A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$.  A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$.  Find the sum of all lonely integers.

[i]Proposed by Evan Chen[/i]
To determine the sum of all lonely integers, we need to analyze the conditions under which a pair \((m, n)\) is compatible. The conditions given are:
\[ m \ge \frac{1}{2} n + 7 \]
\[ n \ge \frac{1}{2} m + 7 \]

A positive integer \( k \ge 1 \) is called lonely if \((k, \ell)\) is not compatible for any integer \(\ell \ge 1\). This means that for a lonely integer \( k \), there does not exist any \(\ell\) such that both conditions are satisfied.

1. **Analyze the conditions for compatibility:**
   Suppose \( k \) is a lonely integer. Then for any \(\ell \ge 1\), at least one of the conditions must fail. We start by assuming that there exists some \(\ell\) such that:
   \[ k \ge \frac{1}{2} \ell + 7 \]
   This implies:
   \[ 2(k - 7) \ge \ell \]

2. **Substitute \(\ell\) into the second condition:**
   We also need to satisfy:
   \[ \ell \ge \frac{1}{2} k + 7 \]
   Substituting \(\ell\) from the first condition, we get:
   \[ 2(k - 7) \ge \frac{1}{2} k + 7 \]

3. **Solve the inequality:**
   \[ 4(k - 7) \ge k + 14 \]
   \[ 4k - 28 \ge k + 14 \]
   \[ 3k - 28 \ge 14 \]
   \[ 3k \ge 42 \]
   \[ k \ge 14 \]

4. **Conclusion for lonely integers:**
   If \( k \ge 14 \), then there exists some \(\ell\) such that \((k, \ell)\) is compatible. Therefore, \( k \) cannot be lonely if \( k \ge 14 \). Hence, the lonely integers must be those for which \( k \le 13 \).

5. **Sum of all lonely integers:**
   The lonely integers are \( 1, 2, 3, \ldots, 13 \). The sum of these integers is:
   \[ \sum_{i=1}^{13} i = \frac{13 \cdot 14}{2} = 91 \]

The final answer is \(\boxed{91}\)