Let $a \ge b \ge c \ge 0$ be three real numbers. Prove that:
$$ (a^2+c^2)(ab+ac+bc)-2ac(a^2+b^2+c^2) \ge 2c(a-b)(a-c)(b-c) $$
Ineq
Re: Ineq
No solution for a long time, so a bruteforce approach
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
[random fun post, don't take seriously]
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
[random fun post, don't take seriously]
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi