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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2017 German National Olympiad
2017 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
6
1
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Consecutive perfect squares adding up to a perfect cube
Prove that there exist infinitely many positive integers
m
m
m
such that there exist
m
m
m
consecutive perfect squares with sum
m
3
m^3
m
3
. Specify one solution with
m
>
1
m>1
m
>
1
.
5
1
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Symmetric inequality with two directions
Prove that for all non-negative numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfying
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
, one has
1
≤
x
1
−
y
z
+
y
1
−
z
x
+
z
1
−
x
y
≤
9
8
.
1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.
1
≤
1
−
yz
x
+
1
−
z
x
y
+
1
−
x
y
z
≤
8
9
.
4
1
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Two circles tangential to sides of a cyclic quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. The point
P
P
P
is chosen on the line
A
B
AB
A
B
such that the circle passing through
C
,
D
C,D
C
,
D
and
P
P
P
touches the line
A
B
AB
A
B
. Similarly, the point
Q
Q
Q
is chosen on the line
C
D
CD
C
D
such that the circle passing through
A
,
B
A,B
A
,
B
and
Q
Q
Q
touches the line
C
D
CD
C
D
.Prove that the distance between
P
P
P
and the line
C
D
CD
C
D
equals the distance between
Q
Q
Q
and
A
B
AB
A
B
.
3
1
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A martial game with two options
General Tilly and the Duke of Wallenstein play "Divide and rule!" (Divide et impera!). To this end, they arrange
N
N
N
tin soldiers in
M
M
M
companies and command them by turns. Both of them must give a command and execute it in their turn.Only two commands are possible: The command "Divide!" chooses one company and divides it into two companies, where the commander is free to choose their size, the only condition being that both companies must contain at least one tin soldier. On the other hand, the command "Rule!" removes exactly one tin soldier from each company.The game is lost if in your turn you can't give a command without losing a company. Wallenstein starts to command.a) Can he force Tilly to lose if they start with
7
7
7
companies of
7
7
7
tin soldiers each?b) Who loses if they start with
M
≥
1
M \ge 1
M
≥
1
companies consisting of
n
1
≥
1
,
n
2
≥
1
,
…
,
n
M
≥
1
n_1 \ge 1, n_2 \ge 1, \dotsc, n_M \ge 1
n
1
≥
1
,
n
2
≥
1
,
…
,
n
M
≥
1
(
n
1
+
n
2
+
…
+
n
M
=
N
)
(n_1+n_2+\dotsc+n_M=N)
(
n
1
+
n
2
+
…
+
n
M
=
N
)
tin soldiers?
2
1
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Two circumcircles intersecting in a fixed point
Let
A
B
C
ABC
A
BC
be a triangle such that
∣
A
B
∣
≠
∣
A
C
∣
\vert AB\vert \ne \vert AC\vert
∣
A
B
∣
=
∣
A
C
∣
. Prove that there exists a point
D
≠
A
D \ne A
D
=
A
on its circumcircle satisfying the following property: For any points
M
,
N
M, N
M
,
N
outside the circumcircle on the rays
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, satisfying
∣
B
M
∣
=
∣
C
N
∣
\vert BM\vert=\vert CN\vert
∣
BM
∣
=
∣
CN
∣
, the circumcircle of
A
M
N
AMN
A
MN
passes through
D
D
D
.
1
1
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A system of two quadratic equations
Given two real numbers
p
p
p
and
q
q
q
, we study the following system of equations with variables
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
: \begin{align*} x^2+py+q&=0,\\ y^2+px+q&=0. \end{align*} Determine the number of distinct solutions
(
x
,
y
)
(x,y)
(
x
,
y
)
in terms of
p
p
p
and
q
q
q
.