MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1966 Swedish Mathematical Competition
1966 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(5)
5
1
Hide problems
limit with f(r), no of lattice points inside the circle radius r
Let
f
(
r
)
f(r)
f
(
r
)
be the number of lattice points inside the circle radius
r
r
r
, center the origin. Show that
lim
r
→
∞
f
(
r
)
r
2
\lim_{r\to \infty} \frac{f(r)}{r^2}
lim
r
→
∞
r
2
f
(
r
)
exists and find it. If the limit is
k
k
k
, put
g
(
r
)
=
f
(
r
)
−
k
r
2
g(r) = f(r) - kr^2
g
(
r
)
=
f
(
r
)
−
k
r
2
. Is it true that
lim
r
→
∞
g
(
r
)
r
h
=
0
\lim_{r\to \infty} \frac{g(r)}{r^h} = 0
lim
r
→
∞
r
h
g
(
r
)
=
0
for any
h
<
2
h < 2
h
<
2
?
4
1
Hide problems
solutions of x = f_n(x), composition, f(x) = 1 + 2/x
Let
f
(
x
)
=
1
+
2
x
f(x) = 1 + \frac{2}{x}
f
(
x
)
=
1
+
x
2
. Put
f
1
(
x
)
=
f
(
x
)
f_1(x) = f(x)
f
1
(
x
)
=
f
(
x
)
,
f
2
(
x
)
=
f
(
f
1
(
x
)
)
f_2(x) = f(f_1(x))
f
2
(
x
)
=
f
(
f
1
(
x
))
,
f
3
(
x
)
=
f
(
f
2
(
x
)
)
f_3(x) = f(f_2(x))
f
3
(
x
)
=
f
(
f
2
(
x
))
,
.
.
.
...
...
. Find the solutions to
x
=
f
n
(
x
)
x = f_n(x)
x
=
f
n
(
x
)
for
n
>
0
n > 0
n
>
0
.
3
1
Hide problems
integer = 7 mod 8 not sum of three squares
Show that an integer
=
7
m
o
d
8
= 7 \mod 8
=
7
mod
8
cannot be sum of three squares.
2
1
Hide problems
a_1 + 2a_2 + 3a_3 + ... + na_n > 0 if a_1 + a_2 + ... + a_n = 0
a
1
+
a
2
+
.
.
.
+
a
n
=
0
a_1 + a_2 + ... + a_n = 0
a
1
+
a
2
+
...
+
a
n
=
0
, for some
k
k
k
we have
a
j
≤
0
a_j \le 0
a
j
≤
0
for
j
≤
k
j \le k
j
≤
k
and
a
j
≥
0
a_j \ge 0
a
j
≥
0
for
j
>
k
j > k
j
>
k
. If ai are not all
0
0
0
, show that
a
1
+
2
a
2
+
3
a
3
+
.
.
.
+
n
a
n
>
0
a_1 + 2a_2 + 3a_3 + ... + na_n > 0
a
1
+
2
a
2
+
3
a
3
+
...
+
n
a
n
>
0
.
1
1
Hide problems
lim {x_n + - y_n} = 0 ? id lim {x_n} = lim {y_n} = 0
Let
{
x
}
\{x\}
{
x
}
denote the fractional part of
x
x
x
,
x
−
[
x
]
x - [x]
x
−
[
x
]
. The sequences
x
1
,
x
2
,
x
3
,
.
.
.
x_1, x_2, x_3, ...
x
1
,
x
2
,
x
3
,
...
and
y
1
,
y
2
,
y
3
,
.
.
.
y_1, y_2, y_3, ...
y
1
,
y
2
,
y
3
,
...
are such that
lim
{
x
n
}
=
lim
{
y
n
}
=
0
\lim \{x_n\} = \lim \{y_n\} = 0
lim
{
x
n
}
=
lim
{
y
n
}
=
0
. Is it true that
lim
{
x
n
+
y
n
}
=
0
\lim \{x_n + y_n\} = 0
lim
{
x
n
+
y
n
}
=
0
?
lim
{
x
n
−
y
n
}
=
0
\lim \{x_n - y_n\} = 0
lim
{
x
n
−
y
n
}
=
0
?