A shifted ratio sum with a reciprocal pair-product correction

Proved

WorkbookSource.plus_23399

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

lean-workbooksource-checked

Let $a,b,c$ be positive numbers satisfying $a+b+c=3$ . Prove that: $\frac{a}{a+3}+\frac{b}{b+3}+\frac{c}{c+3}+\frac{1}{ab+bc+ac}\geq \frac{13}{12}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_23399 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_23399 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a / (a + 3) + b / (b + 3) + c / (c + 3) + 1 / (a * b + b * c + a * c) ≥ 13 / 12   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_23399; Apache-2.0

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