MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1986 Swedish Mathematical Competition
1986 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
no of disjoint intervals of total length >= 1/2 in [0,1]
The interval
[
0
,
1
]
[0,1]
[
0
,
1
]
is covered by a finite number of intervals. Show that one can choose a number of these intervals which are pairwise disjoint and have the total length at least
1
/
2
1/2
1/2
.
5
1
Hide problems
pn numbers in an array, fixed difference wanted
In the arrangement of
p
n
pn
p
n
real numbers below, the difference between the greatest and smallest numbers in each row is at most
d
d
d
,
d
>
0
d > 0
d
>
0
.
a
11
a
12
.
.
.
a
1
n
a
21
a
22
.
.
.
a
2
n
.
.
.
.
.
.
.
.
.
a
n
1
a
n
2
.
.
.
a
n
n
\begin{array}{l} a_{11} \,\, a_{12} \,\, ... \,\, a_{1n}\\ a_{21} \,\, a_{22} \,\, ... \,\, a_{2n}\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ a_{n1} \,\, a_{n2} \,\, ... \,\, a_{nn}\\ \end{array}
a
11
a
12
...
a
1
n
a
21
a
22
...
a
2
n
.
.
.
.
.
.
.
.
.
a
n
1
a
n
2
...
a
nn
Prove that, when the numbers in each column are rearranged in decreasing order, the difference between the greatest and smallest numbers in each row will still be at most d.
4
1
Hide problems
x+y^2 +z^3 = 3, y+z^2 +x^3 = 3, z+x^2 +y^3 = 3
Prove that
x
=
y
=
z
=
1
x = y = z = 1
x
=
y
=
z
=
1
is the only positive solution of the system
{
x
+
y
2
+
z
3
=
3
y
+
z
2
+
x
3
=
3
z
+
x
2
+
y
3
=
3
\left\{ \begin{array}{l} x+y^2 +z^3 = 3\\ y+z^2 +x^3 = 3\\ z+x^2 +y^3 = 3\\ \end{array} \right.
⎩
⎨
⎧
x
+
y
2
+
z
3
=
3
y
+
z
2
+
x
3
=
3
z
+
x
2
+
y
3
=
3
3
1
Hide problems
pairs with q < 2are equally numbered as those with q > 2, if q = b/a
Let
N
≥
3
N \ge 3
N
≥
3
be a positive integer. For every pair
(
a
,
b
)
(a,b)
(
a
,
b
)
of integers with
1
≤
a
<
b
≤
N
1 \le a <b \le N
1
≤
a
<
b
≤
N
consider the quotient
q
=
b
/
a
q = b/a
q
=
b
/
a
. Show that the pairs with
q
<
2
q < 2
q
<
2
are equally numbered as those with
q
>
2
q > 2
q
>
2
.
2
1
Hide problems
\sqrt{S_1} + \sqrt{S_2} <= \sqrt{S} areas in quadrilateral
The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of a quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at
O
O
O
. If
S
1
S_1
S
1
and
S
2
S_2
S
2
are the areas of triangles
A
O
B
AOB
A
OB
and
C
O
D
COD
CO
D
and S that of
A
B
C
D
ABCD
A
BC
D
, show that
S
1
+
S
2
≤
S
\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}
S
1
+
S
2
≤
S
. Prove that equality holds if and only if
A
B
AB
A
B
and
C
D
CD
C
D
are parallel.
1
1
Hide problems
x^6 -x^5 +x^4 -x^3 +x^2 -x+ 3/4 has no real roots
Show that the polynomial
x
6
−
x
5
+
x
4
−
x
3
+
x
2
−
x
+
3
4
x^6 -x^5 +x^4 -x^3 +x^2 -x+\frac34
x
6
−
x
5
+
x
4
−
x
3
+
x
2
−
x
+
4
3
has no real zeroes.