A shifted quartic ratio lower bound at fixed sum six

Proved

WorkbookSource.plus_3298

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

lean-workbooksource-checked

Let $a,b$ and $c$ be real positive numbers such that $a+b+c=6$ . Prove that ${{{\frac{a^4}{1+a+a^2+bc}}+\frac{b^4}{1+b+b^2+ca}}+\frac{c^4}{1+c+c^2+ab}}\ge\frac{48}{11}.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_3298 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_3298 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 6) : a^4 / (1 + a + a^2 + b * c) + b^4 / (1 + b + b^2 + c * a) + c^4 / (1 + c + c^2 + a * b) ≥ 48 / 11   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_3298; Apache-2.0

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