
m
≥
2
m\geq 2
m
≥
2
are integers, prove that
y
m
−
1
y_{m-1}
y
m
−
1
divides
y
r
m
−
1
.
y_{rm-1}.
y
r
m
−
1
.
1
1
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When is an expression prime?
For which non-negative integers
n
n
n
is
K
=
5
2
n
+
3
+
3
n
+
3
⋅
2
n
K=5^{2n+3} + 3^{n+3} \cdot 2^n
K
=
5
2
n
+
3
+
3
n
+
3
⋅
2
n
prime?