Ineq

For discussing Olympiad Level Algebra (and Inequality) problems
yo79
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Ineq

Unread post by yo79 » Sat Mar 02, 2013 2:39 am

Let $a \ge b \ge c \ge 0$ be three real numbers. Prove that:
$$ (a^2+c^2)(ab+ac+bc)-2ac(a^2+b^2+c^2) \ge 2c(a-b)(a-c)(b-c) $$

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*Mahi*
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Re: Ineq

Unread post by *Mahi* » Wed Apr 17, 2013 2:10 am

No solution for a long time, so a bruteforce approach :P

See this Wolframalpha link
Setting $a= c+x+y, b = c+x$, we have the inequality is similar to
$c^2 x^2+4 c^2 x y+c^2 y^2+2 c x^3+4 c x^2 y+2 c x y^2+x^4+3 x^3 y+3 x^2 y^2+x y^3 \geq 0$
which is true
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