MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2020 Federal Competition For Advanced Students, P2
2020 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
Hide problems
polynomial 2 player game. x^n + a_{n-1}x^{n- 1} +... + a_0
The players Alfred and Bertrand put together a polynomial
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
0
x^n + a_{n-1}x^{n- 1} +... + a_0
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
0
with the given degree
n
≥
2
n \ge 2
n
≥
2
. To do this, they alternately choose the value in
n
n
n
moves one coefficient each, whereby all coefficients must be integers and
a
0
≠
0
a_0 \ne 0
a
0
=
0
must apply. Alfred's starts first . Alfred wins if the polynomial has an integer zero at the end. (a) For which
n
n
n
can Alfred force victory if the coefficients
a
j
a_j
a
j
are from the right to the left, i.e. for
j
=
0
,
1
,
.
.
.
,
n
−
1
j = 0, 1,. . . , n - 1
j
=
0
,
1
,
...
,
n
−
1
, be determined? (b) For which
n
n
n
can Alfred force victory if the coefficients
a
j
a_j
a
j
are from the left to the right, i.e. for
j
=
n
−
1
,
n
−
2
,
.
.
.
,
0
j = n -1, n - 2,. . . , 0
j
=
n
−
1
,
n
−
2
,
...
,
0
, be determined?(Theresia Eisenkölbl, Clemens Heuberger)
2
1
Hide problems
2020 points, black and green
In the plane there are
2020
2020
2020
points, some of which are black and the rest are green. For every black point, the following applies: There are exactly two green points that represent the distance
2020
2020
2020
from that black point. Find the smallest possible number of green dots.(Walther Janous)
5
1
Hide problems
fixed angle wanted, 2 mixtlinears created by a perpendicular on semicircle
Let
h
h
h
be a semicircle with diameter
A
B
AB
A
B
. Let
P
P
P
be an arbitrary point inside the diameter
A
B
AB
A
B
. The perpendicular through
P
P
P
on
A
B
AB
A
B
intersects
h
h
h
at point
C
C
C
. The line
P
C
PC
PC
divides the semicircular area into two parts. A circle will be inscribed in each of them that touches
A
B
,
P
C
AB, PC
A
B
,
PC
and
h
h
h
. The points of contact of the two circles with
A
B
AB
A
B
are denoted by
D
D
D
and
E
E
E
, where
D
D
D
lies between
A
A
A
and
P
P
P
. Prove that the size of the angle
D
C
E
DCE
D
CE
does not depend on the choice of
P
P
P
.(Walther Janous)
1
1
Hide problems
concurrent wanted, starting with a cyclic ABCD and two circumcircles
Let
A
B
C
D
ABCD
A
BC
D
be a convex cyclic quadrilateral with the diagonal intersection
S
S
S
. Let further be
P
P
P
the circumcenter of the triangle
A
B
S
ABS
A
BS
and
Q
Q
Q
the circumcenter of the triangle
B
C
S
BCS
BCS
. The parallel to
A
D
AD
A
D
through
P
P
P
and the parallel to
C
D
CD
C
D
through
Q
Q
Q
intersect at point
R
R
R
. Prove that
R
R
R
is on
B
D
BD
B
D
.(Karl Czakler)
4
1
Hide problems
f(xf(y)+1)=y+f(f(x)f(y))
Determine all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
, such that
f
(
x
f
(
y
)
+
1
)
=
y
+
f
(
f
(
x
)
f
(
y
)
)
f(xf(y)+1)=y+f(f(x)f(y))
f
(
x
f
(
y
)
+
1
)
=
y
+
f
(
f
(
x
)
f
(
y
))
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
.(Theresia Eisenkölbl)
3
1
Hide problems
Interesting sequence
Let
a
a
a
be a fixed positive integer and
(
e
n
)
(e_n)
(
e
n
)
the sequence, which is defined by
e
0
=
1
e_0=1
e
0
=
1
and
e
n
=
a
+
∏
k
=
0
n
−
1
e
k
e_n=a + \prod_{k=0}^{n-1} e_k
e
n
=
a
+
k
=
0
∏
n
−
1
e
k
for
n
≥
1
n \geq 1
n
≥
1
.Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence.(Theresia Eisenkölbl)