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Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2023 Austrian MO National Competition
2023 Austrian MO National Competition
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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Equation has 3 distinct rational roots
Does there exist a real number
r
r
r
such that the equation
x
3
−
2023
x
2
−
2023
x
+
r
=
0
x^3-2023x^2-2023x+r=0
x
3
−
2023
x
2
−
2023
x
+
r
=
0
has three distinct rational roots?
5
1
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Well-known orthocenter problem
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
C
≠
B
C
AC\neq BC
A
C
=
BC
,
M
M
M
the midpoint of side
A
B
AB
A
B
,
H
H
H
is the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
,
D
D
D
on
B
C
BC
BC
is the foot of the altitude from
A
A
A
and
E
E
E
on
A
C
AC
A
C
is the foot of the perpendicular from
B
B
B
. Prove that the lines
A
B
,
D
E
AB, DE
A
B
,
D
E
and the perpendicular to
M
H
MH
M
H
through
C
C
C
are concurrent.
4
2
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Version of ISL 2014 C2
The number
2023
2023
2023
is written
2023
2023
2023
times on a blackboard. On one move, you can choose two numbers
x
,
y
x, y
x
,
y
on the blackboard, delete them and write
x
+
y
4
\frac{x+y} {4}
4
x
+
y
instead. Prove that when one number remains, it is greater than
1
1
1
.
NT equation with factorial and power of n
Find all pairs of positive integers
(
n
,
k
)
(n, k)
(
n
,
k
)
satisfying the equation
n
!
+
n
=
n
k
.
n!+n=n^k.
n
!
+
n
=
n
k
.
3
2
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Subsets with odd minimal element
Given a positive integer
n
n
n
, find the proportion of the subsets of
{
1
,
2
,
…
,
2
n
}
\{1,2, \ldots, 2n\}
{
1
,
2
,
…
,
2
n
}
such that their smallest element is odd.
2 player game drawing segments of length 1
Alice and Bob play a game, in which they take turns drawing segments of length
1
1
1
in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?
2
2
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Circumcenter lies on angle bisector
Given is a triangle
A
B
C
ABC
A
BC
. The points
P
,
Q
P, Q
P
,
Q
lie on the extensions of
B
C
BC
BC
beyond
B
,
C
B, C
B
,
C
, respectively, such that
B
P
=
B
A
BP=BA
BP
=
B
A
and
C
Q
=
C
A
CQ=CA
CQ
=
C
A
. Prove that the circumcenter of triangle
A
P
Q
APQ
A
PQ
lies on the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
.
Easy geo about circles with equal circumradiuses
Given is a triangle
A
B
C
ABC
A
BC
with circumcentre
O
O
O
. The circumcircle of triangle
A
O
C
AOC
A
OC
intersects side
B
C
BC
BC
at
D
D
D
and side
A
B
AB
A
B
at
E
E
E
. Prove that the triangles
B
D
E
BDE
B
D
E
and
A
O
C
AOC
A
OC
have circumradiuses of equal length.
1
2
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symmetric R to R FE
Given is a nonzero real number
α
\alpha
α
. Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
f
(
x
+
y
)
)
=
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
+
α
x
y
f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy
f
(
f
(
x
+
y
))
=
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
+
αx
y
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
.
4-variable inequality with square root
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive reals strictly smaller than
1
1
1
, such that
a
+
b
+
c
+
d
=
2
a+b+c+d=2
a
+
b
+
c
+
d
=
2
. Prove that
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
(
1
−
d
)
≤
a
c
+
b
d
2
.
\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}.
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
(
1
−
d
)
≤
2
a
c
+
b
d
.