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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1967 Swedish Mathematical Competition
1967 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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area of triangle with lattice points inside triangle and on the sides
The vertices of a triangle are lattice points. There are no lattice points on the sides (apart from the vertices) and
n
n
n
lattice points inside the triangle. Show that its area is
n
+
1
2
n + \frac12
n
+
2
1
. Find the formula for the general case where there are also
m
m
m
lattice points on the sides (apart from the vertices).
5
1
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a_n \ge C n when C > 0, a_n^2 >= a_1 + a_2 +... + a_{n-1}
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
are positive reals such that
a
n
2
≥
a
1
+
a
2
+
.
.
.
+
a
n
−
1
a_n^2 \ge a_1 + a_2 +... + a_{n-1}
a
n
2
≥
a
1
+
a
2
+
...
+
a
n
−
1
. Show that for some
C
>
0
C > 0
C
>
0
we have
a
n
≥
C
n
a_n \ge C n
a
n
≥
C
n
for all
n
n
n
.
4
1
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lim b_n = 0 and sum a_ib_i$ diverges when sum a_i diverges
The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
of positive reals is such that
∑
a
i
\sum a_i
∑
a
i
diverges. Show that there is a sequence
b
1
,
b
2
,
b
3
,
.
.
.
b_1, b_2, b_3, ...
b
1
,
b
2
,
b
3
,
...
of positive reals such that
lim
b
n
=
0
\lim b_n = 0
lim
b
n
=
0
and
∑
a
i
b
i
\sum a_ib_i
∑
a
i
b
i
diverges.
3
1
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finitely many triples of positive (a, b, c) such that 1/a + 1/b + 1/c = 1/1000
Show that there are only finitely many triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that
1
a
+
1
b
+
1
c
=
1
1000
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}
a
1
+
b
1
+
c
1
=
1000
1
.
2
1
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angle bisector and perp. bisector costruction by a specific ruler
You are given a ruler with two parallel straight edges a distance
d
d
d
apart. It may be used (1) to draw the line through two points, (2) given two points a distance
≥
d
\ge d
≥
d
apart, to draw two parallel lines, one through each point, (3) to draw a line parallel to a given line, a distance d away. One can also (4) choose an arbitrary point in the plane, and (5) choose an arbitrary point on a line. Show how to construct : (A) the bisector of a given angle, and (B) the perpendicular to the midpoint of a given line segment.
1
1
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no of rectangles by p // lines and q perpendicular to them
p
p
p
parallel lines are drawn in the plane and
q
q
q
lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?