Olympiad problems

2016 Iran MO (3rd Round), 1

The sequence $(a_n)$ is defined as: $$a_1=1007$$ $$a_{i+1}\geq a_i+1$$ Prove the inequality: $$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$

2005 Czech And Slovak Olympiad III A, 6

Tags: coloring , monotone , sequence , permutation , combinatorics

Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.

2021 JBMO Shortlist, A2

Tags: shortlist , sequence , algebra

Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for which the following property holds: If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$. Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]

1973 IMO Shortlist, 10

Tags: sequence , inequality , geometric sequence , algebra , construction

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

2018 Pan African, 3

Tags: sequence , algebra , number theory

For any positive integer $x$, we set $$ g(x) = \text{ largest odd divisor of } x, $$ $$ f(x) = \begin{cases} \frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\ 2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.} \end{cases} $$ Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.

2020 Bulgaria National Olympiad, P2

Tags: n-variable inequality , sequence , algebra , inequalities

Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ [hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]

1978 Swedish Mathematical Competition, 4

Tags: sequence , logarithm , convex , algebra

$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.

2020 Francophone Mathematical Olympiad, 3

Tags: sequence , algebra

Let $(a_i)_{i\in \mathbb{N}}$ be a sequence with $a_1=\frac{3}2$ such that $$a_{n+1}=1+\frac{n}{a_n}$$ Find $n$ such that $2020\le a_n <2021$

2023 Polish MO Finals, 1

Tags: sequence , divisibility , number theory

Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.

1982 All Soviet Union Mathematical Olympiad, 328

Tags: fibonacci , sequence , algebra

Every member, starting from the third one, of two sequences $\{a_n\}$ and $\{b_n\}$ equals to the sum of two preceding ones. First members are: $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$. How many natural numbers are encountered in both sequences (may be on the different places)?

2018 Costa Rica - Final Round, F2

Tags: sequence , functional , algebra , functional equation , combinatorics

Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then $$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$

VI Soros Olympiad 1999 - 2000 (Russia), 9.5

Tags: sequence , number theory , algebra

Let b be a given real number. The sequence of integers $a_1, a_2,a_3, ...$ is such that $a_1 =(b]$ and $a_{n+1}=(a_n+b]$ for all $n\ge 1$ Prove that the sum $a_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n}$ is an integer number for any natural $n$ . (In the condition of the problem, $(x]$ denotes the smallest integer that is greater than or equal to $x$)

1981 All Soviet Union Mathematical Olympiad, 313

Tags: sequence , integer sequence , number theory

Find all the sequences of natural $k_n$ with two properties: a) $k_n \le n \sqrt {n}$ for all $n$ b) $(k_n - k_m)$ is divisible by $(m-n)$ for all $m>n$

2022 Vietnam National Olympiad, 1

Tags: sequence , calculus , algebra

Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as: $u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$ a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$

2016 China Second Round Olympiad, 4

Tags: sequence , number theory

Let $p>3$ and $p+2$ are prime numbers,and define sequence $$a_{1}=2,a_{n}=a_{n-1}+\lfloor \dfrac{pa_{n-1}}{n}\rfloor$$ show that:for any $n=3,4,\cdots,p-1$ have $$n|pa_{n-1}+1$$

2002 District Olympiad, 1

Tags: algebra , sequence

Determine the sequence of complex numbers $ \left( x_n\right)_{n\ge 1} $ for which $ 1=x_1, $ and for any natural number $ n, $ the following equality is true: $$ 4\left( x_1x_n+2x_2x_{n-1}+3x_3x_{n-2}+\cdots +nx_nx_1\right) =(1+n)\left( x_1x_2+x_2x_3+\cdots +x_{n-1}x_n +x_nx_{n+1}\right) . $$

2022 Brazil Team Selection Test, 4

Tags: sequence , number theory

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

2017 Vietnamese Southern Summer School contest, Problem 1

Tags: convergence , sequence , real number , analysis

Given a real number $a$ and a sequence $(x_n)_{n=1}^\infty$ defined by: $$\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right.$$ for all positive integers $n$. 1. For $a=0$, prove that $(x_n)$ converges. 2. Determine the largest possible value of $a$ such that $(x_n)$ converges.

1954 Miklós Schweitzer, 1

Tags: sequence , real analysis

[b]1.[/b] Given a positive integer $r>1$, prove that there exists an infinite number of infinite geometrical series, with positive terms, having the sum 1 and satisfying the following condition: for any positive real numbers $S_{1},S_{2},\dots,S_{r}$ such that $S_{1}+S_{2}+\dots+S_{r}=1$, any of these infinite geometrical series can be divided into $r$ infinite series(not necessarily geometrical) having the sums $S_{1},S_{2},\dots,S_{r}$, respectively. [b](S. 6)[/b]

2014 IMC, 1

Tags: sequence

For a positive integer $x$, denote its $n^{\mathrm{th}}$ decimal digit by $d_n(x)$, i.e. $d_n(x)\in \{ 0,1, \dots, 9\}$ and $x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}$. Suppose that for some sequence $(a_n)_{n=1}^{\infty}$, there are only finitely many zeros in the sequence $(d_n(a_n))_{n=1}^{\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence $(a_n)_{n=1}^{\infty}$. (Proposed by Alexander Bolbot, State University, Novosibirsk)

2014 Albania Round 2, 2

Tags: arithmetic progression , sequence , algebra

Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

1998 Tuymaada Olympiad, 7

Tags: sequence , sum , arithmetic sequence , algebra , square

All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.

2005 Grigore Moisil Urziceni, 3

Tags: real analysis , limit , sequence

Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality $$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$ for all natural numbers $ n. $ [b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $ [b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $

2022 Brazil National Olympiad, 3

Tags: finite differences , sequence , algebra

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of integers numbers. Let $\Delta^1a_n=a_{n+1}-a_n$ for a non-negative integer $n$. Define $\Delta^Ma_n= \Delta^{M-1}a_{n+1}- \Delta^{M-1}a_n$. A sequence is [i]patriota[/i] if there are positive integers $k,l$ such that $a_{n+k}=\Delta^Ma_{n+l}$ for all non-negative integers $n$. Determine, with proof, whether exists a sequence that the last value of $M$ for which the sequence is [i]patriota[/i] is $2022$.

1989 Bulgaria National Olympiad, Problem 2

Tags: series , floor function , limit , sequence , algebra

Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.