The base camp is located 5,300 meters above sea level. The hiker observes that the water begins to boil at 82⁰C, much lower than the 100⁰C needed to cook the raw egg. He hopes that just leaving the egg in the boiling water longer will let the egg cook. Will he have hard-boiled eggs for breakfast?

No he will not have hard boiled eggs for breakfast

Explanation:

The temperature required to boil the eggs is about 100⁰C. Unfortunately, at the height of the base camp (5,300 meters), water boils at 82⁰C.

This implies that the egg will not become cooked as at the time when the water begins to boil.

Since the eggs can only become hard boiled at 100⁰C and the temperature at which water boils at 5300 meters is less than that(82⁰C), the eggs can not become hard boiled.