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2021 Turkey Team Selection Test
6
Find positive integers n, i<=x_i<=2i is given
Find positive integers n, i<=x_i<=2i is given
Source: 2021 Turkey TST P6
May 23, 2021
algebra
Turkey
equation
n-variable equation
construction
Problem Statement
For which positive integers
n
n
n
, one can find real numbers
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots ,x_n
x
1
,
x
2
,
⋯
,
x
n
such that
x
1
2
+
x
2
2
+
⋯
+
x
n
2
(
x
1
+
2
x
2
+
⋯
+
n
x
n
)
2
=
27
4
n
(
n
+
1
)
(
2
n
+
1
)
\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}
(
x
1
+
2
x
2
+
⋯
+
n
x
n
)
2
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
4
n
(
n
+
1
)
(
2
n
+
1
)
27
and
i
≤
x
i
≤
2
i
i\leq x_i\leq 2i
i
≤
x
i
≤
2
i
for all
i
=
1
,
2
,
⋯
,
n
i=1,2,\cdots ,n
i
=
1
,
2
,
⋯
,
n
?
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