MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2009 Kurschak Competition
2009 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
Special function on the integers
Find all functions
f
:
Z
→
Q
f:\mathbb{Z}\to \mathbb{Q}
f
:
Z
→
Q
with the following properties: if
f
(
x
)
<
c
<
f
(
y
)
f(x)<c<f(y)
f
(
x
)
<
c
<
f
(
y
)
for some rational
c
c
c
, then
f
f
f
takes on the value of
c
c
c
, and
f
(
x
)
+
f
(
y
)
+
f
(
z
)
=
f
(
x
)
f
(
y
)
f
(
z
)
f(x)+f(y)+f(z)=f(x)f(y)f(z)
f
(
x
)
+
f
(
y
)
+
f
(
z
)
=
f
(
x
)
f
(
y
)
f
(
z
)
whenever
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
.
2
1
Hide problems
Two-part partition of N
Find all positive integer pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
for which the set of positive integers can be partitioned into sets
H
1
H_1
H
1
and
H
2
H_2
H
2
such that neither
a
a
a
nor
b
b
b
can be represented as the difference of two numbers in
H
i
H_i
H
i
for
i
=
1
,
2
i=1,2
i
=
1
,
2
.
1
1
Hide problems
Permutations of 1,2,..,k in the rows of an array
Let
n
,
k
n,k
n
,
k
be arbitrary positive integers. We fill the entries of an
n
×
k
n\times k
n
×
k
array with integers such that all the
n
n
n
rows contain the integers
1
,
2
,
…
,
k
1,2,\dots,k
1
,
2
,
…
,
k
in some order. Add up the numbers in all
k
k
k
columns – let
S
S
S
be the largest of these sums. What is the minimal value of
S
S
S
?