Question 40
Let $n$ be a positive integer. For nonnegative real numbers $a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}$ satisfying $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$, define
$$
A=\left\{\sum_{j=1}^{2n}\frac{a_i b_j}{a_i b_j+1}:1\le i\le2n,\ \sum_{j=1}^{2n}\frac{a_i b_j}{a_i b_j+1}\ne0\right\},
$$
$$
B=\left\{\sum_{i=1}^{2n}\frac{a_i b_j}{a_i b_j+1}:1\le j\le2n,\ \sum_{i=1}^{2n}\frac{a_i b_j}{a_i b_j+1}\ne0\right\}.
$$
Let $m$ be the minimum element of $A\cup B$. Determine the largest possible value of $m$.
$$
A=\left\{\sum_{j=1}^{2n}\frac{a_i b_j}{a_i b_j+1}:1\le i\le2n,\ \sum_{j=1}^{2n}\frac{a_i b_j}{a_i b_j+1}\ne0\right\},
$$
$$
B=\left\{\sum_{i=1}^{2n}\frac{a_i b_j}{a_i b_j+1}:1\le j\le2n,\ \sum_{i=1}^{2n}\frac{a_i b_j}{a_i b_j+1}\ne0\right\}.
$$
Let $m$ be the minimum element of $A\cup B$. Determine the largest possible value of $m$.