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Contests
International Contests
Nordic
2006 Nordic
2006 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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Nordic MC 2006 Q4
Each square of a
100
×
100
100\times 100
100
×
100
board is painted with one of
100
100
100
different colours, so that each colour is used exactly
100
100
100
times. Show that there exists a row or column of the chessboard in which at least
10
10
10
colours are used.
3
1
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Nordic MC 2006 Q3
A sequence
(
a
n
)
(a_n)
(
a
n
)
of positive integers is defined by
a
0
=
m
a_0=m
a
0
=
m
and
a
n
+
1
=
a
n
5
+
487
a_{n+1}= a_n^5 +487
a
n
+
1
=
a
n
5
+
487
for all
n
≥
0
n\ge 0
n
≥
0
. Find all positive integers
m
m
m
such that the sequence contains the maximum possible number of perfect squares.
2
1
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Nordic MC 2006 Q2
Real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
are not all equal and satisfy
x
+
1
y
=
y
+
1
z
=
z
+
1
x
=
k
x+\frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}=k
x
+
y
1
=
y
+
z
1
=
z
+
x
1
=
k
. Find all possible values of
k
k
k
.
1
1
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Nordic MC 2006 Q1
Points
B
,
C
B,C
B
,
C
vary on two fixed rays emanating from point
A
A
A
such that
A
B
+
A
C
AB+AC
A
B
+
A
C
is constant. Show that there is a point
D
D
D
, other than
A
A
A
, such that the circumcircle of triangle
A
B
C
ABC
A
BC
passes through
D
D
D
for all possible choices of
B
,
C
B, C
B
,
C
.