lean_workbook_plus_11150

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $a;b;c>0$ such that $ab+bc+ca=3$ \nProve: $\sqrt[3]{\dfrac{3a}{b(b+2c)}}+\sqrt[3]{\dfrac{3b}{c(c+2a)}}+\sqrt[3]{\dfrac{3c}{a(a+2b)}} \geq 3$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_11150 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a * b + b * c + c * a = 3) : (3 * a / (b * (b + 2 * c)))^(1 / 3) + (3 * b / (c * (c + 2 * a)))^(1 / 3) + (3 * c / (a * (a + 2 * b)))^(1 / 3) ≥ 3   :=  by sorry

The untyped fractional exponents are natural-number quotients, so the proper fractional exponents become zero and the displayed root expressions become1. The accepted target therefore proves a different, much weaker statement.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Confirmed directly from the Lean text: the numeric-literal exponent carries no type ascription, so it elaborates at type ℕ where division truncates (^(1 / 3) → ^0).

Proposed corrected statement

Use the actual roots and fractional powers in the following source statement: nonnegative real roots for nonnegative radicands, the unique signed real cube root for a possibly negative radicand, and an explicit real type on each other fractional exponent. Preserve the displayed hypotheses, parentheses and inequality direction. Proposed corrected problem: Let $a;b;c>0$ such that $ab+bc+ca=3$ \nProve: $\sqrt[3]{\dfrac{3a}{b(b+2c)}}+\sqrt[3]{\dfrac{3b}{c(c+2a)}}+\sqrt[3]{\dfrac{3c}{a(a+2b)}} \geq 3$

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.