Olympiad problems

2019 Finnish National High School Mathematics Comp, 4

Define a sequence $ a_n = n^n + (n - 1)^{n+1}$ when $n$ is a positive integer. Define all those positive integer $m$ , for which this sequence of numbers is eventually periodic modulo $m$, e.g. there are such positive integers $K$ and $s$ such that $a_k \equiv a_{k+s}$ ($mod \,m$), where $k$ is an integer with $k \ge K$.

2022 Bulgaria EGMO TST, 4

Tags: shortlist , number theory , sequence , recursion , existence , admiration

Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there exists some $i \in \mathbb{N}$ with $a_i = m^2$. [i]Proposed by Nikola Velov, North Macedonia[/i]

2018 China Northern MO, 6

Tags: number theory , sequence

For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$ Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$

1980 All Soviet Union Mathematical Olympiad, 297

Tags: product of digits , number theory , sequence

Let us denote with $P(n)$ the product of all the digits of $n$. Consider the sequence $$n_{k+1} = n_k + P(n_k)$$ Can it be unbounded for some $n_1$?

2009 VJIMC, Problem 4

Tags: sequence

Let $(a_n)_{n=1}^\infty$ be a sequence of real numbers. We say that the sequence $(a_n)_{n=1}^\infty$ covers the set of positive integers if for any positive integer $m$ there exists a positive integer $k$ such that $\sum_{n=1}^\infty a_n^k=m$. a) Does there exist a sequence of real positive numbers which covers the set of positive integers? b) Does there exist a sequence of real numbers which covers the set of positive integers?

2008 Thailand Mathematical Olympiad, 3

Tags: algebra , inequalities , sequence

For each positive integer $n$, define $a_n = n(n + 1)$. Prove that $$n^{1/a_1} + n^{1/a_3} + n^{1/a_5} + ...+ n^{1/a_{2n-1}} \ge n^{a_{3n+2}/a_{3n+1}}$$ .

1983 IMO Longlists, 19

Tags: number theory , sequence , recurrence relation , algebra

Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and \[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\] Show that for each positive integer $n$, $a_n$ is a positive integer.

2008 Estonia Team Selection Test, 4

Tags: sequence , number theory , divisible , consecutive

Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.

1970 Putnam, A4

Tags: sequence , limit

Given a sequence $(x_n )$ such that $\lim_{n\to \infty} x_n - x_{n-2}=0,$ prove that $$\lim_{n\to \infty} \frac{x_n -x_{n-1}}{n}=0.$$

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)

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.

2024 Francophone Mathematical Olympiad, 4

Tags: divisibility , sequence , number theory

Find all integers $n \ge 2$ for which there exists $n$ integers $a_1,a_2,\dots,a_n \ge 2$ such that for all indices $i \ne j$, we have $a_i \mid a_j^2+1$.

1977 IMO, 2

Tags: linear algebra , algebra , sequence , maximization , extremal combinatorics

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

2001 Austrian-Polish Competition, 6

Tags: floor function , sequence , integer , algebra , number theory

Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.

1989 Greece National Olympiad, 3

Tags: limit , real analysis , sequence

Find the limit of the sequence $x_n$ defined by recurrence relation $$x_{n+2}=\frac{1}{12}x_{n+1}+\frac{1}{2}x_{n}+1$$ where $n=0,1,2,...$ for any initial values $x_2,x_1$.

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 $S_{2023}$.

2018 Pan-African Shortlist, N3

Tags: number theory , sequence , algebra

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.

1971 IMO Shortlist, 9

Tags: algebra , sequence , recurrence relation , linear recurrence

Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and \[T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).\] Show that for all $k$, \[1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],\] where $[x]$ denotes the greatest integer not exceeding $x.$

2002 Switzerland Team Selection Test, 6

Tags: sequence , inequalities , algebra

A sequence $x_1,x_2,x_3,...$ has the following properties: (a) $1 = x_1 < x_2 < x_3 < ...$ (b) $x_{n+1} \le 2n$ for all $n \in N$. Prove that for each positive integer $k$ there exist indices $i$ and $j$ such that $k =x_i -x_j$.

1997 Singapore Team Selection Test, 2

Tags: sequence , consecutive , number theory

Let $a_n$ be the number of n-digit integers formed by $1, 2$ and $3$ which do not contain any consecutive $1$’s. Prove that $a_n$ is equal to $$\left( \frac12 + \frac{1}{\sqrt3}\right)(\sqrt{3} + 1)^n$$ rounded off to the nearest integer.

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 $n$.

1995 French Mathematical Olympiad, Problem 2

Tags: algebra , sequence

Study the convergence of a sequence defined by $u_0\ge0$ and $u_{n+1}=\sqrt{u_n}+\frac1{n+1}$ for all $n\in\mathbb N_0$.

2005 Bosnia and Herzegovina Junior BMO TST, 3

Tags: sequence , algebra

Rational numbers are written in the following sequence: $\frac{1}{1},\frac{2}{1},\frac{1}{2},\frac{3}{1},\frac{2}{2},\frac{1}{3},\frac{4}{1},\frac{3}{2},\frac{2}{3},\frac{1}{4}, . . .$ In which position of this sequence is $\frac{2005}{2004}$ ?

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , sequence

Show that: a) There is a sequence of non-zero natural numbers $a_1, a_2, ...$ uniquely determined, so that: $n = \sum _ {d | n} a _ d$ for whatever $n \in N ^ {*}$ . b) There is a sequence of non-zero natural numbers $b_1, b_2, ...$ uniquely determined, so that: $n = \prod _ {d | n} b _ d$ for whatever $n \in N ^ {*}$ . Note: The sum from a), respectively the product from b), are made after all the natural divisors $d$ of the number $n$ , including $1$ and $n$ .

2000 Saint Petersburg Mathematical Olympiad, 10.1

Tags: algebra , sequence

Sequences $x_1,x_2,\dots,$ and $y_1,y_2,\dots,$ are defined with $x_1=\dfrac{1}{8}$, $y_1=\dfrac{1}{10}$ and $x_{n+1}=x_n+x_n^2$, $y_{n+1}=y_n+y_n^2$. Prove that $x_m\neq y_n$ for all $m,n\in\mathbb{Z}^{+}$. [I]Proposed by A. Golovanov[/i]