MathNet 2020 — Math Olympiad Vault — Part 03 Question 41

Let $x$ and $y$ be non-negative real numbers that sum to $1$. Compute the number of ordered pairs $(a, b)$ with $a, b \in \{0,1,2,3,4\}$ such that the expression $x^{a} y^{b} + y^{a} x^{b}$ has maximum value $2^{1-a-b}$.