MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1983 Federal Competition For Advanced Students, P2
1983 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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cube
Planes
π
1
\pi _1
π
1
and
π
2
\pi _2
π
2
in Euclidean space
R
3
\mathbb{R} ^3
R
3
partition S\equal{}\mathbb{R} ^3 \setminus (\pi _1 \cup \pi _2) into several components. Show that for any cube in
R
3
\mathbb{R} ^3
R
3
, at least one of the components of
S
S
S
meets at least three faces of the cube.
5
1
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find all positive integers
Given positive integers
a
,
b
,
a,b,
a
,
b
,
find all positive integers
x
,
y
x,y
x
,
y
satisfying the equation: x^{a\plus{}b}\plus{}y\equal{}x^a y^b.
4
1
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determine the general term
The sequence
(
x
n
)
n
∈
N
(x_n)_{n \in \mathbb{N}}
(
x
n
)
n
∈
N
is defined by x_1\equal{}2, x_2\equal{}3, and x_{2m\plus{}1}\equal{}x_{2m}\plus{}x_{2m\minus{}1} for
m
≥
1
;
m \ge 1;
m
≥
1
;
x_{2m}\equal{}x_{2m\minus{}1}\plus{}2x_{2m\minus{}2} for
m
≥
2.
m \ge 2.
m
≥
2.
Determine
x
n
x_n
x
n
as a function of
n
n
n
.
3
1
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areas
Let
P
P
P
be a point in the plane of a triangle
A
B
C
ABC
A
BC
. Lines
A
P
,
B
P
,
C
P
AP,BP,CP
A
P
,
BP
,
CP
respectively meet lines
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at points
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
. Points
A
′
′
,
B
′
′
,
C
′
′
A'',B'',C''
A
′′
,
B
′′
,
C
′′
are symmetric to
A
,
B
,
C
A,B,C
A
,
B
,
C
with respect to
A
′
,
B
′
,
C
′
,
A',B',C',
A
′
,
B
′
,
C
′
,
respectively. Show that: S_{A''B''C''}\equal{}3S_{ABC}\plus{}4S_{A'B'C'}.
2
1
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polynomial
Let
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
be the roots of: x^3\minus{}6x^2\plus{}ax\plus{}a\equal{}0. Find all real numbers
a
a
a
for which (x_1\minus{}1)^3\plus{}(x_2\minus{}1)^3\plus{}(x_3\minus{}1)^3\equal{}0. Also, for each such
a
a
a
, determine the corresponding values of
x
1
,
x
2
,
x_1,x_2,
x
1
,
x
2
,
and
x
3
x_3
x
3
.
1
1
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infinitely many values
For every natural number
x
x
x
, let
Q
(
x
)
Q(x)
Q
(
x
)
be the sum and
P
(
x
)
P(x)
P
(
x
)
the product of the (decimal) digits of
x
x
x
. Show that for each
n
∈
N
n \in \mathbb{N}
n
∈
N
there exist infinitely many values of
x
x
x
such that: Q(Q(x))\plus{}P(Q(x))\plus{}Q(P(x))\plus{}P(P(x))\equal{}n.