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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2012 Swedish Mathematical Competition
2012 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
5
1
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3 vertrices have same color, also isosceles, when 3 colors for 13-gon
The vertices of a regular
13
13
13
-gon are colored in three different colors. Show that there are three vertices which have the same color and are also the vertices of an isosceles triangle.
4
1
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a=1 or -1<a<0 if a is real root of nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0
Given that
a
a
a
is a real solution to the polynomial equation
n
x
n
−
x
n
−
1
−
x
n
−
2
−
⋯
−
x
−
1
=
0
nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0
n
x
n
−
x
n
−
1
−
x
n
−
2
−
⋯
−
x
−
1
=
0
where
n
n
n
is a positive integer, show that
a
=
1
a=1
a
=
1
or
−
1
<
a
<
0
-1<a<0
−
1
<
a
<
0
.
2
1
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201212200619 has a factor m such that 6 x10^9 <m <6.5 x10^9
The number
201212200619
201212200619
201212200619
has a factor
m
m
m
such that
6
⋅
1
0
9
<
m
<
6.5
⋅
1
0
9
6 \cdot 10^9 <m <6.5 \cdot 10^9
6
⋅
1
0
9
<
m
<
6.5
⋅
1
0
9
. Find
m
m
m
.
1
1
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f(2012)=? if f (x + 1) =(1 + f (x))/ (1 - f (x)) , f(1000)=2012
The function
f
f
f
satisfies the condition
f
(
x
+
1
)
=
1
+
f
(
x
)
1
−
f
(
x
)
f (x + 1) = \frac{1 + f (x)}{1 - f (x)}
f
(
x
+
1
)
=
1
−
f
(
x
)
1
+
f
(
x
)
for all real
x
x
x
, for which the function is defined. Determine
f
(
2012
)
f(2012)
f
(
2012
)
, if we known that
f
(
1000
)
=
2012
f(1000)=2012
f
(
1000
)
=
2012
.
3
1
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computational, circle tangent to hypotenuse with center on right angle
The catheti
A
C
AC
A
C
and
B
C
BC
BC
in a right-angled triangle
A
B
C
ABC
A
BC
have lengths
b
b
b
and
a
a
a
, respectively. A circle centered at
C
C
C
is tangent to hypotenuse
A
B
AB
A
B
at point
D
D
D
. The tangents to the circle through points
A
A
A
and
B
B
B
intersect the circle at points
E
E
E
and
F
F
F
, respectively (where
E
E
E
and
F
F
F
are both different from
D
D
D
). Express the length of the segment
E
F
EF
EF
in terms of
a
a
a
and
b
b
b
.
6
1
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intersection of diagonals in a tangential trapezoid lies on a diameter _|_ bases
A circle is inscribed in an trapezoid. Show that the diagonals of the trapezoid intersect at a point on the diameter of the circle perpendicular to the two parallel sides.