5Question: Let \( a, b, c \) be positive real numbers such that \( a + b + c = 1 \). Find the minimum value of \( \frac{a}{1 - a} + \frac{b}{1 - b} + \frac{c}{1 - c} \).

["Title: Find the Minimum Value of ( \frac{a}{1 - a} + \frac{b}{1 - b} + \frac{c}{1 - c} ) Given ( a, b, c > 0 ) and ( a + b + c = 1 )", "Meta Description:\nExplore the minimum value of the expression ( \frac{a}{1 - a} + \frac{b}{1 - b} + \frac{c}{1 - c} ) under the constraint ( a + b + c = 1 ), where ( a, b, c ) are positive real numbers. Learn optimization techniques using calculus and inequalities.", "---", "Introduction\nWhen faced with an optimization problem involving symmetric variables constrained by a linear sum, elegant solutions often emerge through inequality analysis or calculus. Here, we aim to find the minimum value of\n[\nS = \frac{a}{1 - a} + \frac{b}{1 - b} + \frac{c}{1 - c}\n]\ngiven that ( a, b, c > 0 ) and ( a + b + c = 1 ).", "Given the symmetry of the expression and constraint, we suspect the minimum occurs when ( a = b = c ). We now justify this intuition mathematically.", "---", "Step 1: Use the Constraint to Simplify the Expression\nSince ( a + b + c = 1 ), we observe that ( 1 - a = b + c ), ( 1 - b = a + c ), and ( 1 - c = a + b ). Thus,\n[\nS = \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}\n]\nNow the problem becomes finding the minimum of\n[\nS = \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}\n]\nunder ( a + b + c = 1 ), ( a, b, c > 0 ).", "---", "Step 2: Apply the Nesbitt’s Inequality\nThe expression ( \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} ) is well-known in inequalities and is addressed by Nesbitt’s Inequality, which states that for positive real numbers ( a, b, c ):\n[\n\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} \geq \frac{3}{2}\n]\nwith equality if and only if ( a = b = c ).", "Applying Nesbitt’s Inequality directly to our transformed expression, we conclude:\n[\nS \geq \frac{3}{2}\n]\nwith equality when ( a = b = c ).", "---", "Step 3: Verify Equality Case Under Given Constraint\nIf ( a = b = c ), then from ( a + b + c = 1 ), we get ( 3a = 1 \Rightarrow a = b = c = \frac{1}{3} ). Substituting into ( S ):\n[\nS = 3 \cdot \frac{\frac{1}{3}}{1 - \frac{1}{3}} = 3 \cdot \frac{\frac{1}{3}}{\frac{2}{3}} = 3 \cdot \frac{1}{2} = \frac{3}{2}\n]\nThis confirms that the minimum value ( \frac{3}{2} ) is indeed achievable.", "---", "Why This Is Best: Mathematical Intuition and Rigor Combined\nWhile calculus-based optimization (e.g., Lagrange multipliers) could be used, the symmetry and structure of the problem make inequality methods both simpler and more illuminating. By applying Nesbitt’s Inequality — a powerful tool developed specifically for symmetric rational sums — we bypass lengthy derivative calculations and arrive at a definitive, elegant solution.", "Moreover, this problem demonstrates how constraints can reshape expressions: rewriting ( \frac{a}{1 - a} ) using ( b + c ) reveals deeper algebraic symmetry, guiding our analytical approach.", "---", "Conclusion\nGiven ( a + b + c = 1 ) and ( a, b, c > 0 ), the expression\n[\n\frac{a}{1 - a} + \frac{b}{1 - b} + \frac{c}{1 - c} = \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}\n]\nattains its minimum value of ( \frac{3}{2} ) when ( a = b = c = \frac{1}{3} ). This minimum is guaranteed by Nesbitt’s Inequality and verified by direct substitution.", "Key Takeaways:\n- Symmetric optimization problems often benefit from elegant inequality tools.\n- Reducing denominators using constraints simplifies expressions.\n- Equality conditions in inequalities like Nesbitt’s provide clear candidate solutions.", "---", "Keywords: minimum of ( \frac{a}{1 - a} + \frac{b}{1 - b} + \frac{c}{1 - c} ), ( a + b + c = 1 ), inequality optimization, Nesbitt’s Inequality, symmetric variables, calculus-free solution, minimum value substitution, positive reals.", "---", "References for Further Reading:\n- Nesbitt’s Inequality: Classic inequality in olympiad and analysis literature\n- Inequalities by HitOMat and Graph Theory (for advanced learners)\n- Optimization using Lagrange multipliers – as an alternative method", "---", "Share this insightful journey through symmetric optimization — where simplicity and rigor combine to reveal minimal values effortlessly."]