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National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2017 Austria Beginners' Competition
2017 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
4
1
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algebra problem
How many solutions does the equation:
[
x
20
]
=
[
x
17
]
[\frac{x}{20}]=[\frac{x}{17}]
[
20
x
]
=
[
17
x
]
have over the set of positve integers?
[
a
]
[a]
[
a
]
denotes the largest integer that is less than or equal to
a
a
a
. Proposed by Karl Czakler
3
1
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Problem 3
. Anthony denotes in sequence all positive integers which are divisible by
2
2
2
. Bertha denotes in sequence all positive integers which are divisible by
3
3
3
. Claire denotes in sequence all positive integers which are divisible by
4
4
4
. Orderly Dora denotes all numbers written by the other three. Thereby she puts them in order by size and does not repeat a number. What is the
2017
t
h
2017th
2017
t
h
number in her list? ¨Proposed by Richard Henner
2
1
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geometry problem
. In the isosceles triangle
A
B
C
ABC
A
BC
with
A
C
=
B
C
AC = BC
A
C
=
BC
we denote by
D
D
D
the foot of the altitude through
C
C
C
. The midpoint of
C
D
CD
C
D
is denoted by
M
M
M
. The line
B
M
BM
BM
intersects
A
C
AC
A
C
in
E
E
E
. Prove that the length of
A
C
AC
A
C
is three times that of
C
E
CE
CE
.
1
1
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inequality
The nonnegative real numbers
a
a
a
and
b
b
b
satisfy
a
+
b
=
1
a + b = 1
a
+
b
=
1
. Prove that:
1
2
≤
a
3
+
b
3
a
2
+
b
2
≤
1
\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1
2
1
≤
a
2
+
b
2
a
3
+
b
3
≤
1
When do we have equality in the right inequality and when in the left inequality? Proposed by Walther Janous