MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2019 Federal Competition For Advanced Students, P2
2019 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
3
1
Hide problems
two countries, different stamp values, n cents total cost
In Oddland there are stamps with values of
1
1
1
cent,
3
3
3
cents,
5
5
5
cents, etc., each for odd number there is exactly one stamp type. Oddland Post dictates: For two different values on a letter must be the number of stamps of the lower one value must be at least as large as the number of tokens of the higher value. In Squareland, on the other hand, there are stamps with values of
1
1
1
cent,
4
4
4
cents,
9
9
9
cents, etc. there is exactly one stamp type for each square number. Brands can be found in Squareland can be combined as required without further regulations. Prove for every positive integer
n
n
n
: there are the same number in the two countries possibilities to send a letter with stamps worth a total of
n
n
n
cents. It makes no difference if you have the same stamps on arrange a letter differently.(Stephan Wagner)
5
1
Hide problems
concyclic wanted, equal ratio of segments, altitudes related
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. Let
D
D
D
and
E
E
E
be the feet of the altitudes on the sides
B
C
BC
BC
or
A
C
AC
A
C
. Points
F
F
F
and
G
G
G
are located on the lines
A
D
AD
A
D
and
B
E
BE
BE
in such a way that
A
F
F
D
=
B
G
G
E
\frac{AF}{FD}=\frac{BG}{GE}
F
D
A
F
=
GE
BG
. The line passing through
C
C
C
and
F
F
F
intersects
B
E
BE
BE
at point
H
H
H
, and the line passing through
C
C
C
and
G
G
G
intersects
A
D
AD
A
D
at point
I
I
I
. Prove that points
F
,
G
,
H
F, G, H
F
,
G
,
H
and
I
I
I
lie on a circle.(Walther Janous)
2
1
Hide problems
condition for equal angles wanted, isosceles trapezoid, tangents of circumcircle
A (convex) trapezoid
A
B
C
D
ABCD
A
BC
D
is good, if it is inscribed in a circle, sides
A
B
AB
A
B
and
C
D
CD
C
D
are the bases and
C
D
CD
C
D
is shorter than
A
B
AB
A
B
. For a good trapezoid
A
B
C
D
ABCD
A
BC
D
the following terms are defined:
∙
\bullet
∙
The parallel to
A
D
AD
A
D
passing through
B
B
B
intersects the extension of side
C
D
CD
C
D
at point
S
S
S
.
∙
\bullet
∙
The two tangents passing through
S
S
S
on the circumircle of the trapezoid touch the circle at
E
E
E
and
F
F
F
, where
E
E
E
lies on the same side of the straight line
C
D
CD
C
D
as
A
A
A
. Give the simplest possible equivalent condition (expressed in side lengths and / or angles of the trapezoid) so that with a good trapezoid
A
B
C
D
ABCD
A
BC
D
the two angles
∠
B
S
E
\angle BSE
∠
BSE
and
∠
F
S
C
\angle FSC
∠
FSC
have the same measure.(Walther Janous)
6
1
Hide problems
if x | y^3 , y | z^3, z | x^3, and xyz | (x + y + z) ^n , find min n
Find the smallest possible positive integer n with the following property: For all positive integers
x
,
y
x, y
x
,
y
and
z
z
z
with
x
∣
y
3
x | y^3
x
∣
y
3
and
y
∣
z
3
y | z^3
y
∣
z
3
and
z
∣
x
3
z | x^3
z
∣
x
3
always to be true that
x
y
z
∣
(
x
+
y
+
z
)
n
xyz| (x + y + z) ^n
x
yz
∣
(
x
+
y
+
z
)
n
.(Gerhard J. Woeginger)
1
1
Hide problems
f(2x + f(y)) = x + y + f(x)
Determine all functions
f
:
R
→
R
f: R\to R
f
:
R
→
R
, such that
f
(
2
x
+
f
(
y
)
)
=
x
+
y
+
f
(
x
)
f (2x + f (y)) = x + y + f (x)
f
(
2
x
+
f
(
y
))
=
x
+
y
+
f
(
x
)
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.(Gerhard Kirchner)
4
1
Hide problems
Simple inequality
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the positive real numbers such that
a
+
b
+
c
+
2
=
a
b
c
.
a+b+c+2=abc .
a
+
b
+
c
+
2
=
ab
c
.
Prove that
(
a
+
1
)
(
b
+
1
)
(
c
+
1
)
≥
27.
(a+1)(b+1)(c+1)\geq 27.
(
a
+
1
)
(
b
+
1
)
(
c
+
1
)
≥
27.