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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2001 Swedish Mathematical Competition
2001 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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a chessboard with 32 dominos
A chessboard is covered with
32
32
32
dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.
5
1
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p'(x)^2 = c p(x) p''(x)
Find all polynomials
p
(
x
)
p(x)
p
(
x
)
such that
p
′
(
x
)
2
=
c
p
(
x
)
p
′
′
(
x
)
p'(x)^2 = c p(x) p''(x)
p
′
(
x
)
2
=
c
p
(
x
)
p
′′
(
x
)
for some constant
c
c
c
.
4
1
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<ABC wanted, EF bisects <AFD and <ADC = 80^o, circle
A
B
C
ABC
A
BC
is a triangle. A circle through
A
A
A
touches the side
B
C
BC
BC
at
D
D
D
and intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
again at
E
,
F
E, F
E
,
F
respectively.
E
F
EF
EF
bisects
∠
A
F
D
\angle AFD
∠
A
F
D
and
∠
A
D
C
=
8
0
o
\angle ADC = 80^o
∠
A
D
C
=
8
0
o
. Find
∠
A
B
C
\angle ABC
∠
A
BC
.
3
1
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cos(A--C) + 4 cos B = 3 if b = (a+c)/2
Show that if
b
=
a
+
c
2
b = \frac{a+c}{2}
b
=
2
a
+
c
in the triangle
A
B
C
ABC
A
BC
, then
cos
(
A
−
C
)
+
4
cos
B
=
3
\cos (A-C) + 4 \cos B = 3
cos
(
A
−
C
)
+
4
cos
B
=
3
.
2
1
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\sqrt[3]{(\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5} is rational
Show that
52
+
5
3
−
52
−
5
3
\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}
3
52
+
5
−
3
52
−
5
is rational.
1
1
Hide problems
any 6 noumbers from a 6x6 ray have equal product
Show that if we take any six numbers from the following array, one from each row and column, then the product is always the same: 4 6 10 14 22 26 6 9 15 21 33 39 10 15 25 35 55 65 16 24 40 56 88 104 18 27 45 63 99 117 20 30 50 70 110 130