Question 7
Find all nonnegative real quadruples $(a,b,c,d)$ satisfying
$$\begin{aligned}a^2(b+c)(b+c+d)&=\sqrt{b+c}\sqrt[3]{b+c+d},\\b^2(c+d)(c+d+a)&=\sqrt{c+d}\sqrt[3]{c+d+a},\\c^2(d+a)(d+a+b)&=\sqrt{d+a}\sqrt[3]{d+a+b},\\d^2(a+b)(a+b+c)&=\sqrt{a+b}\sqrt[3]{a+b+c}.\end{aligned}$$
$$\begin{aligned}a^2(b+c)(b+c+d)&=\sqrt{b+c}\sqrt[3]{b+c+d},\\b^2(c+d)(c+d+a)&=\sqrt{c+d}\sqrt[3]{c+d+a},\\c^2(d+a)(d+a+b)&=\sqrt{d+a}\sqrt[3]{d+a+b},\\d^2(a+b)(a+b+c)&=\sqrt{a+b}\sqrt[3]{a+b+c}.\end{aligned}$$