MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1979 Kurschak Competition
1979 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
remaining table ommiting a column in nxn table has no two rows the same
An
n
×
n
n \times n
n
×
n
array of letters is such that no two rows are the same. Show that it must be possible to omit a column, so that the remaining table has no two rows the same.
2
1
Hide problems
f(x) = x if f(x) <= x and f(x + y) <= f(x) + f(y)
f
f
f
is a real-valued function defined on the reals such that
f
(
x
)
≤
x
f(x) \le x
f
(
x
)
≤
x
and
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
f(x + y) \le f(x) + f(y)
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
for all
x
,
y
x, y
x
,
y
. Prove that
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
for all
x
x
x
.
1
1
Hide problems
base of a convex pyramid must be a regular polygon
The base of a convex pyramid has an odd number of edges. The lateral edges of the pyramid are all equal, and the angles between neighbouring faces are all equal. Show that the base must be a regular polygon.