Olympiad problems

2018 Iran Team Selection Test, 6

Tags: combinatorics , sequence

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $) [i]Proposed by Mohsen Jamali[/i]

2021 Middle European Mathematical Olympiad, 1

Tags: algebra , sequence , constant

Determine all real numbers A such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying \[ x_{n+1}=A-\frac{1}{x_n} \] for every integer $n \ge 1$, has only finitely many negative terms.

2024 China Western Mathematical Olympiad, 2

Tags: algebra , sequence

Find all integers $k$, such that there exists an integer sequence ${\{a_n\}}$ satisfies two conditions below (1) For all positive integers $n$,$a_{n+1}={a_n}^3+ka_n+1$ (2) $|a_n| \leq M$ holds for some real $M$

2024 Taiwan TST Round 3, 6

Tags: algebra , polynomial , sequence

Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold: 1. $a_n\neq 0$; 2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$. [i] Proposed by usjl[/i]

2006 IMO Shortlist, 2

Tags: integration , calculus , inequalities , sequence , summation , algebra

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

2021 Francophone Mathematical Olympiad, 1

Tags: number theory , sequence , divide , recurrence relation

Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.

2016 China Team Selection Test, 4

Tags: sequence , prime number , number theory

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

1980 Austrian-Polish Competition, 6

Tags: inequalities , sequence , algebra , absolute value

Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality \[ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. \] Show that the following inequality holds for all positive integers $k,m$ \[ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. \]

2013 Regional Competition For Advanced Students, 3

Tags: inequalities , arithmetic mean-geometric mean , sequence

For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$ (a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$). (b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$

2023 South East Mathematical Olympiad, 1

Tags: algebra , sequence

The positive sequence $\{a_n\}$ satisfies:$a_1=1$ and $$a_n=2+\sqrt{a_{n-1}}-2 \sqrt{1+\sqrt{a_{n-1}}}(n\geq 2)$$ Let $S_n=\sum\limits_{k=1}^{n}{2^ka_k}$. Find the value of $S_{2023}$.

2024 Belarus Team Selection Test, 2.1

Tags: number theory , algebra , sequence

A sequence $\{y_i\}$ is given, where $y_0=-\frac{1}{4},y_1=0$. For every positive integer $n$ the following equality holds: $$y_{n-1}+y_{n+1}=4y_n+1$$ Prove that for every positive integer $n$ the number $2y_{2n}+\frac{3}{2}$ a) is a positive integer b) is a square of a positive integer [i]D. Zmiaikou[/i]

2022 Greece Team Selection Test, 3

Tags: inequalities , algebra , sequence

Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions : i) $a_0=1$, $a_1=3$ ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$ to be true that $$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.

2021 Macedonian Mathematical Olympiad, Problem 5

Tags: number theory , sequence

Let $(x_{n})_{n=1}^{+\infty}$ be a sequence defined recursively with $x_{n+1} = x_{n}(x_{n}-2)$ and $x_{1} = \frac{7}{2}$. Let $x_{2021} = \frac{a}{b}$, where $a,b \in \mathbb{N}$ are coprime. Show that if $p$ is a prime divisor of $a$, then either $3|p-1$ or $p=3$. [i]Authored by Nikola Velov[/i]

2023 Brazil National Olympiad, 1

Tags: number theory , sequence

Show an infinite sequence $a_1, a_2, \ldots$ of integers with both of the following properties: • $a_i \neq 0$ for every positive integer $i$, that is, no term in the sequence is equal to zero; • for all positive integer $n$, $a_n + a_{2n} + \ldots + a_{2023n} = 0$.

2024 CIIM, 1

Tags: integration , function , recurrence relation , calculus , sequence

Let $(a_n)_{n \geq 1}$ be a sequence of real numbers. We define a sequence of real functions $(f_n)_{n \geq 0}$ such that for all $x \in \mathbb{R}$, the following holds: \[ f_0(x) = 1 \quad \text{and} \quad f_n(x) = \int_{a_n}^{x} f_{n-1}(t) \, dt \quad \text{for } n \geq 1. \] Find all possible sequences $(a_n)_{n \geq 1}$ such that $f_n(0) = 0$ for all $n \geq 2$.\\ [b]Note:[/b] It is not necessarily true that $f_1(0) = 0$.

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.

2015 IMC, 3

Tags: series , sequence

Let $F(0)=0$, $F(1)=\frac32$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$ for $n\ge2$. Determine whether or not $\displaystyle{\sum_{n=0}^{\infty}\, \frac{1}{F(2^n)}}$ is a rational number. (Proposed by Gerhard Woeginger, Eindhoven University of Technology)

2011 Indonesia TST, 4

Tags: number theory , recurrence relation , number theory with sequences , sequence

Let $a, b$, and $c$ be positive integers such that $gcd(a, b) = 1$. Sequence $\{u_k\}$, is given such that $u_0 = 0$, $u_1 = 1$, and u$_{k+2} = au_{k+1} + bu_k$ for all $k \ge 0$. Let $m$ be the least positive integer such that $c | u_m$ and $n$ be an arbitrary positive integer such that $c | u_n$. Show that $m | n$. [hide=PS.] There was a typo in the last line, as it didn't define what n does. Wording comes from [b]tst-2011-1.pdf[/b] from [url=https://sites.google.com/site/imoidn/idntst/2011tst]here[/url]. Correction was made according to #2[/hide]

2018 Thailand TSTST, 4

Tags: algebra , recurrence relation , sequence , inequality

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

2016 IMC, 1

Tags: summation , sequence

Let $(x_1,x_2,\ldots)$ be a sequence of positive real numbers satisfying ${\displaystyle \sum_{n=1}^{\infty}\frac{x_n}{2n-1}=1}$. Prove that $$ \displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le2. $$ (Proposed by Gerhard J. Woeginger, The Netherlands)

2015 BMT Spring, 4

Tags: algebra , sequence

Let $\{a_n\}$ be a sequence of real numbers with $a_1=-1$, $a_2=2$ and for all $n\ge3$, $$a_{n+1}-a_n-a_{n+2}=0.$$ Find $a_1+a_2+a_3+\ldots+a_{2015}$.

1984 IMO Shortlist, 6

Tags: algebra , sequence , recurrence relation , linear recurrence

Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: \[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\] Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$

2011 SEEMOUS, Problem 4

Tags: integration , calculus , sequence

Let $f:[0,1]\to\mathbb R$ be a twice continuously differentiable increasing function. Define the sequences given by $L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right)$ and $U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right)$, $n\ge1$. 1. The interval $[L_n,U_n]$ is divided into three equal segments. Prove that, for large enough $n$, the number $I=\int^1_0f(x)\text dx$ belongs to the middle one of these three segments.

2006 Petru Moroșan-Trident, 3

Tags: real analysis , sequence

Let be a sequence $ \left( u_n \right)_{n\ge 1} $ given by the recurrence relation $ u_{n+1} =u_n+\sqrt{u_n^2-u_1^2} , $ and the constraints $ u_2\ge u_1>0. $ Calculate $ \lim_{n\to\infty }\frac{2^n}{u_n} . $ [i]Dan Negulescu[/i]

2017 IMO, 1

Tags: number theory , sequence

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]