MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1990 Swedish Mathematical Competition
1990 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
1
1
Hide problems
sum d_i/\sqrt {n} = sum \sqrt {n}/d_i, for pos. divisors of 1990!
Let
d
1
,
d
2
,
.
.
.
,
d
k
d_1, d_2, ... , d_k
d
1
,
d
2
,
...
,
d
k
be the positive divisors of
n
=
1990
!
n = 1990!
n
=
1990
!
. Show that
∑
d
i
n
=
∑
n
d
i
\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}
∑
n
d
i
=
∑
d
i
n
.
6
1
Hide problems
117/158 > m/n > 97/131 , n <= 500
Find all positive integers
m
,
n
m, n
m
,
n
such that
117
158
>
m
n
>
97
131
\frac{117}{158} > \frac{m}{n} > \frac{97}{131}
158
117
>
n
m
>
131
97
and
n
≤
500
n \le 500
n
≤
500
.
5
1
Hide problems
f(xy) f( f(y)/x) = 1 , monotonic
Find all monotonic positive functions
f
(
x
)
f(x)
f
(
x
)
defined on the positive reals such that
f
(
x
y
)
f
(
f
(
y
)
x
)
=
1
f(xy) f\left( \frac{f(y)}{x}\right) = 1
f
(
x
y
)
f
(
x
f
(
y
)
)
=
1
for all
x
,
y
x, y
x
,
y
.
4
1
Hide problems
LM = AL + BM wanted, angle bisectors, ABCD
A
B
C
D
ABCD
A
BC
D
is a quadrilateral. The bisectors of
∠
A
\angle A
∠
A
and
∠
B
\angle B
∠
B
meet at
E
E
E
. The line through
E
E
E
parallel to
C
D
CD
C
D
meets
A
D
AD
A
D
at
L
L
L
and
B
C
BC
BC
at
M
M
M
. Show that
L
M
=
A
L
+
B
M
LM = AL + BM
L
M
=
A
L
+
BM
.
3
1
Hide problems
sin x + sin a >= b cos x
Find all
a
,
b
a, b
a
,
b
such that
sin
x
+
sin
a
≥
b
cos
x
\sin x + \sin a\ge b \cos x
sin
x
+
sin
a
≥
b
cos
x
for all
x
x
x
.
2
1
Hide problems
min of \sum PA_i, when A_i are collinear
The points
A
1
,
A
2
,
.
.
,
A
2
n
A_1, A_2,.. , A_{2n}
A
1
,
A
2
,
..
,
A
2
n
are equally spaced in that order along a straight line with
A
1
A
2
=
k
A_1A_2 = k
A
1
A
2
=
k
.
P
P
P
is chosen to minimise
∑
P
A
i
\sum PA_i
∑
P
A
i
. Find the minimum.