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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1968 Yugoslav Team Selection Test
Problem 5
Problem 5
Part of
1968 Yugoslav Team Selection Test
Problems
(1)
a summation over pairs of integers, polynomial
Source: Yugoslav TST 1968 P5
5/29/2021
Let
n
n
n
be an integer greater than
1
1
1
. Let
x
∈
R
x\in\mathbb R
x
∈
R
.(a) Evaluate
S
(
x
,
n
)
=
∑
(
x
+
p
)
(
x
+
q
)
S(x,n)=\sum(x+p)(x+q)
S
(
x
,
n
)
=
∑
(
x
+
p
)
(
x
+
q
)
, where the summation is over all pairs
(
p
,
q
)
(p,q)
(
p
,
q
)
of different numbers from
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
. (b) Do there exist integers
x
,
n
x,n
x
,
n
for which
S
(
x
,
n
)
=
0
S(x,n)=0
S
(
x
,
n
)
=
0
?
Summation
algebra
polynomial