MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1978 Swedish Mathematical Competition
1978 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
p'(x) = nq(x) if p(x)^2 = (x^2 - 1)q(x)^2 + 1
p
(
x
)
p(x)
p
(
x
)
is a polynomial of degree
n
n
n
with leading coefficient
c
c
c
, and
q
(
x
)
q(x)
q
(
x
)
is a polynomial of degree
m
m
m
with leading coefficient
c
c
c
, such that
p
(
x
)
2
=
(
x
2
−
1
)
q
(
x
)
2
+
1
p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1
p
(
x
)
2
=
(
x
2
−
1
)
q
(
x
)
2
+
1
Show that
p
′
(
x
)
=
n
q
(
x
)
p'(x) = nq(x)
p
′
(
x
)
=
n
q
(
x
)
.
5
1
Hide problems
a,b are in the same subset and a+1,b+1 in the same subset of partion of
k
>
1
k > 1
k
>
1
is fixed. Show that for
n
n
n
sufficiently large for every partition of
{
1
,
2
,
…
,
n
}
\{1,2,\dots,n\}
{
1
,
2
,
…
,
n
}
into
k
k
k
disjoint subsets we can find
a
≠
b
a \neq b
a
=
b
such that
a
a
a
and
b
b
b
are in the same subset and
a
+
1
a+1
a
+
1
and
b
+
1
b+1
b
+
1
are in the same subset. What is the smallest
n
n
n
for which this is true?
4
1
Hide problems
ln b_i is convex if b_0,c b_1, c^2b_2,c^3b_3,... is convex for all c > 0
b
0
,
b
1
,
b
2
,
…
b_0, b_1, b_2, \dots
b
0
,
b
1
,
b
2
,
…
is a sequence of positive reals such that the sequence
b
0
,
c
b
1
,
c
2
b
2
,
c
3
b
3
,
…
b_0,c b_1, c^2b_2,c^3b_3,\dots
b
0
,
c
b
1
,
c
2
b
2
,
c
3
b
3
,
…
is convex for all
c
>
0
c > 0
c
>
0
. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that
ln
b
0
,
ln
b
1
,
ln
b
2
,
…
\ln b_0, \ln b_1, \ln b_2, \dots
ln
b
0
,
ln
b
1
,
ln
b
2
,
…
is convex.
3
1
Hide problems
2 satellites subtend an angle of 90^o
Two satellites are orbiting the earth in the equatorial plane at an altitude
h
h
h
above the surface. The distance between the satellites is always
d
d
d
, the diameter of the earth. For which
h
h
h
is there always a point on the equator at which the two satellites subtend an angle of
9
0
∘
90^\circ
9
0
∘
?
2
1
Hide problems
s_1 + s_2 + ...+ s_ n if s_m is the number $66... 6$ with m digits 6
Let
s
m
s_m
s
m
be the number
66
⋯
6
66\cdots 6
66
⋯
6
with
m
m
m
digits
6
6
6
. Find
s
1
+
s
2
+
⋯
+
s
n
s_1 + s_2 + \cdots + s_n
s
1
+
s
2
+
⋯
+
s
n
1
1
Hide problems
x^a + x^d>= x^b + x^c , for x>0 if a>b>c>d >= q 0 and a + d = b + c
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be real numbers such that
a
>
b
>
c
>
d
≥
0
a>b>c>d\geq 0
a
>
b
>
c
>
d
≥
0
and
a
+
d
=
b
+
c
a + d = b + c
a
+
d
=
b
+
c
. Show that
x
a
+
x
d
≥
x
b
+
x
c
x^a + x^d \geq x^b + x^c
x
a
+
x
d
≥
x
b
+
x
c
for
x
>
0
x>0
x
>
0
.