Olympiad problems

2018 Romanian Master of Mathematics Shortlist, N2

Prove that for each positive integer $k$ there exists a number base $b$ along with $k$ triples of Fibonacci numbers $(F_u,F_v,F_w)$ such that when they are written in base $b$, their concatenation is also a Fibonacci number written in base $b$. (Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.) [i]Proposed by Serbia[/i]

2015 Thailand TSTST, 2

Tags: algebra , polynomial , number theory , 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]

1992 IMO Shortlist, 18

Tags: floor function , algebra , Sequence , Fibonacci , Fibonacci sequence , Inequality , IMO Shortlist

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.$

2017 Baltic Way, 3

Tags: algebra , Fibonacci , Additive Number Theory , Extremal combinatorics

Positive integers $x_1,...,x_m$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_1,...,F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$? (Here $F_1,...,F_{2018}$ are the first $2018$ Fibonacci numbers: $F_1=F_2=1, F_{k+1}=F_k+F_{k-1}$ for $k>1$.)

2021 Azerbaijan IMO TST, 3

Tags: Fibonacci , combinatorics , IMO Shortlist , IMO Shortlist 2020 , 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]

2023 Philippine MO, 7

Tags: number theory , Fibonacci , PMO

A set of positive integers is said to be [i]pilak[/i] if it can be partitioned into 2 disjoint subsets $F$ and $T$, each with at least $2$ elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.

2021 Thailand TST, 2

Tags: Fibonacci , combinatorics , IMO Shortlist , IMO Shortlist 2020 , 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]

2023 Olimphíada, 2

Tags: Fibonacci , Sequence , algebra

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 $$a_n = F_{n+k-2} + (-1)^nc.$$

2012 USAJMO, 2

Tags: USA(J)MO , USAMO , 2012 USAJMO , 2012 USAMO , 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.

ICMC 7, 1

Tags: number theory , Fibonacci , ICMC

Let $F_n{}$ denote the $n{}$-th Fibonacci number. Prove that $3^{2023}$ divides \[3^2\cdot F_4+3^3\cdot F_6+3^4\cdot F_8+\dots+3^{2023}F_{4046}.\][i]Proposed by Dylan Toh[/i]

1992 IMO Longlists, 18

Tags: combinatorics , Fibonacci , Fibonacci sequence , Combinatorics of words , Sequence , IMO Shortlist , IMO Longlist

Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$. Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.

2020 HK IMO Preliminary Selection Contest, 20

Tags: Fibonacci , number theory

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?

1983 IMO Shortlist, 19

Tags: algebra , polynomial , Fibonacci , Fibonacci sequence , Sequence , IMO Shortlist

Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying \[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\] Prove that $P(1983) = F_{1983} - 1.$

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.

2018 Stanford Mathematics Tournament, 4

Tags: Fibonacci , Fibonacci sequence , Fibonacci Numbers , number theory

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 $n \ge 1$.

Russian TST 2021, P2

Tags: Fibonacci , combinatorics , IMO Shortlist , IMO Shortlist 2020 , 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]

2014 BAMO, 4

Tags: Fibonacci , Sequence , Divisibility , number theory

Let $F_1, F_2, F_3 \cdots$ be the Fibonacci sequence, the sequence of positive integers satisfying $$F_1 =F_2=1$$ and $$F_{n+2} = F_{n+1} + F_n$$ for all $n \ge 1$. Does there exist an $n \ge 1$ such that $F_n$ is divisible by $2014$? Prove your answer.

2018 Mathematical Talent Reward Programme, MCQ: P 3

Tags: Fibonacci , algebra

$F_{n}$ denotes the Fibonacci Sequence where $F_{1}=0, F_{2}=1, F_{n}=F_{n-1}+F_{n-2},\ \forall \ n \geq 3$ Find$$\sum\limits_{n=3}^{\infty}\frac{18+999F_n}{F_{n-1}\times F_{n+1}}$$ [list=1] [*] 2016 [*] 2017 [*] 2018 [*] None of these [/list]

2020 Jozsef Wildt International Math Competition, W55

Tags: number theory , Diophantine equation , Fibonacci , Fibonacci Numbers

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]

2019 USA TSTST, 6

Tags: algebra , tstst 2019 , Integer Polynomial , Fibonacci

Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant? (A [i]Fibonacci number[/i] is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$.) [i]Nikolai Beluhov[/i]

KoMaL A Problems 2019/2020, A. 770

Tags: komal , number theory , Fibonacci , factorial

Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.

2009 BAMO, 2

Tags: Fibonacci , Sum

The Fibonacci sequence is the list of numbers that begins $1, 2, 3, 5, 8, 13$ and continues with each subsequent number being the sum of the previous two. Prove that for every positive integer $n$ when the first $n$ elements of the Fibonacci sequence are alternately added and subtracted, the result is an element of the sequence or the negative of an element of the sequence. For example, when $n = 4$ we have $1-2+3-5 = -3$ and $3$ is an element of the Fibonacci sequence.

2023 Olimphíada, 1

Tags: Fibonacci , Sequence , algebra

The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for every integer $n$. Let $k$ be a fixed integer. A sequence $(a_n)$ of integers is said to be $\textit{phirme}$ if $a_n + a_{n+1} = F_{n+k}$ for all $n \geq 1$. Find all $\textit{phirme}$ sequences in terms of $n$ and $k$.

2020 Jozsef Wildt International Math Competition, W14

Tags: Fibonacci , number theory

Let $\{F_n\}_{n\ge1}$ be the Fibonacci sequence defined by $F_1=F_2=1$ and for all $n\ge3$, $F_n=F_{n-1}+F_{n-2}$. Prove that among the first $10000000000000002$ terms of the sequence there is one term that ends up with $8$ zeroes. [i]Proposed by José Luis Díaz-Barrero[/i]

1991 Chile National Olympiad, 5

Tags: Fibonacci , number theory , algebra , Sum

The sequence $(a_k)$, $k> 0$ is Fibonacci, with $a_0 = a_1 = 1$. Calculate the value of $$\sum_{j = 0}^{\infty} \frac{a_j}{2^j}$$