Olympiad problems

2024 Romania National Olympiad, 4

Let $a$ be a given positive integer. We consider the sequence $(x_n)_{n \ge 1}$ defined by $x_n=\frac{1}{1+na},$ for every positive integer $n.$ Prove that for any integer $k \ge 3,$ there exist positive integers $n_1<n_2<\ldots<n_k$ such that the numbers $x_{n_1},x_{n_2},\ldots,x_{n_k}$ are consecutive terms in an arithmetic progression.

1998 Brazil Team Selection Test, Problem 5

Tags: inequalities , sequence , number theory

Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that $$k\le2\lfloor\sqrt n\rfloor.$$

2024 Baltic Way, 18

Tags: sequence , prime number , bound , number theory , divisor

An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.

2001 All-Russian Olympiad Regional Round, 11.5

Tags: sequence , periodic , trigonometry , recurrence relation , algebra

Given a sequence $\{x_k\}$ such that $x_1 = 1$, $x_{n+1} = n \sin x_n+ 1$. Prove that the sequence is non-periodic.

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 $S_{2023}$.

2019 Taiwan APMO Preliminary Test, P7

Tags: combinatorics , sequence , algebra

Let positive integer $k$ satisfies $1<k<100$. For the permutation of $1,2,...,100$ be $a_1,a_2,...,a_{100}$, take the minimum $m>k$ such that $a_m$ is at least less than $(k-1)$ numbers of $a_1,a_2,...,a_k$. We know that the number of sequences satisfies $a_m=1$ is $\frac{100!}{4}$. Find the all possible values of $k$.

1973 Spain Mathematical Olympiad, 3

Tags: complex analysis , recurrence relation , recurrence , sequence , analysis

The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$ Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$.

1995 North Macedonia National Olympiad, 1

Tags: sequence , recurrence relation , algebra

Let $ a_0 $ be a real number. The sequence $ \{a_n \} $ is given by $ a_ {n + 1} = 3 ^ n-5a_n $, $ n = 0,1,2, \ldots $. a) Express the general member $ a_n $ through $ a_0 $ and $ n. $ b) Find such $ a_0, $ that $ a_ {n + 1}> a_n, $ for every $ n. $

1980 IMO Longlists, 2

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 \[ 1 - \frac{1}{n} < a_n < 1.\]

1999 Croatia National Olympiad, Problem 4

Tags: sequence , geometry

On the coordinate plane is given the square with vertices $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$. For every $n\in\mathbb N$, point $T_{n+4}$ is defined as the midpoint of the segment $T_nT_{n+1}$. Determine the coordinates of the limit point of $T_n$ as $n\to\infty$, if it exists.

2014 Contests, A3

Tags: sequence , algebra

$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

Russian TST 2019, P2

Tags: algebra , maximum value , sequence

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

1995 IMO Shortlist, 3

Tags: induction , algebra , polynomial , iteration , sequence

For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.

2012 Balkan MO Shortlist, N2

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

Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$. Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.

2017 Simon Marais Mathematical Competition, A2

Tags: algebra , sequence

Let $a_1,a_2,a_3,\ldots$ be the sequence of real numbers deﬁned by $a_1=1$ and $$a_m=\frac1{a_1^2+a_2^2+\ldots+a_{m-1}^2}\qquad\text{for }m\ge2.$$ Determine whether there exists a positive integer $N$ such that $$a_1+a_2+\ldots+a_N>2017^{2017}.$$

1999 Brazil Team Selection Test, Problem 3

Tags: algebra , sequence , recurrence relation

A sequence $a_n$ is defined by $$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.

2010 Ukraine Team Selection Test, 8

Tags: algebra , sequence , inequalities

Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let $\{a_n\}, n\ge 1$ be such a sequence. We call it [i]consistent [/i] if, for an arbitrary natural $k$ and every natural $n ,m$ such that $a_n <a_m$, the inequality $a_{kn} <a _{km}$ also holds. For example, the sequence $a_n = n$ is consistent . a) Prove that there are consistent sequences other than $a_n = n$. b) Are there consistent sequences for which $a_n \ne n, n\ge 2$ ? c) Are there consistent sequences for which $a n \ne n, n\ge 1$ ?

2019 Taiwan APMO Preliminary Test, P4

Tags: algebra , floor function , sequence , recursive

We define a sequence ${a_n}$: $$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$ (1)Find $\lfloor a_{2019}\rfloor$ (2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$

2017 Irish Math Olympiad, 5

Tags: algebra , sequence , sum

Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies $$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$. (a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero. (b) Find a sequence none of whose terms are zero but which is $2017$-powerful.

2022 Korea National Olympiad, 3

Tags: number theory , sequence , integer sequence , perfect square

Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions: [list] [*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square. [*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$. [/list] Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$. And determine the value of $\sum_{k=N}^{N+2021}a_k$.

2010 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , sequence , divisibility

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

2008 Indonesia TST, 2

Tags: algebra , inequalities , sequence , sum

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

2017 Singapore MO Open, 4

Tags: algebra , inequalities , arithmetic progression , geometric progression , sequence

Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$

1978 Germany Team Selection Test, 3

Tags: algebra , recurrence relation , sequence , equation , floor function

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

2022 Brazil Team Selection Test, 3

Tags: number theory , prime number , sequence

Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$ where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.