MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1998 Swedish Mathematical Competition
1998 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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|2^{1/3} - m/n| > c/n^3
Show that for some
c
>
0
c > 0
c
>
0
, we have
∣
2
3
−
m
n
∣
>
c
n
3
\left|\sqrt[3]{2} - \frac{m}{n}\right | > \frac{c}{n^3}
3
2
−
n
m
>
n
3
c
for all integers
m
,
n
m, n
m
,
n
with
n
≥
1
n \ge 1
n
≥
1
.
5
1
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1/x_1 + 1/x_2 +... + 1/x_n = 1997/1998
Show that for any
n
>
5
n > 5
n
>
5
we can find positive integers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ... , x_n
x
1
,
x
2
,
...
,
x
n
such that
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
=
1997
1998
\frac{1}{x_1} + \frac{1}{x_2} +... + \frac{1}{x_n} = \frac{1997}{1998}
x
1
1
+
x
2
1
+
...
+
x
n
1
=
1998
1997
. Show that in any such equation there must be two of the
n
n
n
numbers with a common divisor (
>
1
> 1
>
1
).
4
1
Hide problems
<B =? , (ABCD)=(a+b)h/2
A
B
C
D
ABCD
A
BC
D
is a quadrilateral with
∠
A
=
90
o
\angle A = 90o
∠
A
=
90
o
,
A
D
=
a
AD = a
A
D
=
a
,
B
C
=
b
BC = b
BC
=
b
,
A
B
=
h
AB = h
A
B
=
h
, and area
(
a
+
b
)
h
2
\frac{(a+b)h}{2}
2
(
a
+
b
)
h
. What can we say about
∠
B
\angle B
∠
B
?
3
1
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moves on a cube 5x5
A cube side
5
5
5
is made up of unit cubes. Two small cubes are adjacent if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).
2
1
Hide problems
c >= (a+b) sin(C/2)
A
B
C
ABC
A
BC
is a triangle. Show that
c
≥
(
a
+
b
)
sin
C
2
c \ge (a+b) \sin \frac{C}{2}
c
≥
(
a
+
b
)
sin
2
C
1
1
Hide problems
(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2
Find all positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
, such that
(
8
a
−
5
b
)
2
+
(
3
b
−
2
c
)
2
+
(
3
c
−
7
a
)
2
=
2
(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2
(
8
a
−
5
b
)
2
+
(
3
b
−
2
c
)
2
+
(
3
c
−
7
a
)
2
=
2
.