Olympiad problems

2007 Germany Team Selection Test, 1

We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

2020 Jozsef Wildt International Math Competition, W53

Tags: recurrence relation , algebra , sequence

Define the sequence $(w_n)_{n\ge0}$ by the recurrence relation $$w_{n+2}=2w_{n+1}+3w_n,\enspace\enspace w_0=1,w_1=i,\enspace n=0,1,\ldots$$ (1) Find the general formula for $w_n$ and compute the first $9$ terms. (2) Show that $|\Re w_n-\Im w_n|=1$ for all $n\ge1$. [i]Proposed by Ovidiu Bagdasar[/i]

2014 SEEMOUS, Problem 2

Tags: limit , sequence

Consider the sequence $(x_n)$ given by $$x_1=2,\enspace x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}2,\enspace n\ge2.$$Prove that the sequence $y_n=\sum_{k=1}^n\frac1{x_k^2-1},\enspace n\ge1$ is convergent and find its limit.

1962 All-Soviet Union Olympiad, 13

Tags: sequence , inequalities , algebra

Given are $a_0,a_1, ... , a_n$, satisfying $a_0=a_n = 0$, and $a_{k-1} - 2a_k+a_{k+1}\ge 0$ for $k=0, 1, ... , n-1$. Prove that all the numbers are negative or zero.

1977 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , sequence , additive number theory

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

1983 Putnam, B4

Tags: number theory , function , sequence

[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.

2024 Saint Petersburg Mathematical Olympiad, 2

Tags: sequence , algebra

Given a sequence $a_n$: \[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots \] (one '1', two '2' and so on) and another sequence $b_n$ such that $a_{b_n}=b_{a_n}$ for all positive integers $n$. It is known that $b_k=1$ for some $k>100$. Prove that $b_m=1$ for all $m>k$.

2021 Middle European Mathematical Olympiad, 1

Tags: sequence , algebra , 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.

2008 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: sequence , function , algebra , functional equation

Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that: $i)$ $a(0)=0$ $ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$. If exists prove: $a)$ $a(k)\geq a(k-1)$ $b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.

2007 Mathematics for Its Sake, 1

Tags: floor function , real analysis , sequence , parity , number theory

Prove that the parity of each term of the sequence $ \left( \left\lfloor \left( \lfloor \sqrt q \rfloor +\sqrt{q} \right)^n \right\rfloor \right)_{n\ge 1} $ is opposite to the parity of its index, where $ q $ is a squarefree natural number.

2013 German National Olympiad, 6

Tags: sequence , number theory , root , real number

Define a sequence $(a_n)$ by $a_1 =1, a_2 =2,$ and $a_{k+2}=2a_{k+1}+a_k$ for all positive integers $k$. Determine all real numbers $\beta >0$ which satisfy the following conditions: (A) There are infinitely pairs of positive integers $(p,q)$ such that $\left| \frac{p}{q}- \sqrt{2} \, \right| < \frac{\beta}{q^2 }.$ (B) There are only finitely many pairs of positive integers $(p,q)$ with $\left| \frac{p}{q}- \sqrt{2} \,\right| < \frac{\beta}{q^2 }$ for which there is no index $k$ with $q=a_k.$

2017 Romania EGMO TST, P2

Tags: number theory , gcd , sequence

Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$

2005 Slovenia National Olympiad, Problem 2

Tags: sequence , algebra

Let $(a_n)$ be a geometrical progression with positive terms. Define $S_n=\log a_1+\log a_2+\ldots+\log a_n$. Prove that if $S_n=S_m$ for some $m\ne n$, then $S_{n+m}=0$.

1996 Tuymaada Olympiad, 6

Tags: algebra , sequence , limit of sequence , limit , trigonometric

Given the sequence $f_1(a)=sin(0,5\pi a)$ $f_2(a)=sin(0,5\pi (sin(0,5\pi a)))$ $...$ $f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)))$ , where $a$ is any real number. What limit aspire the members of this sequence as $n \to \infty$?

2007 Grigore Moisil Intercounty, 4

Tags: inequalities , real analysis , convergence , sequence

Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of positive real numbers, verifying the inequality $ x_n\le \frac{x_{n-1}+x_{n-2}}{2} , $ for any natural number $ n\ge 3. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.

2010 Indonesia TST, 2

Tags: greatest common divisor , number theory , sequence

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

2022 Iran MO (2nd round), 5

Tags: number theory , sequence

define $(a_n)_{n \in \mathbb{N}}$ such that $a_1=2$ and $$a_{n+1}=\left(1+\frac{1}{n}\right)^n \times a_{n}$$ Prove that there exists infinite number of $n$ such that $\frac{a_1a_2 \ldots a_n}{n+1}$ is a square of an integer.

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 $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$

2016 AIME Problems, 9

Tags: sequence

The sequences of positive integers $1,a_2,a_3,\ldots$ and $1,b_2,b_3,\ldots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.

1947 Putnam, A1

Tags: sequence , limit

If $(a_n)$ is a sequence of real numbers such that for $n \geq 1$ $$(2-a_n )a_{n+1} =1,$$ prove that $\lim_{n\to \infty} a_n =1.$

2022 IMO Shortlist, N3

Tags: number theory , sequence

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

2018 IMO Shortlist, A2

Tags: algebra , sequence , system of equations

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]

2021 Argentina National Olympiad, 1

Tags: sequence , combinatorics

An infinite sequence of digits $1$ and $2$ is determined by the following two properties: i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$ ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$, the same sequence is again obtained. In which position is the hundredth digit $1$? What is the thousandth digit of the sequence?

1996 Cono Sur Olympiad, 2

Tags: sequence

Consider a sequence of real numbers defined by: $a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$ Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.

1973 Bulgaria National Olympiad, Problem 2

Tags: arithmetic sequence , algebra , sequence , geometric sequence

Let the numbers $a_1,a_2,a_3,a_4$ form an arithmetic progression with difference $d\ne0$. Prove that there are no exists geometric progressions $b_1,b_2,b_3,b_4$ and $c_1,c_2,c_3,c_4$ such that: $$a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.$$