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International Contests
Nordic
2004 Nordic
2004 Nordic
Part of
Nordic
Subcontests
(4)
3
1
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From a sequence, define other sequences
Given a finite sequence
x
1
,
1
,
x
2
,
1
,
…
,
x
n
,
1
x_{1,1}, x_{2,1}, \dots , x_{n,1}
x
1
,
1
,
x
2
,
1
,
…
,
x
n
,
1
of integers
(
n
≥
2
)
(n\ge 2)
(
n
≥
2
)
, not all equal, define the sequences
x
1
,
k
,
…
,
x
n
,
k
x_{1,k}, \dots , x_{n,k}
x
1
,
k
,
…
,
x
n
,
k
by x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k}) \text{where }x_{n+1,k}=x_{1,k}. Show that if
n
n
n
is odd, then not all
x
j
,
k
x_{j,k}
x
j
,
k
are integers. Is this also true for even
n
n
n
?
4
1
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Geometric inequality 1/ab+1/bc+/ca>1/R^2
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the sides and
R
R
R
be the circumradius of a triangle. Prove that
1
a
b
+
1
b
c
+
1
c
a
≥
1
R
2
.
\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.
ab
1
+
b
c
1
+
c
a
1
≥
R
2
1
.
1
1
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How many balls in each bowl, given the averages
Twenty-seven balls labelled from
1
1
1
to
27
27
27
are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue and yellow bowl are
15
15
15
,
3
3
3
, and
18
18
18
, respectively?
2
1
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Nordic M.C.
Show that there exist strictly increasing infinite arithmetic sequence of integers which has no numbers in common with the Fibonacci sequence.