SCHWEITZER 1975 Problem 11

Let $X_1,X_2,...,X_n$ be (not necessary independent) discrete random variables. Prove that there exist at least $n^2/2$ pairs $(i,j)$ such that $$H(X_i+X_j) \geq \frac 13 \min_{1 \leq k \leq n} \{ H(X_k) \},$$ where $H(X)$ denotes the Shannon entropy of $X$ .

GY. Katona