A cubic consequence of a rational constraint

Proved

WorkbookSource.problem_32022

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that $\frac{a}{b+c}+\frac{b}{c+a}=\frac{7}{3}.$ Prove that

a^3+b^3+c^3\geq 5abc

Source: InternLM Lean-Workbook, record lean_workbook_32022 (Apache-2.0). The complete source proposition and its explicit variable declarations are preserved.

Preamble

import Mathlib
open Real

Formal statement

theorem WorkbookSource.problem_32022 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a) = 7 / 3 → a ^ 3 + b ^ 3 + c ^ 3 >= 5 * a * b * c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32022; Apache-2.0

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