MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1970 Swedish Mathematical Competition
1970 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
(n - m)!/m! <= (n/2 + 1/2)^{n-2m} if 0< 2m < n
Show that
(
n
−
m
)
!
m
!
≤
(
n
2
+
1
2
)
n
−
2
m
\frac{(n - m)!}{m!} \le \left(\frac{n}{2} + \frac{1}{2}\right)^{n-2m}
m
!
(
n
−
m
)!
≤
(
2
n
+
2
1
)
n
−
2
m
for positive integers
m
,
n
m, n
m
,
n
with
2
m
≤
n
2m \le n
2
m
≤
n
.
5
1
Hide problems
6 holes in 3x1 paper, folded twice to give a square side 1
A
3
×
1
3\times 1
3
×
1
paper rectangle is folded twice to give a square side
1
1
1
. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making
6
6
6
holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?
4
1
Hide problems
p(x) p''(x) <= p'(x)^2
Let
p
(
x
)
=
(
x
−
x
1
)
(
x
−
x
2
)
(
x
−
x
3
)
p(x) = (x- x_1)(x- x_2)(x- x_3)
p
(
x
)
=
(
x
−
x
1
)
(
x
−
x
2
)
(
x
−
x
3
)
, where
x
1
,
x
2
x_1, x_2
x
1
,
x
2
and
x
3
x_3
x
3
are real. Show that
p
(
x
)
p
′
′
(
x
)
≤
p
′
(
x
)
2
p(x) p''(x) \le p'(x)^2
p
(
x
)
p
′′
(
x
)
≤
p
′
(
x
)
2
for all
x
x
x
.
3
1
Hide problems
polynomial does not take the value 9 at any integer
A polynomial with integer coefficients takes the value
5
5
5
at five distinct integers. Show that it does not take the value
9
9
9
at any integer.
2
1
Hide problems
no point lies inside all 6 disks
6
6
6
open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all
6
6
6
disks.
1
1
Hide problems
sum of three fourth powers of integers ar finite many
Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers.