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a-1-gt-a-2-gt-a-3-gt-gt-a-n-gt-0-b-1-gt-b-2-gt-b-3-gt-gt-b-n-gt-0-prove-i-1-n-a-i-b-i-i-1-n-a-i-b-n-i-1-




Question Number 208440 by liuxinnan last updated on 16/Jun/24
a_1 >a_2 >a_3 >...>a_n >0  b_1 >b_2 >b_3 >...>b_n >0  prove  Σ_(i=1) ^n a_i b_i ≥Σ_(i=1) ^n a_i b_(n−i+1)
$${a}_{\mathrm{1}} >{a}_{\mathrm{2}} >{a}_{\mathrm{3}} >…>{a}_{{n}} >\mathrm{0} \\ $$$${b}_{\mathrm{1}} >{b}_{\mathrm{2}} >{b}_{\mathrm{3}} >…>{b}_{{n}} >\mathrm{0} \\ $$$${prove} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{i}} {b}_{{i}} \geqslant\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{i}} {b}_{{n}−{i}+\mathrm{1}} \\ $$

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