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Korea Summer Program Practice Test
2016 Korea Summer Program Practice Test
7
7
Part of
2016 Korea Summer Program Practice Test
Problems
(1)
Infinite sequence with parallelograms
Source: Korean Summer Program Practice Test 2016 7
8/17/2016
A infinite sequence
{
a
n
}
n
≥
0
\{ a_n \}_{n \ge 0}
{
a
n
}
n
≥
0
of real numbers satisfy
a
n
≥
n
2
a_n \ge n^2
a
n
≥
n
2
. Suppose that for each
i
,
j
≥
0
i, j \ge 0
i
,
j
≥
0
there exist
k
,
l
k, l
k
,
l
with
(
i
,
j
)
≠
(
k
,
l
)
(i,j) \neq (k,l)
(
i
,
j
)
=
(
k
,
l
)
,
l
−
k
=
j
−
i
l - k = j - i
l
−
k
=
j
−
i
, and
a
l
−
a
k
=
a
j
−
a
i
a_l - a_k = a_j - a_i
a
l
−
a
k
=
a
j
−
a
i
. Prove that
a
n
≥
(
n
+
2016
)
2
a_n \ge (n + 2016)^2
a
n
≥
(
n
+
2016
)
2
for some
n
n
n
.
algebra
geometry
parallelogram