MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2017 Federal Competition For Advanced Students
2017 Federal Competition For Advanced Students
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
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number theory
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of non-negative integers such that:
201
7
a
=
b
6
−
32
b
+
1
2017^a=b^6-32b+1
201
7
a
=
b
6
−
32
b
+
1
proposed by Walther Janous
3
1
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wining strategy
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with
k
≤
n
2
k \le \frac{n}{2}
k
≤
2
n
or an odd number with
n
2
≤
k
≤
n
\frac{n}{2}\le k \le n
2
n
≤
k
≤
n
. A player wins the game if she removes the last marble from the table. Determine the smallest number
N
≥
100000
N\ge100000
N
≥
100000
which Berta has wining strategy.proposed by Gerhard Woeginger
2
1
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Regular Pentagon
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a regular pentagon with center
M
M
M
. A point
P
P
P
(different from
M
M
M
) is chosen on the line segment
M
D
MD
M
D
. The circumcircle of
A
B
P
ABP
A
BP
intersects the line segment
A
E
AE
A
E
in
A
A
A
and
Q
Q
Q
and the line through
P
P
P
perpendicular to
C
D
CD
C
D
in
P
P
P
and
R
R
R
. Prove that
A
R
AR
A
R
and
Q
R
QR
QR
have same length. proposed by Stephan Wagner
1
1
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Easy Polynomial
Determine all polynomials
P
(
x
)
∈
R
[
x
]
P(x) \in \mathbb R[x]
P
(
x
)
∈
R
[
x
]
satisfying the following two conditions : (a)
P
(
2017
)
=
2016
P(2017) = 2016
P
(
2017
)
=
2016
and (b)
(
P
(
x
)
+
1
)
2
=
P
(
x
2
+
1
)
(P(x) + 1)^2 = P(x^2 + 1)
(
P
(
x
)
+
1
)
2
=
P
(
x
2
+
1
)
for all real numbers
x
x
x
. proposed by Walther Janous