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Gheorghe Vranceanu
2008 Gheorghe Vranceanu
2008 Gheorghe Vranceanu
Part of
Gheorghe Vranceanu
Subcontests
(4)
4
1
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Finite partition of naturals in NT
Find the largest natural number
k
k
k
which has the property that there is a partition of the natural numbers
⋃
1
≤
j
≤
k
V
j
,
\bigcup_{1\le j\le k} V_j,
⋃
1
≤
j
≤
k
V
j
,
an index
i
∈
{
1
,
…
,
k
}
i\in\{ 1,\ldots ,k \}
i
∈
{
1
,
…
,
k
}
and three natural numbers
a
,
b
,
c
∈
V
i
,
a,b,c\in V_i,
a
,
b
,
c
∈
V
i
,
satisfying
a
+
2
b
=
4
c
.
a+2b=4c.
a
+
2
b
=
4
c
.
3
1
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Homothety problem IMO-like
If the circumradius of any three consecutive vertices of a convex polygon is at most
1
,
1,
1
,
show that the discs of radius
1
1
1
centered at each vertex cover the polygon and its interior.
1
5
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2
5
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