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Problems
Contests
International Contests
Austrian-Polish
1979 Austrian-Polish Competition
1979 Austrian-Polish Competition
Part of
Austrian-Polish
Subcontests
(9)
9
1
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greatest power of 2 that divides a_n = [(3+\sqrt{11} )^{2n+1}]
Find the greatest power of
2
2
2
that divides
a
n
=
[
(
3
+
11
)
2
n
+
1
]
a_n = [(3+\sqrt{11} )^{2n+1}]
a
n
=
[(
3
+
11
)
2
n
+
1
]
, where
n
n
n
is a given positive integer.
7
1
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hexagon with analytic geometry, partitioned in squares
Let
n
n
n
and
m
m
m
be fixed positive integers. The hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with vertices
A
=
(
0
,
0
)
A = (0,0)
A
=
(
0
,
0
)
,
B
=
(
n
,
0
)
B = (n,0)
B
=
(
n
,
0
)
,
C
=
(
n
,
m
)
C = (n,m)
C
=
(
n
,
m
)
,
D
=
(
n
−
1
,
m
)
D = (n-1,m)
D
=
(
n
−
1
,
m
)
,
E
=
(
n
−
1
,
1
)
E = (n-1,1)
E
=
(
n
−
1
,
1
)
,
F
=
(
0
,
1
)
F = (0,1)
F
=
(
0
,
1
)
has been partitioned into
n
+
m
−
1
n+m-1
n
+
m
−
1
unit squares. Find the number of paths from
A
A
A
to
C
C
C
along grid lines, passing through every grid node at most once.
6
1
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system of equations \sum_{i=1}^{n} x_i^k= a^k
A positive integer
n
n
n
and a real number
a
a
a
are given. Find all
n
n
n
-tuples
(
x
1
,
.
.
.
,
x
n
)
(x_1, ... ,x_n)
(
x
1
,
...
,
x
n
)
of real numbers that satisfy the system of equations
∑
i
=
1
n
x
i
k
=
a
k
f
o
r
k
=
1
,
2
,
.
.
.
,
n
\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n
i
=
1
∑
n
x
i
k
=
a
k
f
or
k
=
1
,
2
,
...
,
n
4
1
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f (x+y)+ f (x-y)= f (3x)
Determine all functions
f
:
N
0
→
R
f : N_0 \to R
f
:
N
0
→
R
satisfying
f
(
x
+
y
)
+
f
(
x
−
y
)
=
f
(
3
x
)
f (x+y)+ f (x-y)= f (3x)
f
(
x
+
y
)
+
f
(
x
−
y
)
=
f
(
3
x
)
for all
x
,
y
x,y
x
,
y
.
5
1
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circumcenter and incenter of given tetrahedron coincide
The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.
3
1
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Find all positive integers n
Find all positive integers
n
n
n
such that the inequality
(
∑
i
=
1
n
a
i
2
)
(
∑
i
=
1
n
a
i
)
−
∑
i
=
1
n
a
i
3
≥
6
∏
i
=
1
n
a
i
\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i
(
i
=
1
∑
n
a
i
2
)
(
i
=
1
∑
n
a
i
)
−
i
=
1
∑
n
a
i
3
≥
6
i
=
1
∏
n
a
i
holds for any
n
n
n
positive numbers
a
1
,
…
,
a
n
a_1, \dots, a_n
a
1
,
…
,
a
n
.
2
1
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Polynomials of the form P_n(x)=n!x^n+a_{n-1}x^{n-1}+...+a_1x+(-1)^n(n+1)
Find all polynomials of the form
P
n
(
x
)
=
n
!
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
(
−
1
)
n
(
n
+
1
)
P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1)
P
n
(
x
)
=
n
!
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
(
−
1
)
n
(
n
+
1
)
with integer coefficients, having
n
n
n
real roots
x
1
,
…
,
x
n
x_1,\dots,x_n
x
1
,
…
,
x
n
satisfying
k
≤
x
k
≤
k
+
1
k \leq x_k \leq k+1
k
≤
x
k
≤
k
+
1
for
k
=
1
,
…
,
n
k=1, \dots,n
k
=
1
,
…
,
n
.
1
1
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On sides AB and BC of a square ABCD respective points E and F
On sides
A
B
AB
A
B
and
B
C
BC
BC
of a square
A
B
C
D
ABCD
A
BC
D
the respective points
E
E
E
and
F
F
F
have been chosen so that
B
E
=
B
F
BE = BF
BE
=
BF
. Let
B
N
BN
BN
be the altitude in triangle
B
C
E
BCE
BCE
. Prove that
∠
D
N
F
=
90
\angle DNF = 90
∠
D
NF
=
90
.
8
1
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AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2 in 3D, with midpoints
Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be four points in space, and
M
M
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and
B
D
BD
B
D
, respectively. Show that
A
B
2
+
B
C
2
+
C
D
2
+
D
A
2
=
A
C
2
+
B
D
2
+
4
M
N
2
AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2
A
B
2
+
B
C
2
+
C
D
2
+
D
A
2
=
A
C
2
+
B
D
2
+
4
M
N
2