MathDB
Problems
Contests
International Contests
Austrian-Polish
1978 Austrian-Polish Competition
1978 Austrian-Polish Competition
Part of
Austrian-Polish
Subcontests
(9)
9
1
Hide problems
Classic polygon diagonal problem
In a convex polygon
P
P
P
some diagonals have been drawn, without intersections inside
P
P
P
. Show that there exist at least two vertices of
P
P
P
, neither one of them being an endpoint of any one of those diagonals.
8
1
Hide problems
Limit of square root sequence
For any positive integer
k
k
k
consider the sequence
a
n
=
k
+
k
+
⋯
+
k
,
a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},
a
n
=
k
+
k
+
⋯
+
k
,
where there are
n
n
n
square-root signs on the right-hand side.(a) Show that the sequence converges, for every fixed integer
k
≥
1
k\ge 1
k
≥
1
. (b) Find
k
k
k
such that the limit is an integer. Furthermore, prove that if
k
k
k
is odd, then the limit is irrational.
7
1
Hide problems
Existence of domino tiling
Let
M
M
M
be the set of all lattice points in the plane (i.e. points with integer coordinates, in a fixed Cartesian coordinate system). For any point
P
=
(
x
,
y
)
∈
M
P=(x,y)\in M
P
=
(
x
,
y
)
∈
M
we call the points
(
x
−
1
,
y
)
(x-1,y)
(
x
−
1
,
y
)
,
(
x
+
1
,
y
)
(x+1,y)
(
x
+
1
,
y
)
,
(
x
,
y
−
1
)
(x,y-1)
(
x
,
y
−
1
)
,
(
x
,
y
+
1
)
(x,y+1)
(
x
,
y
+
1
)
neighbors of
P
P
P
. Let
S
S
S
be a finite subset of
M
M
M
. A one-to-one mapping
f
f
f
of
S
S
S
onto
S
S
S
is called perfect if
f
(
P
)
f(P)
f
(
P
)
is a neighbor of
P
P
P
, for any
P
∈
S
P\in S
P
∈
S
. Prove that if such a mapping exists, then there exists also a perfect mapping
g
:
S
→
S
g:S\to S
g
:
S
→
S
with the additional property
g
(
g
(
P
)
)
=
P
g(g(P))=P
g
(
g
(
P
))
=
P
for
P
∈
S
P\in S
P
∈
S
.
6
1
Hide problems
Disjoint discs tangent to 6 others
We are given a family of discs in the plane, with pairwise disjoint interiors. Each disc is tangent to at least six other discs of the family. Show that the family is infinite.
4
1
Hide problems
Partition of N avoiding ratio c
Let
c
≠
1
c\neq 1
c
=
1
be a positive rational number. Show that it is possible to partition
N
\mathbb{N}
N
, the set of positive integers, into two disjoint nonempty subsets
A
,
B
A,B
A
,
B
so that
x
/
y
≠
c
x/y\neq c
x
/
y
=
c
holds whenever
x
x
x
and
y
y
y
lie both in
A
A
A
or both in
B
B
B
.
3
1
Hide problems
Tangent inequality
Prove that
tan
1
∘
⋅
tan
2
∘
⋅
⋯
⋅
tan
4
4
∘
44
<
2
−
1
<
tan
1
∘
+
tan
2
∘
+
⋯
+
tan
4
4
∘
44
.
\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.
44
tan
1
∘
⋅
tan
2
∘
⋅
⋯
⋅
tan
4
4
∘
<
2
−
1
<
44
tan
1
∘
+
tan
2
∘
+
⋯
+
tan
4
4
∘
.
2
1
Hide problems
Parallelogram in hexagon
A parallelogram is inscribed into a regular hexagon so that the centers of symmetry of both figures coincide. Prove that the area of the parallelogram does not exceed
2
/
3
2/3
2/3
the area of the hexagon.
1
1
Hide problems
Quick functional equation
Determine all functions
f
:
(
0
;
∞
)
→
R
f:(0;\infty)\to \mathbb{R}
f
:
(
0
;
∞
)
→
R
that satisfy f(x+y)=f(x^2+y^2) \forall x,y\in (0;\infty)
5
1
Hide problems
Please respond fast!!!!!
We are given
1978
1978
1978
sets of size
40
40
40
each. The size of the intersection of any two sets is exactly
1
1
1
. Prove that all the sets have a common element.