MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1986 Federal Competition For Advanced Students, P2
1986 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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find all functions
Given a positive integer
n
n
n
, find all functions
F
:
N
→
R
F: \mathbb{N} \rightarrow \mathbb{R}
F
:
N
→
R
such that F(x\plus{}y)\equal{}F(xy\minus{}n) whenever
x
,
y
∈
N
x,y \in \mathbb{N}
x
,
y
∈
N
satisfy
x
y
>
n
xy>n
x
y
>
n
.
5
1
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inequality
Show that for every convex
n
n
n
-gon
(
n
≥
4
)
( n \ge 4)
(
n
≥
4
)
, the arithmetic mean of the lengths of its sides is less than the arithmetic mean of the lengths of all its diagonals.
4
1
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find n and N
Find the largest
n
n
n
for which there is a natural number
N
N
N
with
n
n
n
decimal digits which are all different such that
n
!
n!
n
!
divides
N
N
N
. Furthermore, for this largest
n
n
n
find all possible numbers
N
N
N
.
3
1
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infinitely many natural numbers
Find all possible values of
x
0
x_0
x
0
and
x
1
x_1
x
1
such that the sequence defined by: x_{n\plus{}1}\equal{}\frac{x_{n\minus{}1} x_n}{3x_{n\minus{}1}\minus{}2x_n} for
n
≥
1
n \ge 1
n
≥
1
contains infinitely many natural numbers.
2
1
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find the number of rhombi
For
s
,
t
∈
N
s,t \in \mathbb{N}
s
,
t
∈
N
, consider the set M\equal{}\{ (x,y) \in \mathbb{N} ^2 | 1 \le x \le s, 1 \le y \le t \}. Find the number of rhombi with the vertices in
M
M
M
and the diagonals parallel to the coordinate axes.
1
1
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interesting
Show that a square can be inscribed in any regular polygon.