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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2023 Moldova Team Selection Test
6
6
Part of
2023 Moldova Team Selection Test
Problems
(1)
x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}
Source: Moldova TST 2023
4/8/2023
Show that if
2023
2023
2023
real numbers
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,
x
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,
…
,
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2023
x_1,x_2,\dots,x_{2023}
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…
,
x
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satisfy
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≥
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0
,
x_1\geq x_2\geq\dots\geq x_{2023}\geq0,
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≥
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≥
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,
then
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5
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⋯
+
(
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2023
−
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⋅
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≤
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.
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.
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⋅
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≤
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2
.
When does the equality take place?
algebra