A shifted cyclic squared-denominator ratio inequality at fixed total four

Proved

WorkbookSource.base_9113

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

lean-workbooksource-checked

Let positive real a,b,c,d which satisfy: $a+b+c+d=4$ Prove that: $\frac{a}{a+3b^2}+\frac{b}{b+3c^2}+\frac{c}{c+3d^2}+\frac{d}{d+3a^2} \ge 1$ .

Source: InternLM Lean-Workbook, record lean_workbook_9113 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_9113 (a b c d : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (habc : a + b + c + d = 4) : a / (a + 3 * b ^ 2) + b / (b + 3 * c ^ 2) + c / (c + 3 * d ^ 2) + d / (d + 3 * a ^ 2) ≥ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_9113; Apache-2.0

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