A cyclic pairwise ratio bound with a quadratic correction

Proved

WorkbookSource.base_2062

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

lean-workbooksource-checked

Let $a$ , $b$ and $c$ positive real numbers such that $a+b+c=3$ . Prove that: $\frac{a+b}{b+c}+\frac{b+c}{c+a}+\frac{c+a}{a+b} \geq \frac{a^2+b^2+c^2+9}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_2062 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2062 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hab : a + b + c = 3) : (a + b) / (b + c) + (b + c) / (c + a) + (c + a) / (a + b) ≥ (a ^ 2 + b ^ 2 + c ^ 2 + 9) / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2062; Apache-2.0

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