A sum of pairwise and shifted ratios is at least nine quarters

Proved

WorkbookSource.base_28648

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers . Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{a}{2a+b+c}+\frac{b}{a+2b+c}+\frac{c}{a+b+2c}\ge\frac{9}{4}.$

Source: InternLM Lean-Workbook, record lean_workbook_28648 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_28648 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a) + c / (a + b) + a / (2 * a + b + c) + b / (a + 2 * b + c) + c / (a + b + 2 * c)) ≥ 9 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_28648; Apache-2.0

View graph