A fixed-sum quadratic reciprocal upper bound

Proved

WorkbookSource.base_5208

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

lean-workbooksource-checked

Prove that: $\dfrac{a}{b^2+c^2+9}+\dfrac{b}{c^2+a^2+9}+\dfrac{c}{a^2+b^2+9} \le \dfrac{1}{11}(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c})$ Given: $a,b,c>0 , a+b+c=3.$

Source: InternLM Lean-Workbook, record lean_workbook_5208 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5208 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a / (b ^ 2 + c ^ 2 + 9) + b / (c ^ 2 + a ^ 2 + 9) + c / (a ^ 2 + b ^ 2 + 9)) ≤ (1 / 11) * (1 / a + 1 / b + 1 / c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5208; Apache-2.0

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