Found problems: 963
2016 Taiwan TST Round 3, 2
Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions:
$(i)$ $a_0=a_n=1$;
$(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$;
$(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
$(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$.
Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.
2023 Estonia Team Selection Test, 5
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
2007 Denmark MO - Mohr Contest, 5
The sequence of numbers $a_0,a_1,a_2,...$ is determined by $a_0 = 0$, and
$$a_n= \begin{cases} 1+a_{n-1} \,\,\, when\,\,\, n \,\,\, is \,\,\, positive \,\,\, and \,\,\, odd \\
3a_{n/2} \,\,\,when \,\,\,n \,\,\,is \,\,\,positive \,\,\,and \,\,\,even\end{cases}$$
How many of these numbers are less than $2007$ ?
2021 Korea Junior Math Olympiad, 2
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
[list]
[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
[/list]
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.
2019 Nigerian Senior MO Round 4, 4
We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$
We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$.
Prove that for every $n>0, y_n$ is the square of an odd integer.
2018 German National Olympiad, 5
We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence.
1982 Brazil National Olympiad, 2
Any positive integer $n$ can be written in the form $n = 2^b(2c+1)$. We call $2c+1$ the[i] odd part[/i] of $n$. Given an odd integer $n > 0$, define the sequence $ a_0, a_1, a_2, ...$ as follows: $a_0 = 2^n-1, a_{k+1} $ is the [i]odd part[/i] of $3a_k+1$. Find $a_n$.
2021 Albanians Cup in Mathematics, 3
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
2009 Postal Coaching, 5
Define a sequence $<x_n>$ by $x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1$.
Consider the polynomial $P_n(x) = x_{n-1}x_{n+1} - x_n^2$, for each $n \ge 2$.
Prove or disprove that the coefficients of $P_n(x)$ are all non-negative, except for the constant term when $n$ is odd.
1989 IMO Shortlist, 16
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
2014 Federal Competition For Advanced Students, 3
Let $a_n$ be a sequence de fined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$.
Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.
2018 Costa Rica - Final Round, F2
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
$$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$
1996 Singapore Team Selection Test, 3
Let $S$ be a sequence $n_1, n_2,..., n_{1995}$ of positive integers such that $n_1 +...+ n_{1995 }=m < 3990$. Prove that for each integer $q$ with $1 \le q \le m$, there is a sequence $n_{i_1} , n_{i_2} , ... , n_{i_k}$ , where $1 \le i_1 < i_2 < ...< i_k \le 1995$, $n_{i_1} + ...+ n_{i_k} = q$ and $k$ depends on $q$.
1987 Greece National Olympiad, 3
There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$
2004 Gheorghe Vranceanu, 1
Define a finite sequence $ \left( s_i \right)_{1\le i\le 2004} $ with $ s_0+2=s_1+1=s_2=2 $ and the recurrence relation
$$ s_n=1+s_{n-1} +s_{n-2} -s_{n-3} . $$
Calculate its last element.
2012 Singapore Senior Math Olympiad, 4
Let $a_1, a_2, ..., a_n, a_{n+1}$ be a finite sequence of real numbers satisfying $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1$ for $k = 1, 2, ..., n$
Prove that for $k=0, 1, ..., n+1,$ $|a_k| \leq \frac{k(n+1-k)}{2}$
2012 Tournament of Towns, 1
Given an infinite sequence of numbers $a_1, a_2, a_3,...$ . For each positive integer $k$ there exists a positive integer $t = t(k)$ such that $a_k = a_{k+t} = a_{k+2t} =...$. Is this sequence necessarily periodic? That is, does a positive integer $T$ exist such that $a_k = a_{k+T}$ for each positive integer k?
2024 IFYM, Sozopol, 3
The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \).
a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \).
b) Does there exist a non-constant bounded sequence with the above properties?
(A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)
2001 Mongolian Mathematical Olympiad, Problem 1
Suppose that a sequence $x_1,x_2,\ldots,x_{2001}$ of positive real numbers satisfies
$$3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.$$Find the maximum possible value of $x_1$.
2021 Thailand TSTST, 3
A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called [i]quadratic[/i] if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$.
$\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$.
$\text{(ii)}$ Find the smallest positive integer $n$ for which there exists a quadratic sequence with $a_0=0$ and $a_n=2021$.
1992 IMO Shortlist, 18
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that
\[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\]
where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$
1996 French Mathematical Olympiad, Problem 2
Let $a$ be an odd natural number and $b$ be a positive integer. We define a sequence of reals $(u_n)$ as follows: $u_0=b$ and, for all $n\in\mathbb N_0$, $u_{n+1}$ is $\frac{u_n}2$ if $u_n$ is even and $a+u_n$ otherwise.
(a) Prove that one can find an element of $u_n$ smaller than $a$.
(b) Prove that the sequence is eventually periodic.
2025 Vietnam National Olympiad, 1
Let $P(x) = x^4-x^3+x$.
a) Prove that for all positive real numbers $a$, the polynomial $P(x) - a$ has a unique positive zero.
b) A sequence $(a_n)$ is defined by $a_1 = \dfrac{1}{3}$ and for all $n \geq 1$, $a_{n+1}$ is the positive zero of the polynomial $P(x) - a_n$. Prove that the sequence $(a_n)$ converges, and find the limit of the sequence.
2011 Germany Team Selection Test, 1
A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$
[i]Proposed by Gerhard Wöginger, Austria[/i]
1973 IMO Shortlist, 10
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] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$