Found problems: 963
2020 Regional Olympiad of Mexico Northeast, 1
Let $a_1=2020$ and let $a_{n+1}=\sqrt{2020+a_n}$ for $n\ge 1$. How much is $\left\lfloor a_{2020}\right\rfloor$?
Note: $\lfloor x\rfloor$ denotes the integer part of a number, that is that is, the immediate integer less than $x$. For example, $\lfloor 2.71\rfloor=2$ and $\lfloor \pi\rfloor=3$.
2017 Kazakhstan NMO, Problem 3
An infinite, strictly increasing sequence $\{a_n\}$ of positive integers satisfies the condition $a_{a_n}\le a_n + a_{n + 3}$ for all $n\ge 1$. Prove that there are infinitely many triples $(k, l, m)$ of positive integers such that $k <l <m$ and $a_k + a_m = 2a_l$.
2015 Bulgaria National Olympiad, 3
The sequence $a_1, a_2,...$ is de fined by the equalities $a_1 = 2, a_2 = 12$ and $a_{n+1} = 6a_n-a_{n-1}$ for every positive integer $n \ge 2$. Prove that no member of this sequence is equal to a perfect power (greater than one) of a positive integer.
2016 AIME Problems, 10
A strictly increasing sequence of positive integers $a_1, a_2, a_3, \ldots$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$. Find $a_1$.
2002 Croatia National Olympiad, Problem 4
Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.
2025 Alborz Mathematical Olympiad, P3
For every positive integer \( n \), do there exist pairwise distinct positive integers \( a_1, a_2, \dots, a_n \) that satisfy the following condition?
For every \( 3 \leq m \leq n \), there exists an \( i \leq m-2 \) such that:
$$
a_m = a_{\gcd(m-1, i)} + \gcd(a_{m-1}, a_i).
$$
Proposed by Alireza Jannati
2005 Czech And Slovak Olympiad III A, 1
Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.
1984 Polish MO Finals, 4
A coin is tossed $n$ times, and the outcome is written in the form ($a_1,a_2,...,a_n$), where $a_i = 1$ or $2$ depending on whether the result of the $i$-th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$, and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$. Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$.
2020 June Advanced Contest, 4
Let \(c\) be a positive real number. Alice wishes to pick an integer \(n\) and a sequence \(a_1\), \(a_2\), \(\ldots\) of distinct positive integers such that \(a_{i} \leq ci\) for all positive integers \(i\) and \[n, \qquad n + a_1, \qquad n + a_1 - a_2, \qquad n + a_1 - a_2 + a_3, \qquad \cdots\] is a sequence of distinct nonnegative numbers. Find all \(c\) such that Alice can fulfil her wish.
1992 IMO Longlists, 18
Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$. Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.
2000 Moldova National Olympiad, Problem 7
The Fibonacci sequence is defined by $F_0=F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge0$. Prove that the sum of $2000$ consecutive terms of the Fibonacci sequence is never a term of the sequence.
2008 IMO Shortlist, 2
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
1978 Czech and Slovak Olympiad III A, 6
Show that the number
\[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\]
is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$
2021 Thailand TST, 1
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
2014 IMO Shortlist, N4
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
1980 Brazil National Olympiad, 3
Given a triangle $ABC$ and a point $P_0$ on the side $AB$. Construct points $P_i, Q_i, R_i $ as follows. $Q_i$ is the foot of the perpendicular from $P_i$ to $BC, R_i$ is the foot of the perpendicular from $Q_i$ to $AC$ and $P_i$ is the foot of the perpendicular from $R_{i-1}$ to $AB$. Show that the points $P_i$ converge to a point $P$ on $AB$ and show how to construct $P$.
1984 IMO Shortlist, 6
Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows:
\[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\]
Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$
2022 Thailand Mathematical Olympiad, 7
Let $d \geq 2$ be a positive integer. Define the sequence $a_1,a_2,\dots$ by
$$a_1=1 \ \text{and} \ a_{n+1}=a_n^d+1 \ \text{for all }n\geq 1.$$
Determine all pairs of positive integers $(p, q)$ such that $a_p$ divides $a_q$.
1999 Brazil Team Selection Test, Problem 3
A sequence $a_n$ is defined by
$$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.
2017 China Second Round Olympiad, 2
Given a sequence $\{a_n\}$: $a_1=1, a_{n+1}=\left\{
\begin{array}{lcr}
a_n+n,\quad a_n\le n, \\
a_n-n,\quad a_n>n,
\end{array}
\right.
\quad n=1,2,\cdots.$
Find the number of positive integers $r$ satisfying $a_r<r\le 3^{2017}$.
2009 Postal Coaching, 1
Let $a_1, a_2, a_3, . . . , a_n, . . . $ be an infinite sequence of natural numbers in which $a_1$ is not divisible by $5$. Suppose $a_{n+1} = a_n + b_n$ where bn is the last digit of $a_n$, for every $n$. Prove that the sequence $\{a_n\}$ contains infinitely many powers of 2.
2011 Dutch IMO TST, 4
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$:
$p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.
2015 Dutch IMO TST, 3
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
2023 South East Mathematical Olympiad, 1
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 $S_{2023}$.
1978 Germany Team Selection Test, 4
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.