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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1989 Swedish Mathematical Competition
1989 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
5
1
Hide problems
1/(x_1 +x_3)^a+1/(x_2 +x_4)^a+1/(x_2 +x_5)^a < ...
Assume
x
1
,
x
2
,
.
.
,
x
5
x_1,x_2,..,x_5
x
1
,
x
2
,
..
,
x
5
are positive numbers such that
x
1
<
x
2
x_1 < x_2
x
1
<
x
2
and
x
3
,
x
4
,
x
5
x_3,x_4, x_5
x
3
,
x
4
,
x
5
are all greater than
x
2
x_2
x
2
. Prove that if
a
>
0
a > 0
a
>
0
, then
1
(
x
1
+
x
3
)
a
+
1
(
x
2
+
x
4
)
a
+
1
(
x
2
+
x
5
)
a
<
1
(
x
1
+
x
2
)
a
+
1
(
x
2
+
x
3
)
a
+
1
(
x
4
+
x
5
)
a
\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}
(
x
1
+
x
3
)
a
1
+
(
x
2
+
x
4
)
a
1
+
(
x
2
+
x
5
)
a
1
<
(
x
1
+
x
2
)
a
1
+
(
x
2
+
x
3
)
a
1
+
(
x
4
+
x
5
)
a
1
6
1
Hide problems
4n points on circle, alternately colored yellow and blue, colored segments
On a circle
4
n
4n
4
n
points are chosen (
n
≥
1
n \ge 1
n
≥
1
). The points are alternately colored yellow and blue. The yellow points are divided into
n
n
n
pairs and the points in each pair are connected with a yellow line segment. In the same manner the blue points are divided into
n
n
n
pairs and the points in each pair are connected with a blue segment. Assume that no three of the segments pass through a single point. Show that there are at least
n
n
n
intersection points of blue and yellow segments.
4
1
Hide problems
locus wanted, edge tangent to sphere, regular terahedron ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a regular tetrahedron. Find the positions of point
P
P
P
on the edge
B
D
BD
B
D
such that the edge
C
D
CD
C
D
is tangent to the sphere with diameter
A
P
AP
A
P
.
3
1
Hide problems
n^3 - 18n^2 + 115n - 391 a perfect cube
Find all positive integers
n
n
n
such that
n
3
−
18
n
2
+
115
n
−
391
n^3 - 18n^2 + 115n - 391
n
3
−
18
n
2
+
115
n
−
391
is the cube of a positive intege
2
1
Hide problems
f(x)+ f(x^2) = 0
Find all continuous functions
f
f
f
such that
f
(
x
)
+
f
(
x
2
)
=
0
f(x)+ f(x^2) = 0
f
(
x
)
+
f
(
x
2
)
=
0
for all real numbers
x
x
x
.
1
1
Hide problems
n^2(n^2 + 2)^2 , n^4(n^2 + 2)^2, in base n^2 +1, same digits
Let
n
n
n
be a positive integer. Prove that the numbers
n
2
(
n
2
+
2
)
2
n^2(n^2 + 2)^2
n
2
(
n
2
+
2
)
2
and
n
4
(
n
2
+
2
)
2
n^4(n^2 + 2)^2
n
4
(
n
2
+
2
)
2
are written in base
n
2
+
1
n^2 +1
n
2
+
1
with the same digits but in opposite order.