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4

Part of 2021 Mediterranean Mathematics Olympiad

Problems(1)

\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5} <= 20

Source: 2021 Mediterranean Mathematical Olympiad P4 MMC

9/11/2021
Let x1,x2,x3,x4,x5x_1,x_2,x_3,x_4,x_5 ve non-negative real numbers, so that x1≤4x_1\le4 and x1+x2≤13x_1+x_2\le13 and x1+x2+x3≤29x_1+x_2+x_3\le29 and x1+x2+x3+x4≤54x_1+x_2+x_3+x_4\le54 and x1+x2+x3+x4+x5≤90x_1+x_2+x_3+x_4+x_5\le90. Prove that x1+x2+x3+x4+x5≤20\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5}\le20.
inequalitiesalgebra