Question 1
Let $a,b,c$ be nonnegative real numbers with $a+b+c=3$. Prove
$$\begin{aligned}&\left(a^2b+\frac23\right)\left(b^2c+\frac23\right)+\left(b^2c+\frac23\right)\left(c^2a+\frac23\right)\\&\quad+\left(c^2a+\frac23\right)\left(a^2b+\frac23\right)\le\frac{75}{4}.\end{aligned}$$
$$\begin{aligned}&\left(a^2b+\frac23\right)\left(b^2c+\frac23\right)+\left(b^2c+\frac23\right)\left(c^2a+\frac23\right)\\&\quad+\left(c^2a+\frac23\right)\left(a^2b+\frac23\right)\le\frac{75}{4}.\end{aligned}$$