MathDB
Problems
Contests
International Contests
Austrian-Polish
1984 Austrian-Polish Competition
1984 Austrian-Polish Competition
Part of
Austrian-Polish
Subcontests
(9)
8
1
Hide problems
a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b, f_0(x) = 2x and f_1(x) =x/(x-1)
The functions
f
0
,
f
1
:
(
1
,
∞
)
→
(
1
,
∞
)
f_0,f_1 : (1,\infty) \to (1,\infty)
f
0
,
f
1
:
(
1
,
∞
)
→
(
1
,
∞
)
are given by
f
0
(
x
)
=
2
x
f_0(x) = 2x
f
0
(
x
)
=
2
x
and
f
1
(
x
)
=
x
x
−
1
f_1(x) =\frac{x}{x-1}
f
1
(
x
)
=
x
−
1
x
. Show that for any real numbers
a
,
b
a, b
a
,
b
with
1
≤
a
<
b
1 \le a < b
1
≤
a
<
b
there exist a positive integer
k
k
k
and indices
i
1
,
i
2
,
.
.
.
,
i
k
∈
{
0
,
1
}
i_1,i_2,...,i_k \in \{0,1\}
i
1
,
i
2
,
...
,
i
k
∈
{
0
,
1
}
such that
a
<
f
i
k
(
f
i
k
−
1
(
.
.
.
(
f
i
j
(
2
)
)
.
.
.
)
)
<
b
a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b
a
<
f
i
k
(
f
i
k
−
1
(
...
(
f
i
j
(
2
))
...
))
<
b
.
7
1
Hide problems
permutation of numbers on a matrix so that \sum b_{ij} < 2
A
m
×
n
m\times n
m
×
n
matrix
(
a
i
j
)
(a_{ij})
(
a
ij
)
of real numbers satisfies
∣
a
i
j
∣
<
1
|a_{ij}| <1
∣
a
ij
∣
<
1
and
∑
i
=
1
m
a
i
j
=
0
\sum_{i=1}^m a_{ij}= 0
∑
i
=
1
m
a
ij
=
0
for all
j
j
j
. Show that one can permute the entries in each column in such a way that the obtained matrix
(
b
i
j
)
(b_{ij})
(
b
ij
)
satisfies
∑
j
=
1
n
b
i
j
<
2
\sum_{j=1}^n b_{ij} < 2
∑
j
=
1
n
b
ij
<
2
for all
i
i
i
.
5
1
Hide problems
a_nx_1 + a_1x_2 +...-+ a_{n-1}x_n = yx_n , system
Given
n
>
2
n > 2
n
>
2
nonnegative distinct integers
a
1
,
.
.
.
,
a
n
a_1,...,a_n
a
1
,
...
,
a
n
, find all nonnegative integers
y
y
y
and
x
1
,
.
.
.
,
x
n
x_1,...,x_n
x
1
,
...
,
x
n
satisfying
g
c
d
(
x
1
,
.
.
.
,
x
n
)
=
1
gcd(x_1,...,x_n) = 1
g
c
d
(
x
1
,
...
,
x
n
)
=
1
and
{
a
1
x
1
+
a
2
x
2
+
.
.
.
+
a
n
x
n
=
y
x
1
a
2
x
1
+
a
3
x
2
+
.
.
.
+
a
1
x
n
=
y
x
2
.
.
.
a
n
x
1
+
a
1
x
2
+
.
.
.
+
a
n
−
1
x
n
=
y
x
n
\begin{cases} a_1x_1 + a_2x_2 +...+ a_nx_n = yx_1 \\ a_2x_1 + a_3x_2 +...+ a_1x_n = yx_2 \\ ... \\ a_nx_1 + a_1x_2 +...+ a_{n-1}x_n = yx_n \end{cases}
⎩
⎨
⎧
a
1
x
1
+
a
2
x
2
+
...
+
a
n
x
n
=
y
x
1
a
2
x
1
+
a
3
x
2
+
...
+
a
1
x
n
=
y
x
2
...
a
n
x
1
+
a
1
x
2
+
...
+
a
n
−
1
x
n
=
y
x
n
9
1
Hide problems
f (x + y) = f (x)f (y) - f(xy) + 1 for all x,y \in Q
Find all functions
f
:
Q
→
R
f: Q \to R
f
:
Q
→
R
satisfying
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
−
f
(
x
y
)
+
1
f (x + y) = f (x)f (y) - f(xy) + 1
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
−
f
(
x
y
)
+
1
for all
x
,
y
∈
Q
x,y \in Q
x
,
y
∈
Q
6
1
Hide problems
n girls and n boys in a dancing hall
In a dancing hall, there are
n
n
n
girls standing in one row and
n
n
n
boys in the other row across them (so that all
2
n
2n
2
n
dancers form a
2
×
n
2 \times n
2
×
n
board). Each dancer gives her / his left hand to a neighboring person standing to the left, across, or diagonally to the left. The analogous rule applies for right hands. No dancer gives both hands to the same person. In how many ways can the dancers do this?
2
1
Hide problems
4 digits number using only 2 disticnt digits problem
Let
A
A
A
be the set of four-digit natural numbers having exactly two distinct digits, none of which is zero. Interchanging the two digits of
n
∈
A
n\in A
n
∈
A
yields a number
f
(
n
)
∈
A
f (n) \in A
f
(
n
)
∈
A
(for instance,
f
(
3111
)
=
1333
f (3111) = 1333
f
(
3111
)
=
1333
). Find those
n
∈
A
n \in A
n
∈
A
with
n
>
f
(
n
)
n > f (n)
n
>
f
(
n
)
for which
g
c
d
(
n
,
f
(
n
)
)
gcd(n, f (n))
g
c
d
(
n
,
f
(
n
))
is the largest possible.
4
1
Hide problems
PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6 in regular heptagon
A regular heptagon
A
1
A
2
.
.
.
A
7
A_1A_2... A_7
A
1
A
2
...
A
7
is inscribed in circle
C
C
C
. Point
P
P
P
is taken on the shorter arc
A
7
A
1
A_7A_1
A
7
A
1
. Prove that
P
A
1
+
P
A
3
+
P
A
5
+
P
A
7
=
P
A
2
+
P
A
4
+
P
A
6
PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6
P
A
1
+
P
A
3
+
P
A
5
+
P
A
7
=
P
A
2
+
P
A
4
+
P
A
6
.
1
1
Hide problems
criterion for a tetrahedron to be regular, feet of altitudes are incenters
Prove that if the feet of the altitudes of a tetrahedron are the incenters of the corresponding faces, then the tetrahedron is regular.
3
1
Hide problems
An inequality with n-variables
Show that for
n
>
1
n>1
n
>
1
and any positive real numbers
k
,
x
1
,
x
2
,
.
.
.
,
x
n
k,x_{1},x_{2},...,x_{n}
k
,
x
1
,
x
2
,
...
,
x
n
then
f
(
x
1
−
x
2
)
x
1
+
x
2
+
f
(
x
2
−
x
3
)
x
2
+
x
3
+
.
.
.
+
f
(
x
n
−
x
1
)
x
n
+
x
1
≥
n
2
2
(
x
1
+
x
2
+
.
.
.
+
x
n
)
\frac{f(x_{1}-x_{2})}{x_{1}+x_{2}}+\frac{f(x_{2}-x_{3})}{x_{2}+x_{3}}+...+\frac{f(x_{n}-x_{1})}{x_{n}+x_{1}}\geq \frac{n^2}{2(x_{1}+x_{2}+...+x_{n})}
x
1
+
x
2
f
(
x
1
−
x
2
)
+
x
2
+
x
3
f
(
x
2
−
x
3
)
+
...
+
x
n
+
x
1
f
(
x
n
−
x
1
)
≥
2
(
x
1
+
x
2
+
...
+
x
n
)
n
2
Where
f
(
x
)
=
k
x
f(x)=k^x
f
(
x
)
=
k
x
. When does equality hold.