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 (1+1/a)^a is greater than 2. Using Newtons formula (also called the Binomial Theorem)

(1+  1 a )a = 1+a·  1 a +other positiveterms > 2

for every integer a>2.

For a=1000000, the problem is solved.

(b): a bit more complex. I'll use the formula (1+1/a)^a < 3 , which derives from (1+1/a)^a → e . It is possible to demonstrate it without using infinitesimal calculus.

I will now use the last inequality to prove by induction that n! > (n/3)ⁿ for every integer n where n>1.

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

n! >

1 3n

nn

nn > 1/3(n+1)n, which is the lemma above  1 3n nn >  1 3n  1 3 (n+1)n n! >  1 3n nn >  1 3n  1 3 (n+1)n n! >  1 3n+1 (n+1)n n!(n+1) >  1 3n+1 (n+1)n(n+1) (n+1)! >  1 3n+1 (n+1)n+1

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