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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2018 Federal Competition For Advanced Students, P2
2018 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
1
1
Hide problems
f(f(x) + y) = ax + 1/f (1/y) , when x,y >0
Let
a
≠
0
a \ne 0
a
=
0
be a real number. Find all functions
f
:
R
>
0
→
R
>
0
f : R_{>0}\to R_{>0}
f
:
R
>
0
→
R
>
0
with
f
(
f
(
x
)
+
y
)
=
a
x
+
1
f
(
1
y
)
f(f(x) + y) = ax + \frac{1}{f\left(\frac{1}{y}\right)}
f
(
f
(
x
)
+
y
)
=
a
x
+
f
(
y
1
)
1
for all
x
,
y
∈
R
>
0
x, y \in R_{>0}
x
,
y
∈
R
>
0
.(Proposed by Walther Janous)
6
1
Hide problems
find digits z , when decimal representation of n^9 ends with at least k digits z
Determine all digits
z
z
z
such that for each integer
k
≥
1
k \ge 1
k
≥
1
there exists an integer
n
≥
1
n\ge 1
n
≥
1
with the property that the decimal representation of
n
9
n^9
n
9
ends with at least
k
k
k
digits
z
z
z
.(Proposed by Walther Janous)
5
1
Hide problems
2018 numbers on a circle, labeled with an integer, > sum of 2 preceding, max
On a circle
2018
2018
2018
points are marked. Each of these points is labeled with an integer. Let each number be larger than the sum of the preceding two numbers in clockwise order. Determine the maximal number of positive integers that can occur in such a configuration of
2018
2018
2018
integers.(Proposed by Walther Janous)
2
1
Hide problems
AB + BD > AC + CD in cyclic ABCD, AB is the longest side
Let
A
,
B
,
C
A, B, C
A
,
B
,
C
and
D
D
D
be four different points lying on a common circle in this order. Assume that the line segment
A
B
AB
A
B
is the (only) longest side of the inscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
. Prove that the inequality
A
B
+
B
D
>
A
C
+
C
D
AB + BD > AC + CD
A
B
+
B
D
>
A
C
+
C
D
holds.(Proposed by Karl Czakler)
3
1
Hide problems
distributing pieces of candy in rounds in children, relative prime related
There are
n
n
n
children in a room. Each child has at least one piece of candy. In Round
1
1
1
, Round
2
2
2
, etc., additional pieces of candy are distributed among the children according to the following rule: In Round
k
k
k
, each child whose number of pieces of candy is relatively prime to
k
k
k
receives an additional piece. Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy.(Proposed by Theresia Eisenkölbl)
4
1
Hide problems
DE = AC + BC , related to collinear points, and 2 intersecting circumcircles
Let
A
B
C
ABC
A
BC
be a triangle and
P
P
P
a point inside the triangle such that the centers
M
B
M_B
M
B
and
M
A
M_A
M
A
of the circumcircles
k
B
k_B
k
B
and
k
A
k_A
k
A
of triangles
A
C
P
ACP
A
CP
and
B
C
P
BCP
BCP
, respectively, lie outside the triangle
A
B
C
ABC
A
BC
. In addition, we assume that the three points
A
,
P
A, P
A
,
P
and
M
A
M_A
M
A
are collinear as well as the three points
B
,
P
B, P
B
,
P
and
M
B
M_B
M
B
. The line through
P
P
P
parallel to side
A
B
AB
A
B
intersects circles
k
A
k_A
k
A
and
k
B
k_B
k
B
in points
D
D
D
and
E
E
E
, respectively, where
D
,
E
≠
P
D, E \ne P
D
,
E
=
P
. Show that
D
E
=
A
C
+
B
C
DE = AC + BC
D
E
=
A
C
+
BC
.(Proposed by Walther Janous)