MathDB
Problems
Contests
International Contests
Austrian-Polish
1982 Austrian-Polish Competition
1982 Austrian-Polish Competition
Part of
Austrian-Polish
Subcontests
(9)
3
1
Hide problems
\prod \tan \pi/3 (1+3^k/(3^n-1)]=\prod \cot pi/3 ]1- 3^k/(3^n-1)]
If
n
≥
2
n \ge 2
n
≥
2
is an integer, prove the equality
∏
k
=
1
n
tan
π
3
(
1
+
3
k
3
n
−
1
)
=
∏
k
=
1
n
cot
π
3
(
1
−
3
k
3
n
−
1
)
\prod_{k=1}^n \tan \frac{\pi}{3}\left(1+\frac{3^k}{3^n-1}\right)=\prod_{k=1}^n \cot \frac{\pi}{3}\left(1-\frac{3^k}{3^n-1}\right)
k
=
1
∏
n
tan
3
π
(
1
+
3
n
−
1
3
k
)
=
k
=
1
∏
n
cot
3
π
(
1
−
3
n
−
1
3
k
)
9
1
Hide problems
n<= S_n <= Cn , where S_n= \frac{1}{\sqrt{j^2+k^2}}
Define
S
n
=
∑
j
,
k
=
1
n
1
j
2
+
k
2
S_n=\sum_{j,k=1}^{n} \frac{1}{\sqrt{j^2+k^2}}
S
n
=
∑
j
,
k
=
1
n
j
2
+
k
2
1
. Find a positive constant
C
C
C
such that the inequality
n
≤
S
n
≤
C
n
n\le S_n \le Cn
n
≤
S
n
≤
C
n
holds for all
n
≥
3
n \ge 3
n
≥
3
. (Note. The smaller
C
C
C
, the better the solution.)
7
1
Hide problems
x^y = a^b = c^d, z = ab = cd , x + y = a + b, x > a > c , diophantine
Find the triple of positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
with
z
z
z
least possible for which there are positive integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
with the following properties: (i)
x
y
=
a
b
=
c
d
x^y = a^b = c^d
x
y
=
a
b
=
c
d
and
x
>
a
>
c
x > a > c
x
>
a
>
c
(ii)
z
=
a
b
=
c
d
z = ab = cd
z
=
ab
=
c
d
(iii)
x
+
y
=
a
+
b
x + y = a + b
x
+
y
=
a
+
b
.
6
1
Hide problems
f(x+y) = f (x) f (y) for all x,y >= a with x + y >= a.
An integer
a
a
a
is given. Find all real-valued functions
f
(
x
)
f (x)
f
(
x
)
defined on integers
x
≥
a
x \ge a
x
≥
a
, satisfying the equation
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
f (x+y) = f (x) f (y)
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
for all
x
,
y
≥
a
x,y \ge a
x
,
y
≥
a
with
x
+
y
≥
a
x + y \ge a
x
+
y
≥
a
.
4
1
Hide problems
x_{n+1} = x_n + P(x_n), P(x) is product of digitss, x_n bounded?
Let
P
(
x
)
P(x)
P
(
x
)
denote the product of all (decimal) digits of a natural number
x
x
x
. For any positive integer
x
1
x_1
x
1
, define the sequence
(
x
n
)
(x_n)
(
x
n
)
recursively by
x
n
+
1
=
x
n
+
P
(
x
n
)
x_{n+1} = x_n + P(x_n)
x
n
+
1
=
x
n
+
P
(
x
n
)
. Prove or disprove that the sequence
(
x
n
)
(x_n)
(
x
n
)
is necessarily bounded.
2
1
Hide problems
at each point one can draw two rays tangent to F of angle of 60^o, locus
Let
F
F
F
be a closed convex region inside a circle
C
C
C
with center
O
O
O
and radius
1
1
1
. Furthermore, assume that from each point of
C
C
C
one can draw two rays tangent to
F
F
F
which form an angle of
6
0
o
60^o
6
0
o
. Prove that
F
F
F
is the disc centered at
O
O
O
with radius
1
/
2
1/2
1/2
.
1
1
Hide problems
gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1
Find all pairs
(
n
,
m
)
(n, m)
(
n
,
m
)
of positive integers such that
g
c
d
(
(
n
+
1
)
m
−
n
,
(
n
+
1
)
m
+
3
−
n
)
>
1
gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1
g
c
d
((
n
+
1
)
m
−
n
,
(
n
+
1
)
m
+
3
−
n
)
>
1
.
8
1
Hide problems
d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD)>=3/2\sqrt2, r.tetrahedron
Let
P
P
P
be a point inside a regular tetrahedron ABCD with edge length
1
1
1
. Show that
d
(
P
,
A
B
)
+
d
(
P
,
A
C
)
+
d
(
P
,
A
D
)
+
d
(
P
,
B
C
)
+
d
(
P
,
B
D
)
+
d
(
P
,
C
D
)
≥
3
2
2
d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2
d
(
P
,
A
B
)
+
d
(
P
,
A
C
)
+
d
(
P
,
A
D
)
+
d
(
P
,
BC
)
+
d
(
P
,
B
D
)
+
d
(
P
,
C
D
)
≥
2
3
2
, with equality only when
P
P
P
is the centroid of
A
B
C
D
ABCD
A
BC
D
. Here
d
(
P
,
X
Y
)
d(P,XY)
d
(
P
,
X
Y
)
denotes the distance from point
P
P
P
to line
X
Y
XY
X
Y
.
5
1
Hide problems
[0,1]=AUB
Show that [0,1] cannot be partitioned into two disjoints sets A and B such that B=A+a for some real a.