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ISI Entrance Examination
2024 ISI Entrance UGB
P6
ISI 2024 P6
ISI 2024 P6
Source: college entrance
May 12, 2024
algebra
inequalities
ISI 2024
Problem Statement
Let
x
1
,
…
,
x
2024
x_1 , \dots , x_{2024}
x
1
,
…
,
x
2024
be non negative real numbers with
∑
i
=
1
2024
x
i
=
1
\displaystyle{\sum_{i=1}^{2024}}x_i = 1
i
=
1
∑
2024
x
i
=
1
. Find, with proof, the minimum and maximum possible values of the following expression
∑
i
=
1
1012
x
i
+
∑
i
=
1013
2024
x
i
2
.
\sum_{i=1}^{1012} x_i + \sum_{i=1013}^{2024} x_i^2 .
i
=
1
∑
1012
x
i
+
i
=
1013
∑
2024
x
i
2
.
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