Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.8 AIME, harder Prove it

1 Given $k>a>b>c>0$ , prove:
$k^{2}-(a+b+c) k+(a b+b c+c a)>0$

Solution

1. Consider the identity $(k-a)(k-b)(k-c)=k^{3}-(a+b+c) k^{2}+(a b+b c+$ ca) $k-a b c$ .

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