A comparison of pairwise and mixed quadratic reciprocal sums

Proved

WorkbookSource.base_14806

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

lean-workbooksource-checked

Let $a,$ $b,$ $c$ be positive real numbers. Prove that $\frac {1}{b + c} + \frac {1}{c + a} + \frac {1}{a + b} \ge \frac {2a}{3a^2 + bc} + \frac {2b}{3b^2 + ca} + \frac {2c}{3c^2 + ab}.$

Source: InternLM Lean-Workbook, record lean_workbook_14806 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_14806 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (b + c) + 1 / (c + a) + 1 / (a + b)) ≥ (2 * a / (3 * a ^ 2 + b * c) + 2 * b / (3 * b ^ 2 + c * a) + 2 * c / (3 * c ^ 2 + a * b))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_14806; Apache-2.0

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