MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2019 Austrian Junior Regional Competition
2019 Austrian Junior Regional Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
2
1
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60^o angle wanted, equilateral on a square
A square
A
B
C
D
ABCD
A
BC
D
is given. Over the side
B
C
BC
BC
draw an equilateral triangle
B
C
S
BCS
BCS
on the outside. The midpoint of the segment
A
S
AS
A
S
is
N
N
N
and the midpoint of the side
C
D
CD
C
D
is
H
H
H
. Prove that
∠
N
H
C
=
6
0
o
\angle NHC = 60^o
∠
N
H
C
=
6
0
o
. . (Karl Czakler)
1
1
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(x^2 + y^2)/(x + y)= 10 diophantine
Let
x
x
x
and
y
y
y
be integers with
x
+
y
≠
0
x + y \ne 0
x
+
y
=
0
. Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
x
2
+
y
2
x
+
y
=
10.
\frac{x^2 + y^2}{x + y}= 10.
x
+
y
x
2
+
y
2
=
10.
(Walther Janous)
3
1
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2player game, adding numbers, whoever reaches no >= 2019 wins
Alice and Bob are playing a year number game. There will be two game numbers
19
19
19
and
20
20
20
and one starting number from the set
{
9
,
10
}
\{9, 10\}
{
9
,
10
}
used. Alice chooses independently her game number and Bob chooses the starting number. The other number is given to Bob. Then Alice adds her game number to the starting number, Bob adds his game number to the result, Alice adds her number of games to the result, etc. The game continues until the number
2019
2019
2019
is reached or exceeded. Whoever reaches the number
2019
2019
2019
wins. If
2019
2019
2019
is exceeded, the game ends in a draw.
∙
\bullet
∙
Show that Bob cannot win.
∙
\bullet
∙
What starting number does Bob have to choose to prevent Alice from winning?(Richard Henner)
4
1
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sum of 4 primes with 5 <p <q <r <s <p + 10 is divisible by 60
Let
p
,
q
,
r
p, q, r
p
,
q
,
r
and
s
s
s
be four prime numbers such that
5
<
p
<
q
<
r
<
s
<
p
+
10.
5 <p <q <r <s <p + 10.
5
<
p
<
q
<
r
<
s
<
p
+
10.
Prove that the sum of the four prime numbers is divisible by
60
60
60
.(Walther Janous)