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For positive real numbers, let $c_2+a_2>a_1$, $c_2+b_2>b_1$, $a_2+b_2>a_1$, $a_2+b_2>b_1$, $a_1>a_2$ and $b_1>b_2$. Under these conditions, can we claim that $c_2+a_2+b_2>a_1+b_1$ ?

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Let $a_2=b_2=c_2=1$ , $a_1=b_1=2-e$ , $e$-is small number . As we can see $c_2+a_2>a_1$... but $c_2+a_2+b_2=3$,$a_1+b_1=4-2e$ .Take $e=\frac{1}{4} $ and you will see that your inequality is false .

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