HMMT 2008 S3

By a tropical polynomial we mean a function of the form

$$p(x)=a_{n} \odot x^{n} \oplus a_{n-1} \odot x^{n-1} \oplus \cdots \oplus a_{1} \odot x \oplus a_{0}$$

where exponentiation is as defined in the previous problem. Let $p$ be a tropical polynomial. Prove that

$$p\left(\frac{x+y}{2}\right) \geq \frac{p(x)+p(y)}{2}$$

for all $x, y \in \mathbb{R} \cup\{\infty\}$. (This means that all tropical polynomials are concave.)