Which is larger:

(a) 1.000001 ^1000000 or 2?
(b) 100 ³⁰⁰ or 300! (i.e.factorial 300)

Congratulations to Federico Poloni, age 17, Casirate d'Adda (Italy) for the following solution.

(a): It's possible to prove that every number in the form is greater than 2. Using Newton's formula (also called the Binomial Theorem)

for every integer a>2.

For a=1000000, the problem is solved.

(b): a bit more complex. I'll use the formula , which derives from . It is possible to demonstrate it without using infinitesimal calculus.

I will now use the last inequality to prove by induction that for every integer n where n>1.

The formula is obviously correct for n=2. Now,
Hypothesis:

The demonstration by induction is complete. In particular, for n=300, the formula solves the given problem.