The problem is like this:
Let $A$ be a finite subset of $\mathbb{R}$.
Define $A+B = \{ a+ b \mid a \in A,\ b \in B\}$ and $A\cdot B = \{a \times\ b \mid a\in A, \ b\in B\}$
Prove that $$|A+A|\cdot|A\cdot A| \ge \frac{1}{64}|A|^{\frac{5}{2}}$$
My attempts
The only hint I have is to solve via probabilistic method. Another one is that $|A+A|$ is maximized when elements of $A$ in A.P and then $|A+A| \ge 2|A|-3$ and same for GP. But I could not use this property and the fact that it cant be an AP and GP simultaneously to show that their product is greater then what is to be proved.
Any help would be appreciated.