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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2005 Swedish Mathematical Competition
2005 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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Tetrahedron Projection
A regular tetrahedron of edge length
1
1
1
is orthogonally projected onto a plane. Find the largest possible area of its image.
4
1
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Arithmetic Progression of Roots
The zeroes of a fourth degree polynomial
f
(
x
)
f(x)
f
(
x
)
form an arithmetic progression. Prove that the three zeroes of the polynomial
f
′
(
x
)
f'(x)
f
′
(
x
)
also form an arithmetic progression.
3
1
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Angle Bisectors
In a triangle
A
B
C
ABC
A
BC
the bisectors of angles
A
A
A
and
C
C
C
meet the opposite sides at
D
D
D
and
E
E
E
respectively. Show that if the angle at
B
B
B
is greater than
6
0
∘
60^\circ
6
0
∘
, then
A
E
+
C
D
<
A
C
AE +CD <AC
A
E
+
C
D
<
A
C
.
2
1
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Line at a Bank
There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?
1
1
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Two Variable Diophantine
Find all integer solutions
x
x
x
,
y
y
y
of the equation
(
x
+
y
2
)
(
x
2
+
y
)
=
(
x
+
y
)
3
(x+y^2)(x^2+y)=(x+y)^3
(
x
+
y
2
)
(
x
2
+
y
)
=
(
x
+
y
)
3
.
5
1
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A 2005 x 2005 board
Every cell of a
2005
×
2005
2005 \times 2005
2005
×
2005
square board is colored white or black so that every
2
×
2
2 \times 2
2
×
2
subsquare contains an odd number of black cells. Show that among the corner cells there is an even number of black ones. How many ways are there to color the board in this manner?