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Home » The Best Valentines Presents for Teachers Every Student Deserves – You’ll Keep Them Guilty! - Blask

The Best Valentines Presents for Teachers Every Student Deserves – You’ll Keep Them Guilty! - Blask

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 given that \( x > y > z \).

📌 We are to minimize the expression

📌 S = \frac{x + y}{x - y} + \frac{y + z}{y - z} + \frac{z + x}{z - x}

📌 under the condition \( x > y > z > 0 \). Note that all denominators are positive since \( x > y > z \), so each fraction is positive and unbounded as any variable approaches another.

📌 We analyze the structure of each term. Consider the identity:

📌 \frac{x + y}{x - y} = \frac{(x - y) + 2y}{x - y} = 1 + \frac{2y}{x - y}

📌 \frac{y + z}{y - z} = 1 + \frac{2z}{y - z}, \quad \frac{z + x}{z - x} = \frac{z + x}{-(x - z)} = -\frac{z + x}{x - z} = -\left( \frac{(x - z) + 2z}{x - z} \right) = -1 - \frac{2z}{x - z}

📌 Adding the three:

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