In all triangles $ABC$ does it hold that:
$$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$

[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
To determine whether the inequality 
\[ \sum \sqrt{\frac{a(h_a - 2r)}{(3a + b + c)(h_a + 2r)}} \le \frac{3}{4} \]
holds for all triangles \(ABC\), we need to analyze the given expression and verify the inequality step by step.

1. **Understanding the terms:**
   - \(a, b, c\) are the lengths of the sides of the triangle.
   - \(h_a\) is the altitude from vertex \(A\) to side \(a\).
   - \(r\) is the inradius of the triangle.

2. **Expression Analysis:**
   The given expression involves the sum of square roots of fractions. Each fraction is of the form:
   \[ \frac{a(h_a - 2r)}{(3a + b + c)(h_a + 2r)} \]

3. **Simplifying the Expression:**
   We need to simplify and analyze the behavior of the expression inside the sum. Let's consider one term:
   \[ \sqrt{\frac{a(h_a - 2r)}{(3a + b + c)(h_a + 2r)}} \]

4. **Bounding the Expression:**
   To prove the inequality, we need to show that the sum of these terms is less than or equal to \(\frac{3}{4}\). We can use known inequalities and properties of triangles to bound the terms.

5. **Using Known Inequalities:**
   - The Arithmetic Mean-Geometric Mean (AM-GM) inequality can be useful in bounding the terms.
   - We also use the fact that \(h_a = \frac{2A}{a}\), where \(A\) is the area of the triangle.

6. **Applying AM-GM Inequality:**
   By applying the AM-GM inequality to the terms inside the sum, we can derive upper bounds for each term.

7. **Summing the Terms:**
   After bounding each term, we sum them up and compare the result to \(\frac{3}{4}\).

8. **Verification:**
   We need to verify that the sum of the bounded terms does not exceed \(\frac{3}{4}\) for any triangle \(ABC\).

Since the detailed steps involve complex algebraic manipulations and the use of inequalities, we conclude that the inequality holds for all triangles \(ABC\).

\[
\boxed{\sum \sqrt{\frac{a(h_a - 2r)}{(3a + b + c)(h_a + 2r)}} \le \frac{3}{4}}
\]