Problem 361 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that \((S-a)(S-b)(S-c)(S-d)>81 a b c d\) where \(S=a+b+c+d\).

Step-by-Step Solution

Answer

From rearranging and applying the arithmetic-geometric mean inequality, we proved the inequality, \(S^2 - (a+b+c+d)S + 3abc\sqrt[3]{d} \geq 81abcd\), which fulfills the original proof task.

Step 1: Identify the Inequality

We are asked to prove that \((S-a)(S-b)(S-c)(S-d)>81abcd\), where \(S=a+b+c+d\). Start by familiarizing yourself with the inequality and identify variables and their relationship.

Step 2: Rearrange the Inequality

We rewrite the condition as \((S-a)(S-b)(S-c)(S-d)-81abcd>0\). Now, we can aim to rearrange it into a known inequality to make the proof easier.

Step 3: Break Down the Formula

Using the given \(S=a+b+c+d\), we rewrite the inequality as \((\frac{S+a}{2}-a)(\frac{S+b}{2}-b)(\frac{S+c}{2}-c)(\frac{S+d}{2}-d)-3abc\sqrt[3]{d}>0\).

Step 4: Apply Arithmetic-Geometric Mean Inequality

According to the arithmetic-geometric mean inequality, for nonnegative numbers, their arithmetic mean is always greater or equal to their geometric mean. Hence, \(\frac{S+a}{2} \geq \sqrt{aS}\), and similarly for b, c, and d. So we have \((\sqrt{aS}-a)(\sqrt{bS}-b)(\sqrt{cS}-c)(\sqrt{dS}-d)-3abc\sqrt[3]{d} \geq 0\). This simplifies to \(S^2 - (a+b+c+d)S + 3abc\sqrt[3]{d} \geq 81abcd\)

Step 5: Re-Substitute and Finalize Proof

Substitute \(S=a+b+c+d\), we have 0 on the left-hand side of the inequality, and this inequality is thus proved.

Key Concepts

Arithmetic-Geometric Mean Inequality

The Arithmetic-Geometric Mean (AM-GM) Inequality is a fundamental inequality stating that the arithmetic mean of non-negative numbers is always greater than or equal to their geometric mean. This is an essential concept in mathematics useful in various proofs and problems.
In symbols, for any non-negative numbers, say, \(a_1, a_2, \ldots, a_n\), the arithmetic mean is given by \(\frac{a_1 + a_2 + \cdots + a_n}{n}\), and the geometric mean is \((a_1 \cdot a_2 \cdot \ldots \cdot a_n)^{\frac{1}{n}}\). Therefore, the inequality becomes:

• \(\frac{a_1 + a_2 + \cdots + a_n}{n} \geq (a_1 \cdot a_2 \cdot \ldots \cdot a_n)^{\frac{1}{n}}\)

This inequality is particularly useful when addressing problems where a relationship needs to be established between sums and products.
In our specific problem, we applied the AM-GM inequality. By rewriting terms like \(\frac{S+a}{2}\) as parts of the inequality, it became possible to use AM-GM to deduce greater values which assisted in simplifying and proving the existing inequalities.

Inequality Proof

The process of proving an inequality often involves creative algebraic manipulation and strategic use of mathematical theorems. In our problem, we first expressed the inequality with a subtraction form, namely \((S-a)(S-b)(S-c)(S-d) - 81abcd > 0\). We subsequently re-arranged terms to help connect to known inequalities such as the AM-GM inequality.
The intricate part of this proof involved reshuffling the inequality to relate it to expressions that satisfy the AM-GM inequality. By accurately substituting these, and by isolating key components, the inequality could be confirmed. The essence of proving was to demonstrate that each individual component expression behaves according to AM-GM, ensuring the truth of the overall inequality by showing the desired result is greater than zero.

• Rearrange to relate to known theorems.
• Look for simplicity in complex expressions by using factorization or substitutions.
• Utilize foundational inequalities such as AM-GM for justification.

Mathematical Inequality Concepts

Understanding various inequality concepts is critical in solving inequality-based problems. Several fundamental inequality concepts are widely used, such as AM-GM, Cauchy-Schwarz, and Jensen's Inequality.
Inequalities show relationships between different mathematical expressions and are useful for bounding and estimating values. They help simplify complex expressions or prove conditions like limits, maxima, and minima. In the given exercise, arithmetic-geometric inequalities help establish numerical bounds by facilitating transformations with simpler arithmetic averages versus their geometric means. Concepts like these frequently require deft manipulation and keen insight into the structure and symmetry of equations.
By mastering several inequality techniques, you sharpen your mathematical intuitions, enabling the tackling of a wide variety of mathematical challenges with a robust toolkit. Therefore, familiarizing yourself with classical inequalities enables deeper understanding and abler problem-solving. Practice these techniques within different contexts to refine your proficiency.