Berkeley Math Circle: Monthly Contest 2

Solution:

Jacob should pick $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\left(\frac{1}{2}, \frac{1}{3}, \frac{1}{7}, \frac{1}{42}\right)$. More generally, suppose the weights are $p_{1}, \ldots, p_{4}$. Then Jacob's expected payoff is

$$10+\sum_{i=1}^{4} p_{i} \log \left(x_{i}\right)=10+\sum_{i=1}^{4} p_{i} \log p_{i}+\sum_{i=1}^{4} p_{i} \log \left(\frac{x_{i}}{p_{i}}\right)$$

Now, by JENSEN's INEQUALITY on the concave function $\log x$, we obtain

$$\sum_{i=1}^{4} p_{i} \log \left(\frac{x_{i}}{p_{i}}\right) \leq \log \left(\sum_{i=1}^{4} p_{i} \cdot \frac{x_{i}}{p_{i}}\right)=\log 1=0$$

and equality occurs exactly when $\frac{x_{1}}{p_{1}}=\frac{x_{2}}{p_{2}}=\frac{x_{3}}{p_{3}}=\frac{x_{4}}{p_{4}}$; that is, when $x_{i}=p_{i}$ for every $i$.