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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1983 Swedish Mathematical Competition
1983 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
x(x+y)^2 = 9 , x(y^3 - x^3) = 7
Show that the only real solution to
{
x
(
x
+
y
)
2
=
9
x
(
y
3
−
x
3
)
=
7
\left\{ \begin{array}{l} x(x+y)^2 = 9 \\ x(y^3 - x^3) = 7 \\ \end{array} \right.
{
x
(
x
+
y
)
2
=
9
x
(
y
3
−
x
3
)
=
7
is
x
=
1
x = 1
x
=
1
,
y
=
2
y = 2
y
=
2
.
5
1
Hide problems
unit square covered with 3 equal disks with radius <= 1/sqrt2
Show that a unit square can be covered with three equal disks with radius less than
1
2
\frac{1}{\sqrt{2}}
2
1
. What is the smallest possible radius?
4
1
Hide problems
rectangle with max area and sides on 2 concentric circles
C
C
C
,
C
′
C'
C
′
are concentric circles with radii
R
R
R
,
R
′
R'
R
′
. A rectangle has two adjacent vertices on
C
C
C
and the other two vertices on
C
′
C'
C
′
. Find its sides if its area is as large as possible.
3
1
Hide problems
diophantine system nx(n-1) has a solution, therefor n is even
The systems of equations
{
2
x
1
−
x
2
=
1
−
x
1
+
2
x
2
−
x
3
=
1
−
x
2
+
2
x
3
−
x
4
=
1
−
x
3
+
3
x
4
−
x
5
=
1
⋯
⋯
⋯
⋯
−
x
n
−
2
+
2
x
n
−
1
−
x
n
=
1
−
x
n
−
1
+
2
x
n
=
1
\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right.
⎩
⎨
⎧
2
x
1
−
x
2
=
1
−
x
1
+
2
x
2
−
x
3
=
1
−
x
2
+
2
x
3
−
x
4
=
1
−
x
3
+
3
x
4
−
x
5
=
1
⋯⋯⋯⋯
−
x
n
−
2
+
2
x
n
−
1
−
x
n
=
1
−
x
n
−
1
+
2
x
n
=
1
has a solution in positive integers
x
i
x_i
x
i
. Show that
n
n
n
must be even.
2
1
Hide problems
cos x^2 + cos y^2 - cos xy < 3
Show that
cos
x
2
+
cos
y
2
−
cos
x
y
<
3
\cos x^2 + \cos y^2 - \cos xy < 3
cos
x
2
+
cos
y
2
−
cos
x
y
<
3
for reals
x
x
x
,
y
y
y
.
1
1
Hide problems
n-th sum if pos. integers are grouped as 1, 2+3, 4+5+6, 7+8+9+10,...
The positive integers are grouped as follows:
1
,
2
+
3
,
4
+
5
+
6
,
7
+
8
+
9
+
10
,
…
1, 2+3, 4+5+6, 7+8+9+10,\dots
1
,
2
+
3
,
4
+
5
+
6
,
7
+
8
+
9
+
10
,
…
. Find the value of the
n
n
n
-th sum.