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)

$(1 + \frac{1}{a})^{a} = 1 + a \cdot \frac{1}{a} + other positive
terms > 2$

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 $(1 + 1 / a)^{a} \to e$ . It is possible to demonstrate it without using infinitesimal calculus.

\begin{matrix} (1 + 1 / a)^{a} < 3 \\ (\frac{a + 1}{a})^{a} < 3 \\ (a + 1)^{a} < 3 a^{a} \end{matrix}

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: $n! > \frac{1}{3^{n}} n^{n}$

$\begin{matrix} n^{n} & > & 1 / 3 (n + 1)^{n}, & which is the lemma above \\ \frac{1}{3^{n}} n^{n} & > & \frac{1}{3^{n}} \frac{1}{3} (n + 1)^{n} \\ n! & > & \frac{1}{3^{n}} n^{n} & > \frac{1}{3^{n}} \frac{1}{3} (n + 1)^{n} \\ n! & > & \frac{1}{3^{n + 1}} (n + 1)^{n} \\ n! (n + 1) & > & \frac{1}{3^{n + 1}} (n + 1)^{n} (n + 1) \\ (n + 1)! & > & \frac{1}{3^{n + 1}} (n + 1)^{n + 1} \end{matrix}$

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