A shifted cyclic quadratic ratio sum bounds a normalized total

Proved

WorkbookSource.base_39922

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

lean-workbooksource-checked

Show that a/(1+b^2)+b/(1+c^2)+c/(1+a^2) \geq 3(a+b+c)/(3+a^2+b^2+c^2) for all positive a, b, c.

(AOPS is weird and doesn't let me use latex)

Let $a, b, c$ be positive reals. Prove that

\\frac{a}{1+b^2}+\\frac{b}{1+c^2}+\\frac{c}{1+a^2}\\geq \\frac{3(a+b+c)}{3+a^2+b^2+c^2}

Source: InternLM Lean-Workbook, record lean_workbook_39922 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_39922 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (1 + b ^ 2) + b / (1 + c ^ 2) + c / (1 + a ^ 2)) ≥ 3 * (a + b + c) / (3 + a ^ 2 + b ^ 2 + c ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_39922; Apache-2.0

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