MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2018 India IMO Training Camp
2
Sequence becomes positive halfway through
Sequence becomes positive halfway through
Source: India TST 2018, D4 P2
July 18, 2018
algebra
Sequences
Problem Statement
Let
n
≥
2
n\ge 2
n
≥
2
be a natural number. Let
a
1
≤
a
2
≤
a
3
≤
⋯
≤
a
n
a_1\le a_2\le a_3\le \cdots \le a_n
a
1
≤
a
2
≤
a
3
≤
⋯
≤
a
n
be real numbers such that
a
1
+
a
2
+
⋯
+
a
n
>
0
a_1+a_2+\cdots +a_n>0
a
1
+
a
2
+
⋯
+
a
n
>
0
and
n
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
=
2
(
a
1
+
a
2
+
⋯
+
a
n
)
2
.
n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2.
n
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
=
2
(
a
1
+
a
2
+
⋯
+
a
n
)
2
.
If
m
=
⌊
n
/
2
⌋
+
1
m=\lfloor n/2\rfloor+1
m
=
⌊
n
/2
⌋
+
1
, the smallest integer larger than
n
/
2
n/2
n
/2
, then show that
a
m
>
0.
a_m>0.
a
m
>
0.
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