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International Contests
Nordic
2007 Nordic
2007 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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Nordic MC 2007 Q4
A line through
A
A
A
intersects a circle at points
B
,
C
B,C
B
,
C
with
B
B
B
between
A
,
C
A,C
A
,
C
. The two tangents from
A
A
A
intersect the circle at
S
,
T
S,T
S
,
T
.
S
T
ST
ST
and
A
C
AC
A
C
intersect at
P
P
P
. Show that
A
P
P
C
=
2
A
B
B
C
\frac{AP}{PC}=2\frac{AB}{BC}
PC
A
P
=
2
BC
A
B
.
2
1
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Nordic MC 2007 Q2
Three given rectangles cover the sides of a triangle completely and each rectangle has a side parallel to a given line. Show that the rectangles also cover the interior of the triangle.
3
1
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Nordic MC 2007 Q3
The number
1
0
2007
10^{2007}
1
0
2007
is written on the blackboard. Anne and Berit play a two player game in which the player in turn performs one of the following operations: 1) replace a number
x
x
x
on the blackboard with two integers
a
,
b
>
1
a,b>1
a
,
b
>
1
such that
a
b
=
x
ab=x
ab
=
x
. 2) strike off one or both of two equal numbers on the blackboard.The person who cannot perform any operation loses. Who has the winning strategy if Anne starts?
1
1
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Nordic MC 2007 Q1
Find a solution to the equation
x
2
−
2
x
−
2007
y
2
=
0
x^2-2x-2007y^2=0
x
2
−
2
x
−
2007
y
2
=
0
in positive integers.