A shifted cyclic quadratic ratio bounds the quadratic sum

Proved

WorkbookSource.base_53094

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 $\frac{a(a+2)}{a(b+3)+2}+\frac{b(b+2)}{b(c+3)+2}+\frac{c(c+2)}{c(a+3)+2}\le\frac{a^2+b^2+c^2}{2}$ Happy New Year to all Mathlinkers ! With the greatest respect, oldbeginner

Source: InternLM Lean-Workbook, record lean_workbook_53094 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_53094 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a * (a + 2) / (a * (b + 3) + 2) + b * (b + 2) / (b * (c + 3) + 2) + c * (c + 2) / (c * (a + 3) + 2) ≤ (a ^ 2 + b ^ 2 + c ^ 2) / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_53094; Apache-2.0

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