[poper]

$$2(\sqrt{a-1}+2\sqrt{b-4}+3\sqrt{c-9}+4\sqrt{d-16})=a+b+c+d$$
$$2\sqrt{a-1}+4\sqrt{b-4}+6\sqrt{c-9}+8\sqrt{d-16}=a+b+c+d$$
$$(a-2\sqrt{a-1})+(b-4\sqrt{b-4})+(c-6\sqrt{c-9})+(d-8\sqrt{d-16})=0$$
$$(\sqrt{a-1}-1)^2+(\sqrt{b-4}-2)^2+(\sqrt{c-9}-3)^2+(\sqrt{d-16}-4)^2=0$$
$$a=2,b=8,c=18,d=32$$
$$a+b+c+d=60$$