Let $F_1,F_2,F_3,...$ be the [i]Fibonacci sequence[/i], the sequence of positive integers with $F_1 =F_2 =1$ and $F_{n+2}=F_{n+1}+F_n$ for all $n \ge 1$. A [i]Fibonacci number[/i] is by definition a number appearing in this sequence. Let $P_1,P_2,P_3,...$ be the sequence consisting of all the integers that are products of two Fibonacci numbers (not necessarily distinct) in increasing order. The first few terms are $1,2,3,4,5,6,8,9,10,13,...$ since, for example $3 = 1 \cdot 3, 4 = 2 \cdot 2$, and $10 = 2 \cdot 5$. Consider the sequence $D_n$ of [i]successive [/i] differences of the $P_n$ sequence, where $D_n = P_{n+1}-P_n$ for $n \ge 1$. The first few terms of D_n are $1,1,1,1,1,2,1,1,3, ...$ . Prove that every number in $D_n$ is a [i]Fibonacci number[/i].

The writer Arthur has $n \ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having written with Arthur. Inspecting the numbers on the notes, they discover that the numbers written down are the first $n$ Fibonacci numbers (defined by $F_1 = F_2 = 1$ and $F_{k+2}= F_{k+1} + F_k$). For which $n$ is it possible that none of the co-authors had a lapse of memory?

Let $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$, for all positive integer $n$, be the Fibonacci sequence. Prove that for any positive integer $m$ there exist infinitely many positive integers $n$ such that $F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2}$ mod $m$ .

The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$. (a) Prove that there are Fibonacci numbes that end in a $9$ in the decimal system. (b) Determine for which $n$ can a Fibonacci number end in $n$ $9$-s in the decimal system.

Let $F_k$ denote the series of Fibonacci numbers shifted back by one index, so that $F_0 = 1$, $F_1 = 1,$ and $F_{k+1} = F_k +F_{k-1}$. It is known that for any fixed $n \ge 1$ there exist real constants $b_n$, $c_n$ such that the following recurrence holds for all $k \ge 1$: $$F_{n\cdot (k+1)} = b_n \cdot F_{n \cdot k} + c_n \cdot F_{n\cdot (k-1)}.$$ Prove that $|c_n| = 1$ for all $n \ge 1$.

The Fibonacci numbers $f_n$ are defined for each natural number $n$ as follows: $f_0=f_1=1$ and for $n$ greater than or equal to $2$, by recurrence: $f_n=f_{n-1}+f_{n-2}$ Let $S=f_1+f_2+...+f_{2004}+f_{2005}$. Calculate the largest value of $N$, such that the Fibonacci number $f_N$ satisfies $f_N<S$

The Fibonacci numbers $(F_n)_{n=1}^{\infty}$ are defined as follows: $$F_1 = F_2 = 1, F_n = F_{n-2} + F_{n-1}, n = 3, 4, ...$$ Assume $p$ is a prime greater than $3$. With $m$ being a natural number greater than $3$, find all $n$ numbers such that $F_n$ is divisible by $p^m$.

The Fibonacci sequence is defined by $F_0=F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge0$. Prove that the sum of $2000$ consecutive terms of the Fibonacci sequence is never a term of the sequence.

Prove that the equation $$1320x^3=(y_1+y_2+y_3+y_4)(z_1+z_2+z_3+z_4)(t_1+t_2+t_3+t_4+t_5)$$ has infinitely many solutions in the set of Fibonacci numbers. [i]Proposed by Mihály Bencze[/i]

Fibonacci sequences is defined as $f_1=1$,$f_2=2$, $f_{n+1}=f_{n}+f_{n-1}$ for $n \ge 2$. a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number. b) A positive integer is called [i]Fib-unique[/i] if the way to represent it as sum of several distinct Fibonacci number is unique. Example: $13$ is not Fib-unique because $13 = 13 = 8 + 5 = 8 + 3 + 2$. Find all Fib-unique.

Define Fibonacci sequence $\{F\}_{n=0}^{\infty}$ as $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n +F_{n-1}$ for every integer $n > 1$. Determine all quadruples $(a, b, c,n)$ of positive integers with a $< b < c$ such that each of $a, b,c,a + n, b + n,c + 2n$ is a term of the Fibonacci sequence.

Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$. Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .

Consider the function $f (x) = (x - F_1)(x - F_2) ...(x -F_{3030})$ with $(F_n)$ is the Fibonacci sequence, which defined as $F_1 = 1, F_2 = 2$, $F_{n+2 }=F_{n+1} + F_n$, $n \ge 1$. Suppose that on the range $(F_1, F_{3030})$, the function $|f (x)|$ takes on the maximum value at $x = x_0$. Prove that $x_0 > 2^{2018}$.

The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$. Find all $n$ such that $f_n = n^2$.

A sequence of real numbers is called a [i]Fibonacci [/i] sequence if $$t_{n+2} = t_{n+1} + t_n$$ for $n= 1,2,3,. .$ . Two Fibonacci sequences are said to be [i]essentially different[/i] if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences $1,2,3,5,8,...$ and $1,3,4,7,11,...$ are essentially different, but the sequences $1,2,3,5,8,...$ and $2,4,6,10,16,...$ are not. (a) Prove that there exist real numbers $p$ and $q$ such that the sequences $1,p,p^2,p^3,...$ and $1,q,q^2,q^3,...$ are essentially different Fibonacci sequences. (b) Let $a_1,a_2,a_3,...$ and $b_1,b_2,b_3,...$ be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence $t_1,t_2,t_3,...$, there exists exactly one number $\alpha$ and exactly one number $\beta$, such that: $$t_n = \alpha a_n + \beta b_n$$ for $n = 1,2,3,...$ (c) $t_1,t_2,t_3,...$, is the Fibonacci sequence with $t_1 = 1$ and $t_2= 2$. Express $t_n$ in terms of $n$.