A shifted cyclic quadratic ratio sum is at least three halves

Proved

WorkbookSource.plus_60598

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

lean-workbooksource-checked

Let a, b, c be positive numbers such that a + b + c = 3. Prove that: $\frac{a^2}{1+b^2} + \frac{b^2}{1+c^2} + \frac{c^2}{1+a^2} \geq \frac{3}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_60598 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_60598 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a^2 / (1 + b^2) + b^2 / (1 + c^2) + c^2 / (1 + a^2)) ≥ 3 / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_60598; Apache-2.0

View graph