Problem 6 - Algebra for JEE ADVANCED

Question

Minimum value of \((b+c) / a+(c+a) / b+(a+b) / c\) (for real positive numbers \(a, b, c)\) is a. 1 b. 2 c. 4 d. 6

Step-by-Step Solution

Answer

The minimum value is 6.

Step 1: Undefine Variables

Let \( x = \frac{b+c}{a} \), \( y = \frac{c+a}{b} \), and \( z = \frac{a+b}{c} \). The original expression now becomes \( x + y + z \). Our task is to find the minimum value of this new expression under the condition that the variables are real and positive.

Step 2: Apply AM-GM Inequality

According to the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality), for real positive numbers \( x, y, z \), the inequality \( \frac{x + y + z}{3} \geq \sqrt[3]{xyz} \) holds. Multiplying both sides by 3, we have \( x + y + z \geq 3 \sqrt[3]{xyz} \).

Step 3: Calculate the Product of Variables

We know that \( xyz = \left(\frac{b+c}{a}\right)\left(\frac{c+a}{b}\right)\left(\frac{a+b}{c}\right)\). Simplifying the expression, we find \( xyz = \frac{(b+c)(c+a)(a+b)}{abc} \), which equals \( \frac{1}{abc}(abc + a^2b + a^2c + b^2a + b^2c + c^2a + c^2b) \). This product can be simplified further by expansion if needed.

Step 4: Substitution for Simplicity

For simplifying such expressions, symmetry plays a critical role. Here, let us check the scenario when \( a = b = c \). By substituting \( a = b = c \), the expression \( \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} \) becomes \( 2 + 2 + 2 = 6 \).

Step 5: Conclusion from Simplification and AM-GM Inequality

Since \( x + y + z \geq 6 \) when each of \( a, b, c \) are equal, and by the nature of the AM-GM inequality, the minimum value of the expression is guaranteed to be obtained when all terms are equal. This confirms that the minimum value of the expression \( x + y + z \) is indeed 6.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and mathematical operators. In our problem, we're dealing with expressions like \( \frac{b+c}{a} \), \( \frac{c+a}{b} \), and \( \frac{a+b}{c} \). Algebraic expressions allow us to represent complex relationships in a compact form. In general, the simplification process might include:

• Combining like terms.
• Factoring or expanding expressions.
• Performing operations following the order of operations (PEMDAS/BODMAS).

Understanding algebraic expressions is essential to manipulating them into forms that reveal important properties, such as symmetry or inequalities. This is crucial in solving problems that involve optimization, like finding minimum values.

Minimum Value Problems

Minimum value problems involve finding the smallest possible value of a given expression under certain conditions. In this exercise, we aimed to determine the minimum value of the expression \( \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} \) for positive real numbers \(a\), \(b\), and \(c\).

Key techniques for solving minimum value problems include:

• Applying inequalities like the AM-GM inequality to provide bounds on expressions.
• Using substitution to simplify expressions, making apparent underlying symmetry or other conditions that make the problem easier to solve.

By representing the expression in terms of new variables, simplifying using known inequalities, and testing specific cases such as setting \(a = b = c\), we derived that the minimum is 6. Such strategies help identify efficient paths to solutions.

Symmetric Functions

Symmetric functions have properties that remain consistent under any permutation of their variables. Recognizing this property is powerful for simplifying and solving algebraic problems. In our minimum value problem, the expression \( \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} \) is symmetric in \(a\), \(b\), and \(c\).

This symmetry allows us to:

• Reduce the complexity of the problem by assuming certain values, like \( a = b = c \), without loss of generality.
• Apply well-known inequalities effectively, knowing that any rearrangement of variables keeps the expression unchanged.

By leveraging symmetry, we can focus on simpler cases to understand or estimate the behavior of more complex expressions, making symmetry an invaluable property in mathematical problem-solving.