A squared cyclic ratio sum has a symmetric quadratic upper bound

Proved

WorkbookSource.plus_1420

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that: $\left(\frac{a}{a+b}\right)^2+ \left(\frac{b}{b+c}\right)^2+ \left(\frac{c}{c+a}\right)^2\leq \frac{3(a^2+b^2+c^2)}{4(ab+bc+ca)}$ Similar problems: https://artofproblemsolving.com/community/q1h1814384p12099563

Source: InternLM Lean-Workbook, record lean_workbook_plus_1420 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_1420 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (a + b)) ^ 2 + (b / (b + c)) ^ 2 + (c / (c + a)) ^ 2 ≤ (3 * (a ^ 2 + b ^ 2 + c ^ 2)) / (4 * (a * b + b * c + a * c))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_1420; Apache-2.0

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