MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1964 Swedish Mathematical Competition
1964 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(5)
5
1
Hide problems
f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ... , a_n
a
1
,
a
2
,
...
,
a
n
are constants such that
f
(
x
)
=
1
+
a
1
c
o
s
x
+
a
2
c
o
s
2
x
+
.
.
.
+
a
n
c
o
s
n
x
≥
0
f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0
f
(
x
)
=
1
+
a
1
cos
x
+
a
2
cos
2
x
+
...
+
a
n
cos
n
x
≥
0
for all
x
x
x
. We seek estimates of
a
1
a_1
a
1
. If
n
=
2
n = 2
n
=
2
, find the smallest and largest possible values of
a
1
a_1
a
1
. Find corresponding estimates for other values of
n
n
n
.
4
1
Hide problems
min of max (PH_i) when H_iH_j <= 1
Points
H
1
,
H
2
,
.
.
.
,
H
n
H_1, H_2, ... , H_n
H
1
,
H
2
,
...
,
H
n
are arranged in the plane so that each distance
H
i
H
j
≤
1
H_iH_j \le 1
H
i
H
j
≤
1
. The point
P
P
P
is chosen to minimise
max
(
P
H
i
)
\max (PH_i)
max
(
P
H
i
)
. Find the largest possible value of
max
(
P
H
i
)
\max (PH_i)
max
(
P
H
i
)
for
n
=
3
n = 3
n
=
3
. Find the best upper bound you can for
n
=
4
n = 4
n
=
4
.
3
1
Hide problems
integer P(x) with roots \sqrt2 + \sqrt3 and \sqrt2 + \sqrt[3]{3}
Find a polynomial with integer coefficients which has
2
+
3
\sqrt2 + \sqrt3
2
+
3
and
2
+
3
3
\sqrt2 + \sqrt[3]{3}
2
+
3
3
as roots.
2
1
Hide problems
n + (n+1) + (n+2) +...+ (n+m) = 1000
Find all positive integers
m
,
n
m, n
m
,
n
such that
n
+
(
n
+
1
)
+
(
n
+
2
)
+
.
.
.
+
(
n
+
m
)
=
1000
n + (n+1) + (n+2) + ...+ (n+m) = 1000
n
+
(
n
+
1
)
+
(
n
+
2
)
+
...
+
(
n
+
m
)
=
1000
.
1
1
Hide problems
min side BC wanted for fixed area S and angle <A in triangle ABC
Find the side lengths of the triangle
A
B
C
ABC
A
BC
with area
S
S
S
and
∠
B
A
C
=
x
\angle BAC = x
∠
B
A
C
=
x
such that the side
B
C
BC
BC
is as short as possible.