Olympiad problems

2019 USA TSTST, 6

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]

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.

1983 IMO Longlists, 52

Tags: algebra , polynomial , fibonacci , fibonacci sequence , sequence

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 $P(1983) = F_{1983} - 1.$

2020 ELMO Problems, P2

Tags: number theory , fibonacci

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 Saudi Arabia IMO TST, 8

Tags: fibonacci , combinatorics , 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]

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?

2021 Azerbaijan IMO TST, 3

Tags: fibonacci , combinatorics , 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]

1997 IMO Shortlist, 26

Tags: inequalities , algebra , minimization , sequence , fibonacci

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.$

2022 Baltic Way, 7

Tags: fibonacci number , combinatorics , fibonacci , fibonacci sequence

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?

1981 IMO Shortlist, 4

Tags: algebra , fibonacci , fibonacci sequence , sequence , number theory

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.

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

Russian TST 2021, P2

Tags: fibonacci , combinatorics , 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]

2021 Taiwan TST Round 2, C

Tags: fibonacci , combinatorics , 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]

1990 Greece Junior Math Olympiad, 1

Tags: fibonacci , algebra

Considee thr positive integers $a_1,a_2,...,a_{10}$ such that from the third and on, each it the sum of it's two previous terms (i.e. $a_3=a_2+a_1$, $a_4=a_3+a_2$, ...). If $a_5=7$, find $a_{10}$.

2020 Brazil National Olympiad, 2

Tags: fibonacci , number theory , analytic number theory

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is [i]fibonatic[/i] when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not [i]fibonatic[/i] integers.

2020/2021 Tournament of Towns, P6

Tags: number theory , fibonacci

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, 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 $k$.

KoMaL A Problems 2019/2020, A. 770

Tags: number theory , fibonacci , factorial

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

1946 Moscow Mathematical Olympiad, 121

Tags: fibonacci , sequence , integer sequence , algebra

Given the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, ... ,$ ascertain whether among its first $(10^8+1)$ terms there is a number that ends with four zeros.

2017 Baltic Way, 3

Tags: fibonacci , additive number theory , extremal combinatorics , algebra

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 $k>1$.)

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 $k=0,1,...,n$.

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.

1992 IMO Longlists, 18

Tags: combinatorics , fibonacci , fibonacci sequence , combinatorics of words , sequence

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.

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: $$x^{2024}=f_{2023}x+f_{2022}.$$