Problem

Source: Romania JBMO TST 2022

Tags: algebra, romania, Romanian TST, inequalities



Let $a\geq b\geq c\geq d$ be real numbers such that $(a-b)(b-c)(c-d)(d-a)=-3.$ If $a+b+c+d=6,$ prove that $d<0,36.$ If $a^2+b^2+c^2+d^2=14,$ prove that $(a+c)(b+d)\leq 8.$ When does equality hold?