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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1996 Yugoslav Team Selection Test
1996 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
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sequence is sum of consecutive squares
The sequence
{
x
n
}
\{x_n\}
{
x
n
}
is given by
x
n
=
1
4
(
(
2
+
3
)
2
n
−
1
+
(
2
−
3
)
2
n
−
1
)
,
n
∈
N
.
x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.
x
n
=
4
1
(
(
2
+
3
)
2
n
−
1
+
(
2
−
3
)
2
n
−
1
)
,
n
∈
N
.
Prove that each
x
n
x_n
x
n
is equal to the sum of squares of two consecutive integers.
Problem 2
1
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1996 circles in plane, circle touching three others
Let there be given a set of
1996
1996
1996
equal circles in the plane, no two of them having common interior points. Prove that there exists a circle touching at most three other circles.
Problem 1
1
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set with conditions, |F|≠1996
Let
F
=
{
A
1
,
A
2
,
…
,
A
n
}
\mathfrak F=\{A_1,A_2,\ldots,A_n\}
F
=
{
A
1
,
A
2
,
…
,
A
n
}
be a collection of subsets of the set
S
=
{
1
,
2
,
…
,
n
}
S=\{1,2,\ldots,n\}
S
=
{
1
,
2
,
…
,
n
}
satisfying the following conditions:(a) Any two distinct sets from
F
\mathfrak F
F
have exactly one element in common; (b) each element of
S
S
S
is contained in exactly
k
k
k
of the sets in
F
\mathfrak F
F
.Can
n
n
n
be equal to
1996
1996
1996
?