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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1968 Swedish Mathematical Competition
1968 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(5)
5
1
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min value of (cos ax + cos bx) <= r < 0
Let
a
,
b
a, b
a
,
b
be non-zero integers. Let
m
(
a
,
b
)
m(a, b)
m
(
a
,
b
)
be the smallest value of
cos
a
x
+
cos
b
x
\cos ax + \cos bx
cos
a
x
+
cos
b
x
(for real
x
x
x
). Show that for some
r
r
r
,
m
(
a
,
b
)
≤
r
<
0
m(a, b) \le r < 0
m
(
a
,
b
)
≤
r
<
0
for all
a
,
b
a, b
a
,
b
.
4
1
Hide problems
f(m-n) , where m, n belong to M cannot exceed n-1.
For
n
≠
0
n\ne 0
n
=
0
, let f(n) be the largest
k
k
k
such that
3
k
3^k
3
k
divides
n
n
n
. If
M
M
M
is a set of
n
>
1
n > 1
n
>
1
integers, show that the number of possible values for
f
(
m
−
n
)
f(m-n)
f
(
m
−
n
)
, where
m
,
n
m, n
m
,
n
belong to
M
M
M
cannot exceed
n
−
1
n-1
n
−
1
.
3
1
Hide problems
sum of squares of sides of a quadrilateral >= sum of squares of diagonals
Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?
2
1
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no to labe faces of a cube with 1, 2,..., 6
How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers
1
,
2
,
.
.
.
,
6
1, 2,..., 6
1
,
2
,
...
,
6
?
1
1
Hide problems
min, max of x^2 + 2y^2 + 3z^2 when x^2 + y^2 + z^2 = 1
Find the maximum and minimum values of
x
2
+
2
y
2
+
3
z
2
x^2 + 2y^2 + 3z^2
x
2
+
2
y
2
+
3
z
2
for real
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfying
x
2
+
y
2
+
z
2
=
1
x^2 + y^2 + z^2 = 1
x
2
+
y
2
+
z
2
=
1
.