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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2015 Federal Competition For Advanced Students, P2
2015 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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Problem 6 — Cash Can
Max has
2015
2015
2015
jars labeled with the numbers
1
1
1
to
2015
2015
2015
and an unlimited supply of coins.Consider the following starting configurations:(a) All jars are empty. (b) Jar
1
1
1
contains
1
1
1
coin, jar
2
2
2
contains
2
2
2
coins, and so on, up to jar
2015
2015
2015
which contains
2015
2015
2015
coins. (c) Jar
1
1
1
contains
2015
2015
2015
coins, jar
2
2
2
contains
2014
2014
2014
coins, and so on, up to jar
2015
2015
2015
which contains
1
1
1
coin.Now Max selects in each step a number
n
n
n
from
1
1
1
to
2015
2015
2015
and adds
n
n
n
to each jar except to the jar
n
n
n
.Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar.(Birgit Vera Schmidt)
5
1
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Problem 5 -- Making A Point Here
Let I be the incenter of triangle
A
B
C
ABC
A
BC
and let
k
k
k
be a circle through the points
A
A
A
and
B
B
B
. The circle intersects* the line
A
I
AI
A
I
in points
A
A
A
and
P
P
P
* the line
B
I
BI
B
I
in points
B
B
B
and
Q
Q
Q
* the line
A
C
AC
A
C
in points
A
A
A
and
R
R
R
* the line
B
C
BC
BC
in points
B
B
B
and
S
S
S
with none of the points
A
,
B
,
P
,
Q
,
R
A,B,P,Q,R
A
,
B
,
P
,
Q
,
R
and
S
S
S
coinciding and such that
R
R
R
and
S
S
S
are interior points of the line segments
A
C
AC
A
C
and
B
C
BC
BC
, respectively.Prove that the lines
P
S
PS
PS
,
Q
R
QR
QR
, and
C
I
CI
C
I
meet in a single point.(Stephan Wagner)
4
1
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Problem 4 -- Three Fractions Not Enough
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers with
x
+
y
+
z
≥
3
x+y+z \ge 3
x
+
y
+
z
≥
3
. Prove that
1
x
+
y
+
z
2
+
1
y
+
z
+
x
2
+
1
z
+
x
+
y
2
≤
1
\frac{1}{x+y+z^2} + \frac{1}{y+z+x^2} + \frac{1}{z+x+y^2} \le 1
x
+
y
+
z
2
1
+
y
+
z
+
x
2
1
+
z
+
x
+
y
2
1
≤
1
When does equality hold?(Karl Czakler)
3
1
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Problem 3 -- Topsy Turvy Bases
We consider the following operation applied to a positive integer: The integer is represented in an arbitrary base
b
≥
2
b \ge 2
b
≥
2
, in which it has exactly two digits and in which both digits are different from
0
0
0
. Then the two digits are swapped and the result in base
b
b
b
is the new number.Is it possible to transform every number
>
10
> 10
>
10
to a number
≤
10
\le 10
≤
10
with a series of such operations?(Theresia Eisenkölbl)
2
1
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Problem 2 -- Some Cyclic Points
We are given a triangle
A
B
C
ABC
A
BC
. Let
M
M
M
be the mid-point of its side
A
B
AB
A
B
.Let
P
P
P
be an interior point of the triangle. We let
Q
Q
Q
denote the point symmetric to
P
P
P
with respect to
M
M
M
.Furthermore, let
D
D
D
and
E
E
E
be the common points of
A
P
AP
A
P
and
B
P
BP
BP
with sides
B
C
BC
BC
and
A
C
AC
A
C
, respectively.Prove that points
A
A
A
,
B
B
B
,
D
D
D
, and
E
E
E
lie on a common circle if and only if
∠
A
C
P
=
∠
Q
C
B
\angle ACP = \angle QCB
∠
A
CP
=
∠
QCB
holds.(Karl Czakler)
1
1
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Problem 1 -- Punctual Functional
Let
f
:
Z
>
0
→
Z
f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}
f
:
Z
>
0
→
Z
be a function with the following properties:(i)
f
(
1
)
=
0
f(1) = 0
f
(
1
)
=
0
(ii)
f
(
p
)
=
1
f(p) = 1
f
(
p
)
=
1
for all prime numbers
p
p
p
(iii)
f
(
x
y
)
=
y
⋅
f
(
x
)
+
x
⋅
f
(
y
)
f(xy) = y \cdot f(x) + x \cdot f(y)
f
(
x
y
)
=
y
⋅
f
(
x
)
+
x
⋅
f
(
y
)
for all
x
,
y
x,y
x
,
y
in
Z
>
0
\mathbb{Z}_{>0}
Z
>
0
Determine the smallest integer
n
≥
2015
n \ge 2015
n
≥
2015
that satisfies
f
(
n
)
=
n
f(n) = n
f
(
n
)
=
n
.(Gerhard J. Woeginger)