MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1981 Kurschak Competition
1981 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
r(k) = r(k -1) for infinitely pos. integers k, sum of remainders n: 1,2...,n
For a positive integer
n
n
n
,
r
(
n
)
r(n)
r
(
n
)
denote the sum of the remainders when
n
n
n
is divided by
1
,
2
,
.
.
.
,
n
1, 2,..., n
1
,
2
,
...
,
n
respectively. Prove that
r
(
k
)
=
r
(
k
−
1
)
r(k) = r(k -1)
r
(
k
)
=
r
(
k
−
1
)
for infinitely many positive integers
k
k
k
.
2
1
Hide problems
1/2 n^2 colors in a nx n board
Let
n
>
2
n > 2
n
>
2
be an even number. The squares of an
n
×
n
n\times n
n
×
n
chessboard are coloured with
1
2
n
2
\frac12 n^2
2
1
n
2
colours in such a way that every colour is used for colouring exactly two of the squares. Prove that one can place
n
n
n
rooks on squares of
n
n
n
different colours such that no two of the rooks can take each other.
1
1
Hide problems
AB + PQ + QR + RP <= AP + AQ + AR + BP + BQ + BR
Prove that
A
B
+
P
Q
+
Q
R
+
R
P
≤
A
P
+
A
Q
+
A
R
+
B
P
+
B
Q
+
B
R
AB + PQ + QR + RP \le AP + AQ + AR + BP + BQ + BR
A
B
+
PQ
+
QR
+
RP
≤
A
P
+
A
Q
+
A
R
+
BP
+
BQ
+
BR
where
A
,
B
,
P
,
Q
A, B, P, Q
A
,
B
,
P
,
Q
and
R
R
R
are any five points in a plane.