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Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2022 Austrian MO Beginners' Competition
2022 Austrian MO Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
1
1
Hide problems
x/(y + 1) + y/(x + 1) >= 2/3 when x + y = 1, x,y>--1
Show that for all real numbers
x
x
x
and
y
y
y
with
x
>
−
1
x > -1
x
>
−
1
and
y
>
−
1
y > -1
y
>
−
1
and
x
+
y
=
1
x + y = 1
x
+
y
=
1
the inequality
x
y
+
1
+
y
x
+
1
≥
2
3
\frac{x}{y + 1} +\frac{y}{x + 1} \ge \frac23
y
+
1
x
+
x
+
1
y
≥
3
2
holds. When does equality apply?(Walther Janous)
4
1
Hide problems
p + q^2 = r^4 , diophantine with primes
Determine all prime numbers
p
,
q
p, q
p
,
q
and
r
r
r
with
p
+
q
2
=
r
4
p + q^2 = r^4
p
+
q
2
=
r
4
.(Karl Czakler)
3
1
Hide problems
PM = PS wanted, semicircle with center M
A semicircle is erected over the segment
A
B
AB
A
B
with center
M
M
M
. Let
P
P
P
be one point different from
A
A
A
and
B
B
B
on the semicircle and
Q
Q
Q
the midpoint of the arc of the circle
A
P
AP
A
P
. The point of intersection of the straight line
B
P
BP
BP
with the parallel to
P
Q
P Q
PQ
through
M
M
M
is
S
S
S
. Prove that
P
M
=
P
S
PM = PS
PM
=
PS
holds.(Karl Czakler)
2
1
Hide problems
2x1 and 3x1 tiles to cover 13x2
You are given a rectangular playing field of size
13
×
2
13 \times 2
13
×
2
and any number of dominoes of sizes
2
×
1
2\times 1
2
×
1
and
3
×
1
3\times 1
3
×
1
. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?(Walther Janous)