\( \left\{\begin{array}{l}a+b+c=1 \\a>-\frac{1}{4} \\b>-\frac{1}{4} \\c>-\frac{1}{4} \\\sqrt {4a+1}+\sqrt {4b+1}+\sqrt {4c+1}<5\end{array}\right. \)

Let's break down the problem step by step. We have the following system of equations and inequalities: 1. \( a + b + c = 1 \) 2. \( a > -\frac{1}{4} \) 3. \( b > -\frac{1}{4} \) 4. \( c > -\frac{1}{4} \) 5. \( \sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1} < 5 \) ### Step 1: Analyze the first equation From the first equation, we can express one variable in terms of the others. Let's express \( c \): \[ c = 1 - a - b \] ### Step 2: Substitute into the inequality Now, we substitute \( c \) into the inequality: \[ \sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4(1 - a - b) + 1} < 5 \] This simplifies to: \[ \sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4 - 4a - 4b + 1} < 5 \] \[ \sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{5 - 4a - 4b} < 5 \] ### Step 3: Define new variables Let \( x = \sqrt{4a + 1} \), \( y = \sqrt{4b + 1} \), and \( z = \sqrt{5 - 4a - 4b} \). From these definitions, we can express \( a \) and \( b \) in terms of \( x \) and \( y \): \[ a = \frac{x^2 - 1}{4}, \quad b = \frac{y^2 - 1}{4} \] Substituting these into the equation for \( c \): \[ c = 1 - \frac{x^2 - 1}{4} - \frac{y^2 - 1}{4} = 1 - \frac{x^2 + y^2 - 2}{4} = \frac{6 - x^2 - y^2}{4} \] ### Step 4: Substitute back into the inequality Now we need to ensure that: \[ x + y + \sqrt{5 - 4\left(\frac{x^2 - 1}{4} + \frac{y^2 - 1}{4}\right)} < 5 \] This simplifies to: \[ x + y + \sqrt{5 - (x^2 + y^2 - 2)} < 5 \] \[ x + y + \sqrt{7 - x^2 - y^2} < 5 \] ### Step 5: Solve the inequality To solve this inequality, we can analyze the possible values of \( x \) and \( y \) under the constraints \( a, b, c > -\frac{1}{4} \). 1. Since \( a > -\frac{1}{4} \), we have \( \sqrt{4a + 1} > 0 \) which implies \( x > 0 \). 2. Similarly, \( y > 0 \). 3. The maximum value of \( x + y \) occurs when \( z \) is minimized. ### Step 6: Find the bounds To find the bounds for \( x + y \), we can set \( z = 0 \): \[ x + y < 5 \] This gives us a region in the \( xy \)-plane where \( x + y < 5 \) and both \( x, y > 0 \). ### Conclusion The solution set consists of all \( (a, b, c) \) such that: - \( a + b + c = 1 \) - \( a, b, c > -\frac{1}{4} \) - \( \sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1} < 5 \) This defines a region in three-dimensional space, bounded by the conditions above. The exact values of \( a, b, c \) can be found by testing values within these constraints.