This is the second part of Cauchy – Schwarz Inequality By Engr. Roy Roque Rivera Jr. The primary application of this topic is proving hard inequality problems. This post is also recommended for PMO and IMO aspirants and other Mathematical Olympiads. Please also check the first part of this topic.

**2.2 Proving Inequalities**

**Problem 1: **Let . Prove that

**Solution: **By using the Cauchy- Schwarz Inequality in Engel Form, we have

Hence,

**Problem 2 (Nesbitt’s Inequality – England, 1903): **Let . Prove that

**Solution: **The left-hand side of the inequality can be written as

Since we have

Or

Hence,

Thus,

**Problem 3 (Belarus IMO Team Selection Test, 1999):** Let such . Prove that

**Solution:** By using the Cauchy- Schwarz Inequality in Engel Form, we have

Or simply

But

Because

Which can be written as

Or obviously

Now,

Hence,

But we know that ; thus,

Or simply

**Problem 4 (Ireland, 1999): **Let such that . Prove that

**Solution: **By using the Cauchy- Schwarz Inequality in Engel Form, we have

But ; thus,

**Problem 5 (Iran, 1999):** Let . Prove that

**Solution: **By using the Cauchy – Schwarz Inequality in Engel Form, we have

Again by Cauchy-Schwarz Inequality in Engel Form

Or

It follows that

Or

Now, since

Hence,

Or simply

**Problem 6: **Let . Prove that

**Solution: **The left-hand side of the inequality can be written as

By using the Cauchy-Schwarz Inequality, we have

Or simply

Or

Hence,

**Problem 7 (International Mathematical Olympiad, 1995): **Let such that . Prove that

**Solution: **By Cauchy- Schwarz Inequality, we have

Or

Thus, it suffices to show that

Since

From AM-GM Inequality, we have

But since ; hence,

Which implies that

It follows that

And we are done.

**Problem 8 (GDR, 1967): **Prove that if and are positive real numbers whose sum is , then

**Solution: **The left-hand side of the inequality can be written as

Thus, by Cauchy-Schwarz Inequality in Engel Form, we have

or

Recall the Quadratic Mean- Arithmetic Mean (QM-AM) Inequality

Hence,

Or

It follows that

Hence,

**Problem 9 (Asia Pacific Mathematics Olympiad, 1990):** Let and be positive real numbers such that

Prove that

**Solution: **From Cauchy – Schwarz Inequality in Engel Form, we have

But

Therefore,

Or simply

**Problem 10 (Crux Mathematicorum): **Let be positive real numbers. Prove that

**Solution: **From Cauchy- Schwarz Inequality in Engel Form, we have

It suffices to solve that

Now,

But, from AM-GM Inequality, we know that

Or simply

Also,

If follows that

Hence,

Which implies that

**Problem 11 (Phan Kim Hung): **Let be positive real numbers. Prove that

**Solution:** Note that

It suffices to show that

But from Cauchy – Schwarz Inequality in Engel Form observe that

And

Thus, by adding we obtain

Or in shorthand notation we have

**Problem 12 (Iran, 1998): **Let be positive real number such that Prove that

**Solution:** Observe

Or equivalently

By Cauchy – Schwarz Inequality, we have

Or simply

Recalling that , we obtain

It follows that