A cyclic quadratic reciprocal upper bound at fixed sum three

Proved

WorkbookSource.base_8323

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

lean-workbooksource-checked

For $a, b, c>0, a+b+c=3$ prove that $\frac{a}{2a^2+b^2+c^2+1}+\frac{b}{2b^2+c^2+a^2+1}+\frac{c}{2c^2+a^2+b^2+1}\le\frac{3}{4+abc}$

Source: InternLM Lean-Workbook, record lean_workbook_8323 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_8323 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a / (2 * a ^ 2 + b ^ 2 + c ^ 2 + 1) + b / (2 * b ^ 2 + c ^ 2 + a ^ 2 + 1) + c / (2 * c ^ 2 + a ^ 2 + b ^ 2 + 1)) ≤ (3 / (4 + a * b * c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_8323; Apache-2.0

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