Olympiad problems

2005 China Team Selection Test, 3

$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.

2005 China Team Selection Test, 3

Tags: number theory , modular arithmetic , binomial expansion , fermat number , power of 2

$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.

1993 Abels Math Contest (Norwegian MO), 3

Tags: number theory , recurrence relation , fermat number , coprime

The Fermat-numbers are defined by $F_n = 2^{2^n}+1$ for $n\in N$. (a) Prove that $F_n = F_{n-1}F_{n-2}....F_1F_0 +2$ for $n > 0$. (b) Prove that any two different Fermat numbers are coprime

2011 Saudi Arabia Pre-TST, 1.4

Tags: number theory , algebra , inequalities , fermat number

Let $f_n = 2^{2^n}+ 1$, $n = 1,2,3,...$, be the Fermat’s numbers. Find the least real number $C$ such that $$\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C$$ for all positive integers $n$