Found problems: 307
2015 Peru MO (ONEM), 4
Let $b$ be an odd positive integer. The sequence $a_1, a_2, a_3, a_4$, is definedin the next way: $a_1$ and $a_2$ are positive integers and for all $k \ge 2$,
$$a_{k+1}= \begin{cases} \frac{a_k + a_{k-1}}{2} \,\,\, if \,\,\, a_k + a_{k-1} \,\,\, is \,\,\, even \\ \frac{a_k + a_{k-1+b}}{2}\,\,\, if \,\,\, a_k + a_{k-1}\,\,\, is \,\,\,odd\end{cases}$$
a) Prove that if $b = 1$, then after a certain term, the sequence will become constant.
b) For each $b \ge 3$ (odd), prove that there exist values of $a_1$ and $a_2$ for which the sequence will become constant after a certain term.
1985 IMO Shortlist, 6
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
2019 Peru EGMO TST, 5
Define the sequence sequence $a_0,a_1, a_2,....,a_{2018}, a_{2019}$ of real numbers as follows:
$\bullet$ $a_0 = 1$.
$\bullet$ $a_{n + 1} = a_n - \frac{a_n^2}{2019}$ for $n = 0, 1, ...,2018$.
Prove that $a_{2019} < \frac12 <a_{2018}$.
1993 Austrian-Polish Competition, 7
The sequence $(a_n)$ is defined by $a_0 = 0$ and $a_{n+1} = [\sqrt[3]{a_n +n}]^3$ for $n \ge 0$.
(a) Find $a_n$ in terms of $n$.
(b) Find all $n$ for which $a_n = n$.
2020 Tournament Of Towns, 4
For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers.
Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one.
R. Salimov
1976 IMO, 2
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
2022 China Northern MO, 3
Let $\{a_n\}$ be a sequence of positive terms such that $a_{n+1}=a_n+ \frac{n^2}{a_n}$ . Let $b_n =a_n-n$ .
(1) Are there infinitely many $n$ such that $b_n \ge 0$ ?
(2) Prove that there is a positive number $M$ such that $\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M$.
2019 Miklós Schweitzer, 6
Let $d$ be a positive integer and $1 < a \le (d+2)/(d+1)$. For given $x_0, x_1,\dots, x_d \in (0, a-1)$, let $x_{k+1} = x_k (a - x_{k-d})$, $k \ge d$. Prove that $\lim_{k \to \infty} x_k = a-1$.
1998 German National Olympiad, 5
A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$.
Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.
2001 All-Russian Olympiad Regional Round, 11.5
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.
1976 IMO Longlists, 3
Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
\[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\]
Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
1990 IMO Shortlist, 13
An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$
1978 Germany Team Selection Test, 3
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
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
1973 Dutch Mathematical Olympiad, 4
We have an infinite sequence of real numbers $x_0,x_1, x_2, ... $ such that $x_{n+1} = \sqrt{x_n -\frac14}$ holds for all natural $n$ and moreover $x_0 \in \frac12$.
(a) Prove that for every natural $n$ holds: $x_n > \frac12$
(b) Prove that $\lim_{n \to \infty} x_n$ exists. Calculate this limit.
2017 Grand Duchy of Lithuania, 1
The infinite sequence $a_0, a_1, a_2, a_3,... $ is defined by $a_0 = 2$ and
$$a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}$$ , $n = 1, 2, 3, ...$ Prove that $1 < a_n < 1 + \frac{1}{3^n}$ for all $n = 1, 2, 3, . .$
2003 Olympic Revenge, 2
Let $x_n$ the sequence defined by any nonnegatine integer $x_0$ and $x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}$
Show that there exists prime $p$ such that $p\not|x_n$ for any $n$.
2021 Francophone Mathematical Olympiad, 1
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 $n \ge 1$.
2018 Irish Math Olympiad, 9
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$.
Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
Oliforum Contest I 2008, 1
Consider the sequence of integer such that:
$ a_1 = 2$
$ a_2 = 5$
$ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$
Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.
2016 Brazil Undergrad MO, 3
Let it \(k \geq 1\) be an integer. Define the sequence \((a_n)_{n \geq 1}\) by \(a_0=0,a_1=1\) and
\[ a_{n+2} = ka_{n+1}+a_n \]
for \(n \geq 0\).
Let it \(p\) an odd prime number.
Denote \(m(p)\) as the smallest positive integer \(m\) such that \(p | a_m\).
Denote \(T(p)\) as the smallest positive integer \(T\) such that for every natural \(j\) we gave \(p | (a_{T+j}-a_j)\).
[list='i']
[*] Show that \(T(p) \leq (p-1) \cdot m(p)\).
[*] Show that if \(T(p) = (p-1) \cdot m(p)\) then
\[ \prod_{1 \leq j \leq T(p)-1}^{j \not \equiv 0 \pmod{m(p)}}{a_j} \equiv (-1)^{m(p)-1} \pmod{p} \]
[/list]
1992 Tournament Of Towns, (348) 6
Consider the sequence $a(n)$ defined by the following conditions: $$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$
Prove that the sequence contains an infinite number of perfect squares. (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.)
(A Andjans)
1986 ITAMO, 2
Determine the general term of the sequence ($a_n$) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$
.
1969 IMO Longlists, 41
$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system \[x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots\]
2000 All-Russian Olympiad Regional Round, 10.6
Given a natural number $a_0$, we construct the sequence $\{a_n\}$ as follows $a_{n+1} = a^2_n-5$ if $a_n$ is odd, and $\frac{a_n}{2}$ if $a_n$ is even. Prove that for any odd $a_0 > 5$ in the sequence $\{a_n\}$ arbitrarily large numbers will occur.
2018 Peru IMO TST, 5
Let $d$ be a positive integer.
The seqeunce $a_1, a_2, a_3,...$ of positive integers is defined by $a_1 = 1$ and $a_{n + 1} = n\left \lfloor \frac{a_n}{n} \right \rfloor+ d$ for $n = 1,2,3, ...$ .
Prove that there exists a positive integer $N$ so that the terms $a_N,a_{N + 1}, a_{N + 2},...$ form an arithmetic progression.
Note: If $x$ is a real number, $\left \lfloor x \right \rfloor $ denotes the largest integer that is less than or equal to $x$.