A shifted pairwise quadratic reciprocal sum is at most one ninth

Proved

WorkbookSource.plus_29830

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

lean-workbooksource-checked

If $a, b, c>0, a+b+c=3$ prove that $\frac{1}{a^2+b^2+25}+\frac{1}{b^2+c^2+25}+\frac{1}{c^2+a^2+25}\le\frac{1}{9}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_29830 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_29830 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : 1 / (a ^ 2 + b ^ 2 + 25) + 1 / (b ^ 2 + c ^ 2 + 25) + 1 / (c ^ 2 + a ^ 2 + 25) ≤ 1 / 9   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_29830; Apache-2.0

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