MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2022 Austrian MO Regional Competition
2022 Austrian MO Regional Competition
Part of
Austrian MO Regional Competition
Subcontests
(4)
1
1
Hide problems
1/(1-a)+ 1/(1-b) >= 4 if a,b>0 with a^2 + b^2 =1/2
Let
a
a
a
and
b
b
b
be positive real numbers with
a
2
+
b
2
=
1
2
a^2 + b^2 =\frac12
a
2
+
b
2
=
2
1
. Prove that
1
1
−
a
+
1
1
−
b
≥
4.
\frac{1}{1 - a}+\frac{1}{1-b}\ge 4.
1
−
a
1
+
1
−
b
1
≥
4.
When does equality hold?(Walther Janous)
4
1
Hide problems
subsets of {-2^k, 2^k} fot k=0,1,...,2022
We are given the set
M
=
{
−
2
2022
,
−
2
2021
,
.
.
.
,
−
2
2
,
−
2
,
−
1
,
1
,
2
,
2
2
,
.
.
.
,
2
2021
,
2
2022
}
.
M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.
M
=
{
−
2
2022
,
−
2
2021
,
...
,
−
2
2
,
−
2
,
−
1
,
1
,
2
,
2
2
,
...
,
2
2021
,
2
2022
}
.
Let
T
T
T
be a subset of
M
M
M
, such that neighbouring numbers have the same difference when the elements are ordered by size. (a) Determine the maximum number of elements that such a set
T
T
T
can contain. (b) Determine all sets
T
T
T
with the maximum number of elements.(Walther Janous)
3
1
Hide problems
CU, PI concurrent with circumcircle, incircle, 1 more circumcircle
Let
A
B
C
ABC
A
BC
denote a triangle with
A
C
≠
B
C
AC\ne BC
A
C
=
BC
. Let
I
I
I
and
U
U
U
denote the incenter and circumcenter of the triangle
A
B
C
ABC
A
BC
, respectively. The incircle touches
B
C
BC
BC
and
A
C
AC
A
C
in the points
D
D
D
and E, respectively. The circumcircles of the triangles
A
B
C
ABC
A
BC
and
C
D
E
CDE
C
D
E
intersect in the two points
C
C
C
and
P
P
P
. Prove that the common point
S
S
S
of the lines
C
U
CU
C
U
and
P
I
P I
P
I
lies on the circumcircle of the triangle
A
B
C
ABC
A
BC
.(Karl Czakler)
2
1
Hide problems
no of 10-digit numbers wanted, neighbouring digits
Determine the number of ten-digit positive integers with the following properties:
∙
\bullet
∙
Each of the digits
0
,
1
,
2
,
.
.
.
,
8
0, 1, 2, . . . , 8
0
,
1
,
2
,
...
,
8
and
9
9
9
is contained exactly once.
∙
\bullet
∙
Each digit, except
9
9
9
, has a neighbouring digit that is larger than it. (Note. For example, in the number
1230
1230
1230
, the digits
1
1
1
and
3
3
3
are the neighbouring digits of
2
2
2
while
2
2
2
and
0
0
0
are the neighbouring digits of
3
3
3
. The digits
1
1
1
and
0
0
0
have only one neighbouring digit.)(Karl Czakler)