This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 59

2020 Latvia TST, 1.3
Tags: infinitely many solutions , fibonacci , number theory
Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers

2023 Olimphíada, 4
Tags: fibonacci
We all know the Fibonacci sequence. However, a slightly less known sequence is the $k$-bonacci sequence. In it, we have $F_1^{(k)} = F_2^{(k)} = \cdots = F_{k-1}^{(k)} = 0, F_k^{(k)} = 1$ and $$F^{(k)}_{n+k} = F^{(k)}_{n+k-1} + F^{(k)}_{n+k-2} + \cdots + F^{(k)}_n,$$for all $n \geq 1$. Find all positive integers $k$ for which there exists a constant $N$ such that $$F^{(k)}_{n-1}F^{(k)}_{n+1} - (F ^{(k)}_n)^2 = (-1)^n$$ for every positive integer $n \geq N$.

1992 IMO Shortlist, 18
Tags: sequence , fibonacci , fibonacci sequence , inequality , floor function , algebra
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that \[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\] where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$

2020 HK IMO Preliminary Selection Contest, 20
Tags: number theory , fibonacci
Consider the Fibonacci sequence $1$, $1$, $2$, $3$, $5$, $8$, $13$, ... What are the last three digits (from left to right) of the $2020$th term?

1997 Slovenia National Olympiad, Problem 2
Tags: fibonacci sequence , fibonacci
The Fibonacci sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\in\mathbb N$. (a) Show that $f_{1005}$ is divisible by $10$. (b) Show that $f_{1005}$ is not divisible by $100$.

1991 Chile National Olympiad, 5
Tags: algebra , fibonacci , sum , number theory
The sequence $(a_k)$, $k> 0$ is Fibonacci, with $a_0 = a_1 = 1$. Calculate the value of $$\sum_{j = 0}^{\infty} \frac{a_j}{2^j}$$

2012 USAJMO, 2
Tags: fibonacci , geometry
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.

Russian TST 2021, P2
Tags: combinatorics , fibonacci , additive representation
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

2017 Regional Olympiad of Mexico Southeast, 6
Tags: fibonacci , number theory
Consider $f_1=1, f_2=1$ and $f_{n+1}=f_n+f_{n-1}$ for $n\geq 2$. Determine if exists $n\leq 1000001$ such that the last three digits of $f_n$ are zero.

KoMaL A Problems 2019/2020, A. 770
Tags: fibonacci , factorial , number theory
Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.

2020 European Mathematical Cup, 2
Tags: number theory , fibonacci , divisor , sequence
A positive integer $k\geqslant 3$ is called[i] fibby[/i] if there exists a positive integer $n$ and positive integers $d_1 < d_2 < \ldots < d_k$ with the following properties: \\ $\bullet$ $d_{j+2}=d_{j+1}+d_j$ for every $j$ satisfying $1\leqslant j \leqslant k-2$, \\ $\bullet$ $d_1, d_2, \ldots, d_k$ are divisors of $n$, \\ $\bullet$ any other divisor of $n$ is either less than $d_1$ or greater than $d_k$. Find all fibby numbers. \\ \\ [i]Proposed by Ivan Novak.[/i]

2016 Israel National Olympiad, 5
Tags: algebra , number theory , fibonacci , fibonacci sequence , degree , polynomial
The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$. Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.

1981 IMO Shortlist, 4
Tags: fibonacci , fibonacci sequence , sequence , number theory , algebra
Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $ (a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.

2020 ELMO Problems, P2
Tags: fibonacci , number theory
Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$. Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$. Must $k$ be a Fibonacci number? [i]Proposed by Fedir Yudin.[/i]

2021 Azerbaijan IMO TST, 3
Tags: combinatorics , fibonacci , additive representation
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

2012 USAMO, 1
Tags: geometry , fibonacci
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.

2015 Thailand TSTST, 1
Tags: algebra , number theory , fibonacci
Prove that the Fibonacci sequence $\{F_n\}^\infty_{n=1}$ defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for all $n \geq 1$ is a divisibility sequence, that is, if $m\mid n$ then $F_m \mid F_n$ for all positive integers $m$ and $n$.

1957 Moscow Mathematical Olympiad, 363
Tags: number theory , sum , sequence , fibonacci
Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.

1997 IMO Shortlist, 26
Tags: minimization , sequence , fibonacci , algebra , inequalities
For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i\equal{}0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 \equal{} 1,$ $ a_i \leq a_{i\plus{}1} \plus{} a_{i\plus{}2}$ for $ i \equal{} 0, \ldots, n \minus{} 2.$

2023 Regional Olympiad of Mexico Southeast, 4
Tags: algebra , fibonacci
Given the Fibonacci sequence with $f_0=f_1=1$and for $n\geq 1, f_{n+1}=f_n+f_{n-1}$, find all real solutions to the equation: $$x^{2024}=f_{2023}x+f_{2022}.$$

1992 IMO Longlists, 79
Tags: algebra , floor function , sequence , fibonacci , fibonacci sequence , inequality
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that \[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\] where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$

2015 Thailand TSTST, 2
Tags: algebra , number theory , polynomial , fibonacci
Let $\{F_n\}^\infty_{n=1}$ be the Fibonacci sequence and let $f$ be a polynomial of degree $1006$ such that $f(k) = F_k$ for all $k \in \{1008, \dots , 2014\}$. Prove that $$233\mid f(2015)+1.$$ [i]Note: $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$ for all $n\geq 1$.[/i]

2020/2021 Tournament of Towns, P6
Tags: fibonacci , number theory
Find at least one real number $A{}$ such that for any positive integer $n{}$ the distance between $\lceil A^n\rceil$ and the nearest square of an integer is equal to two. [i]Dmitry Krekov[/i]

2023 Olimphíada, 2
Tags: algebra , fibonacci , sequence
The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for every integer $n$. A sequence $(a_n)$ of integers is said to be $\textit{phirme}$ if there is a fixed integer $k$ such that $a_n + a_{n+1} = F_{n+k}$ for all $n \geq 1$. Show that if $(a_n)$ is a $\textit{phirme}$ sequence, then there exists an integer $c$ such that $$a_n = F_{n+k-2} + (-1)^nc.$$

2022 Baltic Way, 7
Tags: combinatorics , fibonacci , fibonacci sequence , fibonacci number
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?