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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2016 Federal Competition For Advanced Students, P2
2016 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
5
1
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k tokens in nxn unit squares board, game conditions, min and max wanted
Consider a board consisting of
n
×
n
n\times n
n
×
n
unit squares where
n
≥
2
n \ge 2
n
≥
2
. Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain
k
k
k
tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists. (a) Find the minimal
k
k
k
such that the game does not end for any starting configuration and choice of cells during the game. (b) Find the maximal
k
k
k
such that the game ends for any starting configuration and choice of cells during the game.Proposed by Theresia Eisenkölbl
3
1
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arrangements no 1 to 64 on the squares of an 8x8 chessboard
Consider arrangements of the numbers
1
1
1
through
64
64
64
on the squares of an
8
×
8
8\times 8
8
×
8
chess board, where each square contains exactly one number and each number appears exactly once. A number in such an arrangement is called super-plus-good, if it is the largest number in its row and at the same time the smallest number in its column. Prove or disprove each of the following statements: (a) Each such arrangement contains at least one super-plus-good number. (b) Each such arrangement contains at most one super-plus-good number.Proposed by Gerhard J. Woeginger
2
1
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austrian midpoint, incircle and perpendiculars related
Let
A
B
C
ABC
A
BC
be a triangle. Its incircle meets the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
in the points
D
,
E
D, E
D
,
E
and
F
F
F
, respectively. Let
P
P
P
denote the intersection point of
E
D
ED
E
D
and the line perpendicular to
E
F
EF
EF
and passing through
F
F
F
, and similarly let
Q
Q
Q
denote the intersection point of
E
F
EF
EF
and the line perpendicular to
E
D
ED
E
D
and passing through
D
D
D
. Prove that
B
B
B
is the mid-point of the segment
P
Q
PQ
PQ
.Proposed by Karl Czakler
6
1
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An integer sum of three fractions
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be three integers for which the sum
a
b
c
+
a
c
b
+
b
c
a
\frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}
c
ab
+
b
a
c
+
a
b
c
is integer. Prove that each of the three numbers \frac{ab}{c}, \frac{ac}{b}, \frac{bc}{a} is integer.(Proposed by Gerhard J. Woeginger)
1
1
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Functional equation over rationals by Walther Janous
Let
α
∈
Q
+
\alpha\in\mathbb{Q}^+
α
∈
Q
+
. Determine all functions
f
:
Q
+
→
Q
+
f:\mathbb{Q}^+\to\mathbb{Q}^+
f
:
Q
+
→
Q
+
that for all
x
,
y
∈
Q
+
x,y\in\mathbb{Q}^+
x
,
y
∈
Q
+
satisfy the equation
f
(
x
y
+
y
)
=
f
(
x
)
f
(
y
)
+
f
(
y
)
+
α
x
.
f\left(\frac{x}{y}+y\right) ~=~ \frac{f(x)}{f(y)}+f(y)+\alpha x.
f
(
y
x
+
y
)
=
f
(
y
)
f
(
x
)
+
f
(
y
)
+
αx
.
Here
Q
+
\mathbb{Q}^+
Q
+
denote the set of positive rational numbers.(Proposed by Walther Janous)
4
1
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Polynomial inequality in three variables
Let
a
,
b
,
c
≥
−
1
a,b,c\ge-1
a
,
b
,
c
≥
−
1
be real numbers with
a
3
+
b
3
+
c
3
=
1
a^3+b^3+c^3=1
a
3
+
b
3
+
c
3
=
1
. Prove that
a
+
b
+
c
+
a
2
+
b
2
+
c
2
≤
4
a+b+c+a^2+b^2+c^2\le4
a
+
b
+
c
+
a
2
+
b
2
+
c
2
≤
4
, and determine the cases of equality.(Proposed by Karl Czakler)