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Home » C. For exactly one $x$, the statement holds - Crosslake

C. For exactly one $x$, the statement holds - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Hence, the number of such functions is $ \boxed{\infty} $, assuming no further restrictions.

📌 Question: Let $ x, y, z $ be positive real numbers such that $ x + y + z = 3 $. Find the minimum value of

📌 \frac{x^2}{y + z} + \frac{y^2}{z + x} + \frac{z^2}{x + y}.

📌 Solution: We are to minimize

📌 under the constraint $ x + y + z = 3 $.

📌 Note that $ y + z = 3 - x $, $ z + x = 3 - y $, $ x + y = 3 - z $. So the expression becomes:

📌 \sum \frac{x^2}{3 - x}

📌 We apply the **Cauchy-Schwarz Inequality** in the following form:

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The next 3 years: $ (a+3d) + (a+4d) + (a+5d) = 3a + 12d = 210 $
We now solve the system:
From (2): $ a = 70 - 4d $
Substitute into (1): $ 2(70 - 4d) + 3d = 60 $ → $ 140 - 8d + 3d = 60 $...
Then $ a = 70 - 4(16) = 70 - 64 = 6 $
So annual retreat sequence:
Year 1: 6
Year 2: 22
Year 3: 38
Year 4: 54

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