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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2022 Austrian MO National Competition
2022 Austrian MO National Competition
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
Hide problems
square 1000x1000 divides into 31 square tiles
(a) Prove that a square with sides
1000
1000
1000
divided into
31
31
31
squares tiles, at least one of which has a side length less than
1
1
1
.(b) Show that a corresponding decomposition into
30
30
30
squares is also possible.(Walther Janous)
5
1
Hide problems
for any choice of P the points D_P , E_P and F are collinear
Let
A
B
C
ABC
A
BC
be an isosceles triangle with base
A
B
AB
A
B
. We choose a point
P
P
P
inside the triangle on altitude through
C
C
C
. The circle with diameter
C
P
CP
CP
intersects the straight line through
B
B
B
and
P
P
P
again at the point
D
P
D_P
D
P
and the Straight through
A
A
A
and
C
C
C
one more time at point
E
P
E_P
E
P
. Prove that there is a point
F
F
F
such that for any choice of
P
P
P
the points
D
P
,
E
P
D_P , E_P
D
P
,
E
P
and
F
F
F
lie on a straight line.(Walther Janous)
1
2
Hide problems
0 < 1/(x + y + z + 1) - 1/(x + 1)(y + 1)(z + 1)} <= 1/8 for x,y,z>0
Prove that for all positive real numbers
x
,
y
x, y
x
,
y
and
z
z
z
, the double inequality
0
<
1
x
+
y
+
z
+
1
−
1
(
x
+
1
)
(
y
+
1
)
(
z
+
1
)
≤
1
8
0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18
0
<
x
+
y
+
z
+
1
1
−
(
x
+
1
)
(
y
+
1
)
(
z
+
1
)
1
≤
8
1
holds. When does equality hold in the right inequality?(Walther Janous)
find all functions , a - f(b) | af(a) - bf(b)
Find all functions
f
:
Z
>
0
→
Z
>
0
f : Z_{>0} \to Z_{>0}
f
:
Z
>
0
→
Z
>
0
with
a
−
f
(
b
)
∣
a
f
(
a
)
−
b
f
(
b
)
a - f(b) | af(a) - bf(b)
a
−
f
(
b
)
∣
a
f
(
a
)
−
b
f
(
b
)
for all
a
,
b
∈
Z
>
0
a, b \in Z_{>0}
a
,
b
∈
Z
>
0
.(Theresia Eisenkoelbl)
4
2
Hide problems
4q - 1 is prime and (p + q)/(p + r)= r - p, diophantine in primes
Find all triples
(
p
,
q
,
r
)
(p, q, r)
(
p
,
q
,
r
)
of prime numbers for which
4
q
−
1
4q - 1
4
q
−
1
is a prime number and
p
+
q
p
+
r
=
r
−
p
\frac{p + q}{p + r} = r - p
p
+
r
p
+
q
=
r
−
p
holds.(Walther Janous)
infinite many primes divide P(n)
Decide whether for every polynomial
P
P
P
of degree at least
1
1
1
, there exist infinitely many primes that divide
P
(
n
)
P(n)
P
(
n
)
for at least one positive integer
n
n
n
.(Walther Janous)
3
2
Hide problems
one person at each whole number on number line from 0-2022
Each person stands on a whole number on the number line from
0
0
0
to
2022
2022
2022
. In each turn, two people are selected by a distance of at least
2
2
2
. These go towards each other by
1
1
1
. When no more such moves are possible, the process ends. Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.)(Birgit Vera Schmidt)[hide=original wording]Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person. In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang. Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)
2022 -> 210 -> 21 ->6 -> 24 -> 18 -> 33 -> 15 -> 21
Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following: The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. Lisa repeats this until she gets a number for the first time was on the board. (a) Show that the sequence of moves always ends. (b) If Lisa begins with the number
5
3
2022
−
1
53^{2022} - 1
5
3
2022
−
1
, what is the last number on the board?Example: If Lisa starts with the number
2022
2022
2022
, she gets
202
+
4
⋅
2
=
210
202 + 4\cdot 2 = 210
202
+
4
⋅
2
=
210
in the first move and overall the result
2022
→
210
→
21
→
6
→
24
→
18
→
33
→
15
→
21
2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21
2022
→
210
→
21
→
6
→
24
→
18
→
33
→
15
→
21
. Since Lisa gets
21
21
21
for the second time, the turn order ends.(Stephan Pfannerer)
2
2
Hide problems
CD = 2 OF , cyclic ABCD with _|_ diagonals
The points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
lie in this order on a circle with center
O
O
O
. Furthermore, the straight lines
A
C
AC
A
C
and
B
D
BD
B
D
should be perpendicular to each other. The base of the perpendicular from
O
O
O
on
A
B
AB
A
B
is
F
F
F
. Prove
C
D
=
2
O
F
CD = 2 OF
C
D
=
2
OF
.(Karl Czakler)
MS =MF wanted, orthocenter, angle bisector, perpendicular, perp. bisector
Let
A
B
C
ABC
A
BC
be an acute-angled, non-isosceles triangle with orthocenter
H
H
H
,
M
M
M
midpoint of side
A
B
AB
A
B
and
w
w
w
bisector of angle
∠
A
C
B
\angle ACB
∠
A
CB
. Let
S
S
S
be the point of intersection of the perpendicular bisector of side
A
B
AB
A
B
with
w
w
w
and
F
F
F
the foot of the perpendicular from
H
H
H
on
w
w
w
. Prove that the segments
M
S
MS
MS
and
M
F
MF
MF
are equal.(Karl Czakler)