MathDB
Problems
Contests
International Contests
Austrian-Polish
1989 Austrian-Polish Competition
1989 Austrian-Polish Competition
Part of
Austrian-Polish
Subcontests
(9)
3
1
Hide problems
N=aabb, aab and abb are primes
Find all natural numbers
N
N
N
(in decimal system) with the following properties: (i)
N
=
a
a
b
b
‾
N =\overline{aabb}
N
=
aabb
, where
a
a
b
‾
\overline{aab}
aab
and
a
b
b
‾
\overline{abb}
abb
are primes, (ii)
N
=
P
1
P
2
P
3
N = P_1P_2P_3
N
=
P
1
P
2
P
3
, where
P
k
(
k
=
1
,
2
,
3
)
P_k (k = 1,2,3)
P
k
(
k
=
1
,
2
,
3
)
is a prime consisting of
k
k
k
(decimal) digits.
7
1
Hide problems
f_0(x) = x, f_{2k+1} (x) = 3^{f_{2k}(x)}, $f_{2k+2} = 2^{f_{2k+1}(x)}
Functions
f
0
,
f
1
,
f
2
,
.
.
.
f_0, f_1,f_2,...
f
0
,
f
1
,
f
2
,
...
are recursively defined by
f
0
(
x
)
=
x
f_0(x) = x
f
0
(
x
)
=
x
and
f
2
k
+
1
(
x
)
=
3
f
2
k
(
x
)
f_{2k+1} (x) = 3^{f_{2k}(x)}
f
2
k
+
1
(
x
)
=
3
f
2
k
(
x
)
and
f
2
k
+
2
=
2
f
2
k
+
1
(
x
)
f_{2k+2} = 2^{f_{2k+1}(x)}
f
2
k
+
2
=
2
f
2
k
+
1
(
x
)
,
k
=
0
,
1
,
2
,
.
.
.
k = 0,1,2,...
k
=
0
,
1
,
2
,
...
for all
x
∈
R
x \in R
x
∈
R
. Find the greater one of the numbers
f
10
(
1
)
f_{10}(1)
f
10
(
1
)
and
f
9
(
2
)
f_9(2)
f
9
(
2
)
.
9
1
Hide problems
N^2 is sum of odd number of squares of adjacent positive integers
Find the smallest odd natural number
N
N
N
such that
N
2
N^2
N
2
is the sum of an odd number (greater than
1
1
1
) of squares of adjacent positive integers.
6
1
Hide problems
difference a_{n+1} - a_n is a prime or the square of a prime
A sequence
(
a
n
)
n
∈
N
(a_n)_{n \in N}
(
a
n
)
n
∈
N
of squares of nonzero integers is such that for each
n
n
n
the difference
a
n
+
1
−
a
n
a_{n+1} - a_n
a
n
+
1
−
a
n
is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.
4
1
Hide problems
exists circle containing entire convex polygon P
Let
P
P
P
be a convex polygon in the plane. Show that there exists a circle containing the entire polygon
P
P
P
and having at least three adjacent vertices of
P
P
P
on its boundary.
2
1
Hide problems
equilateral triangle with monochromatic vertices (2 colours only)
Each point of the plane is colored by one of the two colors. Show that there exists an equilateral triangle with monochromatic vertices.
5
1
Hide problems
max of AP\cdot AQ , related to a cube circumscribed around a sphere
Let
A
A
A
be a vertex of a cube
ω
\omega
ω
circumscribed about a sphere
k
k
k
of radius
1
1
1
. We consider lines
g
g
g
through
A
A
A
containing at least one point of
k
k
k
. Let
P
P
P
be the intersection point of
g
g
g
and
k
k
k
closer to
A
A
A
, and
Q
Q
Q
be the second intersection point of
g
g
g
and
ω
\omega
ω
. Determine the maximum value of
A
P
⋅
A
Q
AP\cdot AQ
A
P
⋅
A
Q
and characterize the lines
g
g
g
yielding the maximum.
1
1
Hide problems
Hard and useful
Show that
(
∑
i
=
1
n
x
i
y
i
z
i
)
2
≤
(
∑
i
=
1
n
x
i
3
)
(
∑
i
=
1
n
y
i
3
)
(
∑
i
=
1
n
z
i
3
)
(\sum_{i=1}^{n}x_iy_iz_i)^2 \le (\sum_{i=1}^{n}x_i^3) (\sum_{i=1}^{n}y_i^3) (\sum_{i=1}^{n}z_i^3)
(
∑
i
=
1
n
x
i
y
i
z
i
)
2
≤
(
∑
i
=
1
n
x
i
3
)
(
∑
i
=
1
n
y
i
3
)
(
∑
i
=
1
n
z
i
3
)
for any positive reals
x
i
,
y
i
,
z
i
x_i, y_i, z_i
x
i
,
y
i
,
z
i
.
8
1
Hide problems
old problem
A
B
C
ABC
A
BC
is an acute-angled triangle and
P
P
P
a point inside or on the boundary. The feet of the perpendiculars from
P
P
P
to
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
are
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
respectively. Show that if
A
B
C
ABC
A
BC
is equilateral, then
A
C
′
+
B
A
′
+
C
B
′
P
A
′
+
P
B
′
+
P
C
′
\frac{AC'+BA'+CB'}{PA'+PB'+PC'}
P
A
′
+
P
B
′
+
P
C
′
A
C
′
+
B
A
′
+
C
B
′
is the same for all positions of
P
P
P
, but that for any other triangle it is not.