IRL_ABooklet_2023

$$(ABCD)=\sqrt{(\sigma -a)(\sigma -b)(\sigma -c)(\sigma -d)},$$ whence pairing the factors in the expression $\sqrt{(\sigma -a)(\sigma -b)(\sigma -c)(\sigma -d)}$ in three different ways, and noting that by AM-GM, e.g., $$\sqrt{(\sigma -a)(\sigma -b)}\le \frac{2\sigma -a-b}{2}=\frac{d+c}{2},$$ we see that $$(ABCD)\le \frac{(a+b)(c+d)}{4},$$ with equality iff $a=b$ and $c=d$, that $$(ABCD)\le \frac{(a+d)(b+c)}{4},$$ with equality iff $a=d$ and $b=c$, and that $$(ABCD)\le \frac{(a+c)(b+d)}{4},$$ with equality iff $a=c$ and $b=d$. Consequently, combining these, $$3(ABCD)\le \frac{(a+b)(c+d)}{4}+\frac{(a+c)(b+d)}{4}+\frac{(a+d)(b+c)}{4}=\frac{ab+ac+ad+bc+bd+cd}{2},$$ with equality iff $a=b=c=d$.