**Problem 1.**

If a, b, c are non-negative real numbers such that , then prove that the product abc cannot exceed 1.

**Solution I:**

Given that , , , so certainly , , , and .

Now, and hence, , hence we get:

.ย *Clearly, the presence ofย * and reminds us of the AM-GM inequality.

Here it is .

So, .

Also, we can say: . Now, let .

So,

that is, , or , that is, .ย *So, this is a beautiful application of arithmetic mean-geometric mean inequality twice. ๐ ๐*

**Problem 2:**

If a, b, c are three rational numbers, then prove that : is always the square of a rational number.

**Solution 2:**

Let , , . It can be very easily shown that , or . So, the given expression is a perfect square !!!ย *BINGO! ๐ ๐ ๐*

Nalin Pithwa.

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