Question

If \(x\) and \(y\) are positive real numbers satisfying \(x+y=102\), then the minimum possible value of \(2601\bigg(1+\frac{1}{x}\bigg)\bigg(1+\frac{1}{y}\bigg) \)is

Answer 1

Step 1: Recall the Inequalities 

\[ \text{AM} \ge \text{GM} \ge \text{HM} \] \[ \frac{x + y}{2} \ge \sqrt{xy} \ge \frac{2}{\frac{1}{x} + \frac{1}{y}} \]

Step 2: Given

Let \( x + y = 102 \)

To maximize \( xy \), we use the identity:

\[ xy \le \left(\frac{x + y}{2}\right)^2 = \left(\frac{102}{2}\right)^2 = 51^2 = 2601 \] \[ \Rightarrow \frac{1}{xy} \ge \frac{1}{2601} \] \[ \Rightarrow \frac{1}{x} + \frac{1}{y} \ge \frac{2}{51} \]

Step 3: Expression to Minimize

We are asked to find the minimum value of:

\[ 2601 \left(1 + \frac{1}{x}\right)\left(1 + \frac{1}{y}\right) \] Expand it: \[ = 2601 \left(1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{xy}\right) \] Using the inequalities above: \[ \frac{1}{x} + \frac{1}{y} \ge \frac{2}{51}, \quad \frac{1}{xy} \ge \frac{1}{2601} \] Substituting the minimum values: \[ = 2601 \left(1 + \frac{2}{51} + \frac{1}{2601}\right) \]

Step 4: Simplify the Expression

\[ = 2601 \left(1 + \frac{2}{51} + \frac{1}{2601} \right) \] \[ = 2601 \left( \frac{2601 + 102 + 1}{2601} \right) = 2601 \left(\frac{2704}{2601} \right) = 2704 \]

✅ Final Answer:

\[ \boxed{2704} \]

Answer 2

Step 1: Given 

\[ x + y = 102 \]

Step 2: Using AM ≥ GM

By the Arithmetic Mean - Geometric Mean inequality: \[ \frac{x + y}{2} \ge \sqrt{xy} \] Substituting the given value: \[ \frac{102}{2} \ge \sqrt{xy} \Rightarrow 51 \ge \sqrt{xy} \Rightarrow xy \le 2601 \]

Step 3: Expression to Maximize

\[ 2601\left(1 + \frac{1}{x}\right)\left(1 + \frac{1}{y}\right) \] Expand this expression: \[ = 2601\left(1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{xy}\right) \] Use: \[ x + y = 102, \quad xy \le 2601 \] So: \[ \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy} \le \frac{102}{2601}, \quad \frac{1}{xy} \le \frac{1}{2601} \] Now plug values into the expression: \[ 2601\left(1 + \frac{102}{2601} + \frac{1}{2601}\right) = 2601\left(\frac{2601 + 102 + 1}{2601}\right) = 2601 \cdot \frac{2704}{2601} = 2704 \]

✅ Final Answer:

\[ \boxed{2704} \]