A weighted cyclic quadratic ratio sum is at most three fifths

Proved

WorkbookSource.base_24438

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c >0$ . Prove that : (Pham Kim Hung)

$\frac{ab}{a^2+b^2+3c^2}+\frac{bc}{b^2+c^2+3a^2}+\frac{ca}{c^2+a^2+3b^2} \le \frac{3}{5}$

Source: InternLM Lean-Workbook, record lean_workbook_24438 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_24438 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b / (a ^ 2 + b ^ 2 + 3 * c ^ 2) + b * c / (b ^ 2 + c ^ 2 + 3 * a ^ 2) + c * a / (c ^ 2 + a ^ 2 + 3 * b ^ 2)) ≤ 3 / 5  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_24438; Apache-2.0

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