A shifted cyclic quadratic ratio upper bound at fixed sum three

Proved

WorkbookSource.base_21549

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

lean-workbooksource-checked

If $a,b,c$ are positive numbers such that $a+b+c=3$ , then $\frac{a}{3a+b^{2}}+\frac{b}{3b+c^{2}}+\frac{c}{3c+a^{2}}\le \frac 3{4}$ .

Source: InternLM Lean-Workbook, record lean_workbook_21549 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_21549 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a / (3 * a + b ^ 2) + b / (3 * b + c ^ 2) + c / (3 * c + a ^ 2) ≤ 3 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21549; Apache-2.0

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