A cyclic sum of squared pairwise ratios is at least three quarters

Proved

WorkbookSource.base_3268

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

lean-workbooksource-checked

Let $a, b, c$ be positive reals. Prove that $\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$

Source: InternLM Lean-Workbook, record lean_workbook_3268 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_3268 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a / (a + b)) ^ 2 + (b / (b + c)) ^ 2 + (c / (c + a)) ^ 2 ≥ 3 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_3268; Apache-2.0

View graph