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Contests
International Contests
Austrian-Polish
2002 Austrian-Polish Competition
2002 Austrian-Polish Competition
Part of
Austrian-Polish
Subcontests
(10)
1
1
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Austrian-Polish Mathematical Olympiad 1992
Given a circle
G
G
G
with center
O
O
O
and radius
r
r
r
. Let
A
B
AB
A
B
be a fixed diameter of
G
G
G
. Let
K
K
K
be a fixed point of segment
A
O
AO
A
O
. Denote by
t
t
t
the line tangent to at
A
A
A
. For any chord
C
D
CD
C
D
(other than
A
B
AB
A
B
) passing through
K
K
K
. Let
P
P
P
and
Q
Q
Q
be the points of intersection of lines
B
C
BC
BC
and
B
D
BD
B
D
with
t
t
t
. Prove that the product
A
P
⋅
A
Q
AP\cdot AQ
A
P
⋅
A
Q
remains costant as the chord
C
D
CD
C
D
varies.
10
1
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Family of sequences
For all real number
x
x
x
consider the family
F
(
x
)
F(x)
F
(
x
)
of all sequences
(
a
n
)
n
≥
0
(a_{n})_{n\geq 0}
(
a
n
)
n
≥
0
satisfying the equation a_{n+1}=x-\frac{1}{a_{n}} (n\geq 0). A positive integer
p
p
p
is called a minimal period of the family
F
(
x
)
F(x)
F
(
x
)
if (a) each sequence
(
a
n
)
∈
F
(
x
)
\left(a_{n}\right)\in F(x)
(
a
n
)
∈
F
(
x
)
is periodic with the period
p
p
p
, (b) for each
0
<
q
<
p
0<q<p
0
<
q
<
p
there exists
(
a
n
)
∈
F
(
x
)
\left(a_{n}\right)\in F(x)
(
a
n
)
∈
F
(
x
)
such that
q
q
q
is not a period of
(
a
n
)
\left(a_{n}\right)
(
a
n
)
. Prove or disprove that for each positive integer
P
P
P
there exists a real number
x
=
x
(
P
)
x=x(P)
x
=
x
(
P
)
such that the family
F
(
x
)
F(x)
F
(
x
)
has the minimal period
p
>
P
p>P
p
>
P
.
9
1
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Number of acquaintances
A set
P
P
P
of
2002
2002
2002
persons is given. The family of subsets of
P
P
P
containing exactly
1001
1001
1001
persons has the property that the number of acquaintance pairs in each such subset is the same. (It is assumed that the acquaintance relation is symmetric). Find the best lower estimation of the acquaintance pairs in the set
P
P
P
.
8
1
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Number of solutions of a system
Determine the number of real solutions of the system
{
cos
x
1
=
x
2
⋯
cos
x
n
−
1
=
x
n
cos
x
n
=
x
1
\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.
⎩
⎨
⎧
cos
x
1
cos
x
n
−
1
cos
x
n
=
x
2
⋯
=
x
n
=
x
1
4
1
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Find the largest subset
For each positive integer
n
n
n
find the largest subset
M
(
n
)
M(n)
M
(
n
)
of real numbers possessing the property: n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i} \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n) When does the inequality become an equality ?
7
1
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Periodic functions s.t. f(x^2y) = (f(x))^2f(y)
Find all real functions
f
f
f
definited on positive integers and satisying:(a)
f
(
x
+
22
)
=
f
(
x
)
f(x+22)=f(x)
f
(
x
+
22
)
=
f
(
x
)
,(b)
f
(
x
2
y
)
=
(
f
(
x
)
)
2
f
(
y
)
f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)
f
(
x
2
y
)
=
(
f
(
x
)
)
2
f
(
y
)
for all positive integers
x
x
x
and
y
y
y
.
5
1
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Polynomial with integer coefficients
Let
A
A
A
be the set
{
2
,
7
,
11
,
13
}
\{2,7,11,13\}
{
2
,
7
,
11
,
13
}
. A polynomial
f
f
f
with integer coefficients possesses the following property: for each integer
n
n
n
there exists
p
∈
A
p \in A
p
∈
A
such that
p
∣
f
(
n
)
p|f(n)
p
∣
f
(
n
)
. Prove that there exists
p
∈
A
p \in A
p
∈
A
such that
p
∣
f
(
n
)
p|f(n)
p
∣
f
(
n
)
for all integers
n
n
n
.
6
1
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Convex quadrilateral
The diagonals of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect in the point
E
E
E
. Let
U
U
U
be the circumcenter of the triangle
A
B
E
ABE
A
BE
and
H
H
H
be its orthocenter. Similarly, let
V
V
V
be the circumcenter of the triangle
C
D
E
CDE
C
D
E
and
K
K
K
be its orthocenter. Prove that
E
E
E
lies on the line
U
K
UK
U
K
if and only if it lies on the line
V
H
VH
V
H
.
3
1
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Center of gravity of a tetrahedron
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron and let
S
S
S
be its center of gravity. A line through
S
S
S
intersects the surface of
A
B
C
D
ABCD
A
BC
D
in the points
K
K
K
and
L
L
L
. Prove that
1
3
≤
K
S
L
S
≤
3
\frac{1}{3}\leq \frac{KS}{LS}\leq 3
3
1
≤
L
S
K
S
≤
3
2
1
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Diagonal of a convex polygon
Let
P
1
P
2
…
P
2
n
P_{1}P_{2}\dots P_{2n}
P
1
P
2
…
P
2
n
be a convex polygon with an even number of corners. Prove that there exists a diagonal
P
i
P
j
P_{i}P_{j}
P
i
P
j
which is not parallel to any side of the polygon.