A product bound from three symmetric bounds

Proved

WorkbookSource.problem_21991

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

lean-workbooksource-checked

From the condition, we get: $a+b+c \geq 6$ , $ab+bc+ca \geq 12$ and $abc \geq 8$ . Therefore:

\prod_{cyc}(1+a^2) = 1+(a^2+b^2+c^2)+(a^2b^2+b^2c^2+c^2a^2)+(abc)^2 \geq 1+12+48+64=125

Source: InternLM Lean-Workbook, record lean_workbook_21991 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_21991  (a b c : ℝ) (h₀ : a + b + c ≥ 6) (h₁ : a * b + b * c + c * a ≥ 12) (h₂ : a * b * c ≥ 8) :
  (1 + a^2) * (1 + b^2) * (1 + c^2) ≥ 125  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21991; Apache-2.0

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