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National and Regional Contests
Romania Contests
Romania Team Selection Test
2008 Romania Team Selection Test
2008 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(5)
5
1
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Find the gcd 561
Find the greatest common divisor of the numbers 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.
2
5
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3
5
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4
3
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Polygon and set of points inside, triangulation
Prove that there exists a set
S
S
S
of n \minus{} 2 points inside a convex polygon
P
P
P
with
n
n
n
sides, such that any triangle determined by
3
3
3
vertices of
P
P
P
contains exactly one point from
S
S
S
inside or on the boundaries.
Minimum number of edges of complete triangular graphs
Let
G
G
G
be a connected graph with
n
n
n
vertices and
m
m
m
edges such that each edge is contained in at least one triangle. Find the minimum value of
m
m
m
.
Another Turan-type problem
Let
n
n
n
be a nonzero positive integer. A set of persons is called a
n
n
n
-balanced set if in any subset of
3
3
3
persons there exists at least two which know each other and in each subset of
n
n
n
persons there are two which don't know each other. Prove that a
n
n
n
-balanced set has at most (n \minus{} 1)(n \plus{} 2)/2 persons.
1
5
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