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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1996 Swedish Mathematical Competition
1996 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
5
1
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equal sum of the $p$-th powers of numbers from 1,2,...,2^n
Let
n
≥
1
n \ge 1
n
≥
1
. Prove that it is possible to select some of the integers
1
,
2
,
.
.
.
,
2
n
1,2,...,2^n
1
,
2
,
...
,
2
n
so that for each
p
=
0
,
1
,
.
.
.
,
n
−
1
p = 0,1,...,n - 1
p
=
0
,
1
,
...
,
n
−
1
the sum of the
p
p
p
-th powers of the selected numbers is equal to the sum of the
p
p
p
-th powers of the remaining numbers.
3
1
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p_n(x) = 1/2 ((x+\sqrt{x^2-1} )^n+ (x-\sqrt{x^2-1} )^n
For every positive integer
n
n
n
, we define the function
p
n
p_n
p
n
for
x
≥
1
x\ge 1
x
≥
1
by
p
n
(
x
)
=
1
2
(
(
x
+
x
2
−
1
)
n
+
(
x
−
x
2
−
1
)
n
)
.
p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).
p
n
(
x
)
=
2
1
(
(
x
+
x
2
−
1
)
n
+
(
x
−
x
2
−
1
)
n
)
.
Prove that
p
n
(
x
)
≥
1
p_n(x) \ge 1
p
n
(
x
)
≥
1
and that
p
m
n
(
x
)
=
p
m
(
p
n
(
x
)
)
p_{mn}(x) = p_m(p_n(x))
p
mn
(
x
)
=
p
m
(
p
n
(
x
))
.
1
1
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T=(\sqrt{T_1}+\sqrt{T_2}+\sqrt{T_3})^2, areas of triangles
Through an arbitrary point inside a triangle, lines parallel to the sides of the triangle are drawn, dividing the triangle into three triangles with areas
T
1
,
T
2
,
T
3
T_1,T_2,T_3
T
1
,
T
2
,
T
3
and three parallelograms. If
T
T
T
is the area of the original triangle, prove that
T
=
(
T
1
+
T
2
+
T
3
)
2
T=(\sqrt{T_1}+\sqrt{T_2}+\sqrt{T_3})^2
T
=
(
T
1
+
T
2
+
T
3
)
2
.
6
1
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one of side lengths is divisible by 6 if rectangle is tiled with 6x1 rectangles
A rectangle is tiled with rectangles of size
6
×
1
6\times 1
6
×
1
. Prove that one of its side lengths is divisible by
6
6
6
.
4
1
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<C > \pi/2 id <A,<B,<C,<D,<E from an increasinf sequence in cyclic ABCDE
The angles at
A
,
B
,
C
,
D
,
E
A,B,C,D,E
A
,
B
,
C
,
D
,
E
of a pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
inscribed in a circle form an increasing sequence. Show that the angle at
C
C
C
is greater than
π
/
2
\pi/2
π
/2
, and that this lower bound cannot be improved.
2
1
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2 values of stamps in the country of Postonia,
In the country of Postonia, one wants to have only two values of stamps. These values should be integers greater than
1
1
1
with the difference
2
2
2
, and should have the property that one can combine the stamps for any postage which is greater than or equal to the sum of these two values. What values can be chosen?