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Moldova Team Selection Test
2023 Moldova Team Selection Test
6
x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}
x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}
Source: Moldova TST 2023
April 8, 2023
algebra
Problem Statement
Show that if
2023
2023
2023
real numbers
x
1
,
x
2
,
…
,
x
2023
x_1,x_2,\dots,x_{2023}
x
1
,
x
2
,
…
,
x
2023
satisfy
x
1
≥
x
2
≥
⋯
≥
x
2023
≥
0
,
x_1\geq x_2\geq\dots\geq x_{2023}\geq0,
x
1
≥
x
2
≥
⋯
≥
x
2023
≥
0
,
then
x
1
2
+
3
x
2
2
+
5
x
3
2
+
⋯
+
(
2
⋅
2023
−
1
)
⋅
x
2023
2
≤
(
x
1
+
x
2
+
⋯
+
x
2023
)
2
.
x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}\leq(x_1+x_2+\cdots+x_{2023})^2.
x
1
2
+
3
x
2
2
+
5
x
3
2
+
⋯
+
(
2
⋅
2023
−
1
)
⋅
x
2023
2
≤
(
x
1
+
x
2
+
⋯
+
x
2023
)
2
.
When does the equality take place?
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