MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2019 German National Olympiad
2019 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
6
1
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A cyclic system of equations with two possible outcomes
Suppose that real numbers
x
,
y
x,y
x
,
y
and
z
z
z
satisfy the following equations:\begin{align*} x+\frac{y}{z} &=2,\\ y+\frac{z}{x} &=2,\\ z+\frac{x}{y} &=2. \end{align*}Show that
s
=
x
+
y
+
z
s=x+y+z
s
=
x
+
y
+
z
must be equal to
3
3
3
or
7
7
7
.Note: It is not required to show the existence of such numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
.
5
1
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Cutting a rope into pieces with a certain ratio
We are given two positive integers
p
p
p
and
q
q
q
.Step by step, a rope of length
1
1
1
is cut into smaller pieces as follows: In each step all the currently longest pieces are cut into two pieces with the ratio
p
:
q
p:q
p
:
q
at the same time. After an unknown number of such operations, the currently longest pieces have the length
x
x
x
.Determine in terms of
x
x
x
the number
a
(
x
)
a(x)
a
(
x
)
of different lengths of pieces of rope existing at that time.
4
1
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An old polynomial bijection between N and NxN
Show that for each non-negative integer
n
n
n
there are unique non-negative integers
x
x
x
and
y
y
y
such that we have
n
=
(
x
+
y
)
2
+
3
x
+
y
2
.
n=\frac{(x+y)^2+3x+y}{2}.
n
=
2
(
x
+
y
)
2
+
3
x
+
y
.
3
1
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Geometric combinatorics with rectangles in the cartesian plane
In the cartesian plane consider rectangles with sides parallel to the coordinate axes. We say that one rectangle is below another rectangle if there is a line
g
g
g
parallel to the
x
x
x
-axis such that the first rectangle is below
g
g
g
, the second one above
g
g
g
and both rectangles do not touch
g
g
g
.Similarly, we say that one rectangle is to the right of another rectangle if there is a line
h
h
h
parallel to the
y
y
y
-axis such that the first rectangle is to the right of
h
h
h
, the second one to the left of
h
h
h
and both rectangles do not touch
h
h
h
.Show that any finite set of
n
n
n
pairwise disjoint rectangles with sides parallel to the coordinate axes can be enumerated as a sequence
(
R
1
,
…
,
R
n
)
(R_1,\dots,R_n)
(
R
1
,
…
,
R
n
)
so that for all indices
i
,
j
i,j
i
,
j
with
1
≤
i
<
j
≤
n
1 \le i<j \le n
1
≤
i
<
j
≤
n
the rectangle
R
i
R_i
R
i
is to the right of or below the rectangle
R
j
R_j
R
j
2
1
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Geometry with three circles and angular bisectors
Let
a
a
a
and
b
b
b
be two circles, intersecting in two distinct points
Y
Y
Y
and
Z
Z
Z
. A circle
k
k
k
touches the circles
a
a
a
and
b
b
b
externally in the points
A
A
A
and
B
B
B
. Show that the angular bisectors of the angles
∠
Z
A
Y
\angle ZAY
∠
Z
A
Y
and
∠
Y
B
Z
\angle YBZ
∠
Y
BZ
intersect on the line
Y
Z
YZ
Y
Z
.
1
1
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System of equations which is slightly harder than it seems
Determine all real solutions
(
x
,
y
)
(x,y)
(
x
,
y
)
of the following system of equations: \begin{align*} x&=3x^2y-y^3,\\ y &= x^3-3xy^2 \end{align*}