A pair-product bound from a rational inequality

Proved

WorkbookSource.problem_57135

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

lean-workbooksource-checked

From the inequality $\frac{(a+b+c)^2}{a^2+b^2+c^2+3} \leq 1$ , deduce that $ab+bc+ca \leq \frac{3}{2}$ .

Source: InternLM Lean-Workbook, record lean_workbook_57135 (Apache-2.0). The complete source proposition and its explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_57135 (a b c : ℝ) : (a + b + c) ^ 2 / (a ^ 2 + b ^ 2 + c ^ 2 + 3) ≤ 1 → a * b + b * c + c * a ≤ 3 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_57135; Apache-2.0

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