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Romania Team Selection Test
2005 Romania Team Selection Test
2005 Romania Team Selection Test
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Romania Team Selection Test
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(4)
4
2
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sequence of digits interlaced with another sequence of digit
a) Prove that there exists a sequence of digits
{
c
n
}
n
≥
1
\{c_n\}_{n\geq 1}
{
c
n
}
n
≥
1
such that or each
n
≥
1
n\geq 1
n
≥
1
no matter how we interlace
k
n
k_n
k
n
digits,
1
≤
k
n
≤
9
1\leq k_n\leq 9
1
≤
k
n
≤
9
, between
c
n
c_n
c
n
and
c
n
+
1
c_{n+1}
c
n
+
1
, the infinite sequence thus obtained does not represent the fractional part of a rational number. b) Prove that for
1
≤
k
n
≤
10
1\leq k_n\leq 10
1
≤
k
n
≤
10
there is no such sequence
{
c
n
}
n
≥
1
\{c_n\}_{n\geq 1}
{
c
n
}
n
≥
1
. Dan Schwartz
again a polyhedra problem
We consider a polyhedra which has exactly two vertices adjacent with an odd number of edges, and these two vertices are lying on the same edge. Prove that for all integers
n
≥
3
n\geq 3
n
≥
3
there exists a face of the polyhedra with a number of sides not divisible by
n
n
n
.
3
5
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