DanDumitrescu wrote: We consider real positive numbers $a,b,c$ such that $a + b + c = 3.$ Prove that $a^2 + b^2 + c^2 + a^2b + b^2 c + c^2 a \ge 6.$ https://artofproblemsolving.com/community/c6h1653353p10467212 https://artofproblemsolving.com/community/c6h1621344p11151417

$a^2+b^2+c^2+(a^2b+b)+(b^2c+c)+(c^2a+a)\ge(a+b+c)^2$

raigeki173 wrote: $a^2+b^2+c^2+(a^2b+b)+(b^2c+c)+(c^2a+a)\ge(a+b+c)^2$ https://artofproblemsolving.com/community/c6h1653353p10468198