MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1985 Federal Competition For Advanced Students, P2
1985 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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function
Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
satisfying: x^2 f(x)\plus{}f(1\minus{}x)\equal{}2x\minus{}x^4 for all
x
∈
R
x \in \mathbb{R}
x
∈
R
.
5
1
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constant sequence
A sequence
(
a
n
)
(a_n)
(
a
n
)
of positive integers satisfies: a_n\equal{}\sqrt{\frac{a_{n\minus{}1}^2\plus{}a_{n\plus{}1}^2}{2}} for all
n
≥
1
n \ge 1
n
≥
1
. Prove that this sequence is constant.
4
1
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natural numbers
Find all natural numbers
n
n
n
such that the equation: a_{n\plus{}1} x^2\minus{}2x \sqrt{a_1^2\plus{}a_2^2\plus{}...\plus{}a_{n\plus{}1}^2}\plus{}a_1\plus{}a_2\plus{}...\plus{}a_n\equal{}0 has real solutions for all real numbers a_1,a_2,...,a_{n\plus{}1}.
3
1
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collinear points (maybe posted before)
A line meets the lines containing sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
of a triangle
A
B
C
ABC
A
BC
at
A
1
,
B
1
,
C
1
,
A_1,B_1,C_1,
A
1
,
B
1
,
C
1
,
respectively. Points
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
are symmetric to
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
with respect to the midpoints of
B
C
,
C
A
,
A
B
,
BC,CA,AB,
BC
,
C
A
,
A
B
,
respectively. Prove that
A
2
,
B
2
,
A_2,B_2,
A
2
,
B
2
,
and
C
2
C_2
C
2
are collinear.
2
1
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minimum value
For
n
∈
N
n \in \mathbb{N}
n
∈
N
, let f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1. Determine the minimum value of: \frac{f(n\plus{}1)}{f(n)}.
1
1
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quadruples
Determine all quadruples
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
of nonnegative integers satisfying: a^2\plus{}b^2\plus{}c^2\plus{}d^2\equal{}a^2 b^2 c^2.