MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2018 India IMO Training Camp
2
Find sequences with given sum and sum-of-squares
Find sequences with given sum and sum-of-squares
Source: IMOTC PT2 2018 P2, India
July 18, 2018
algebra
inequalities
Problem Statement
For an integer
n
≥
2
n\ge 2
n
≥
2
find all
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
so that (a)
0
≤
a
1
≤
a
2
≤
⋯
≤
a
n
≤
1
≤
b
1
≤
b
2
≤
⋯
≤
b
n
;
0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;
0
≤
a
1
≤
a
2
≤
⋯
≤
a
n
≤
1
≤
b
1
≤
b
2
≤
⋯
≤
b
n
;
(b)
∑
k
=
1
n
(
a
k
+
b
k
)
=
2
n
;
\sum_{k=1}^n (a_k+b_k)=2n;
∑
k
=
1
n
(
a
k
+
b
k
)
=
2
n
;
(c)
∑
k
=
1
n
(
a
k
2
+
b
k
2
)
=
n
2
+
3
n
.
\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.
∑
k
=
1
n
(
a
k
2
+
b
k
2
)
=
n
2
+
3
n
.
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