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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2020 Federal Competition For Advanced Students, P1
2020 Federal Competition For Advanced Students, P1
Part of
Austrian MO National Competition
Subcontests
(4)
2
1
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segment of projections is half as sidelength, right triangle inscribed in right
Let
A
B
C
ABC
A
BC
be a right triangle with a right angle in
C
C
C
and a circumcenter
U
U
U
. On the sides
A
C
AC
A
C
and
B
C
BC
BC
, the points
D
D
D
and
E
E
E
lie in such a way that
∠
E
U
D
=
9
0
o
\angle EUD = 90 ^o
∠
E
U
D
=
9
0
o
. Let
F
F
F
and
G
G
G
be the projection of
D
D
D
and
E
E
E
on
A
B
AB
A
B
, respectively. Prove that
F
G
FG
FG
is half as long as
A
B
AB
A
B
.(Walther Janous)
3
1
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Problem 3 - Greatest Common Divisor
On a blackboard there are three positive integers. In each step the three numbers on the board are denoted as
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
a
>
g
c
d
(
b
,
c
)
a >gcd(b, c)
a
>
g
c
d
(
b
,
c
)
, then
a
a
a
gets replaced by
a
−
g
c
d
(
b
,
c
)
a-gcd(b, c)
a
−
g
c
d
(
b
,
c
)
. The game ends if there is no way to denote the numbers such that
a
>
g
c
d
(
b
,
c
)
a >gcd(b, c)
a
>
g
c
d
(
b
,
c
)
.Prove that the game always ends and that the last three numbers on the blackboard only depend on the starting numbers.(Theresia Eisenkölbl)
1
1
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Problem 1 - Inequality
Let
x
,
y
x, y
x
,
y
and
z
z
z
be positive real numbers such that
x
≥
y
+
z
x \geq y+z
x
≥
y
+
z
. Proof that
x
+
y
z
+
y
+
z
x
+
z
+
x
y
≥
7
\frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7
z
x
+
y
+
x
y
+
z
+
y
z
+
x
≥
7
When does equality occur?(Walther Janous)
4
1
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Problem 4 - Perfect Squares
Determine all positive integers
N
N
N
such that
2
N
−
2
N
2^N-2N
2
N
−
2
N
is a perfect square.(Walther Janous)