Found problems: 963
1946 Moscow Mathematical Olympiad, 121
Given the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, ... ,$ ascertain whether among its first $(10^8+1)$ terms there is a number that ends with four zeros.
1988 IMO Longlists, 1
An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.
1978 IMO Shortlist, 9
Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$
1977 IMO Longlists, 57
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.
2005 Estonia National Olympiad, 4
A sequence of natural numbers $a_1, a_2, a_3,..$ is called [i]periodic modulo[/i] $n$ if there exists a positive integer $k$ such that, for any positive integer $i$, the terms $a_i$ and $a_{i+k}$ are equal modulo $n$. Does there exist a strictly increasing sequence of natural numbers that
a) is not periodic modulo finitely many positive integers and is periodic modulo all the other positive integers?
b) is not periodic modulo infinitely many positive integers and is periodic modulo infinitely many positive integers?
2000 IMO Shortlist, 6
A nonempty set $ A$ of real numbers is called a $ B_3$-set if the conditions $ a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $ a_1 \plus{} a_2 \plus{} a_3 \equal{} a_4 \plus{} a_5 \plus{} a_6$ imply that the sequences $ (a_1, a_2, a_3)$ and $ (a_4, a_5, a_6)$ are identical up to a permutation. Let $A = \{a_0 = 0 < a_1 < a_2 < \cdots \}$, $B = \{b_0 = 0 < b_1 < b_2 < \cdots \}$ be infinite sequences of real numbers with $ D(A) \equal{} D(B),$ where, for a set $ X$ of real numbers, $ D(X)$ denotes the difference set $ \{|x\minus{}y|\mid x, y \in X \}.$ Prove that if $ A$ is a $ B_3$-set, then $ A \equal{} B.$
1982 All Soviet Union Mathematical Olympiad, 344
Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions:
a) not a triple of successive members is chosen,
b) at least one of every triple of successive members is chosen,
c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.
1982 IMO Shortlist, 11
[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes
\[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\]
[b](b)[/b] Find the rearrangement that minimizes $Q.$
1977 IMO Longlists, 28
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
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
2019 Moldova EGMO TST, 8
The sequence $(a_n)_{n\geq1}$ is defined as:
$$a_1=2, a_2=20, a_3=56, a_{n+3}=7a_{n+2}-11a_{n+1}+5a_n-3\cdot2^n.$$ Prove that $a_n$ is positive for every positive integer $n{}$. Find the remainder of the divison of $a_{673}$ to $673$.
1989 Czech And Slovak Olympiad IIIA, 6
Consider a finite sequence $a_1, a_2,...,a_n$ whose terms are natural numbers at most equal to $n$. Determine the maximum number of terms of such a sequence, if you know that every two of its neighboring terms are different and at the same time there is no quartet of terms in it such that $a_p = a_r \ne a_q = a_s$ for $p < q < r < s$.
2018 China Team Selection Test, 1
Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$
2021 Thailand Online MO, P3
Let $a_1,a_2,\cdots$ be an infinity sequence of positive integers such that $a_1=2021$ and
$$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$$
for all positive integers $n$. Prove that for any integer $n\ge 2$, $a_n$ is the product of at least $2n$ (not necessarily distinct) primes.
1962 Dutch Mathematical Olympiad, 2
The $n^{th}$ term of a sequence is $t_n$. For $n \ge 1$, $t_n$ is given by the relation:
$$t_n= n^3+\frac12 n^2+ \frac13 n + \frac14$$
The $n^{th}$ term of a second sequence $T_n$, where $T_n$ represents the smallest integer greater than $t_n$. Calculate: $$(T_1+T_2+...+T_{1014}) -(t_1+t_2+...+t_{1014}) $$
1968 All Soviet Union Mathematical Olympiad, 109
Two finite sequences $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ are just rearranged sequence $1, 1/2, ... , 1/n$ with $$a_1+b_1\ge a_2+b_2\ge...\ge a_n+b_n.$$ Prove that $a_m+a_n\ge 4/m$ for every $m$ ($1\le m\le n$) .
2021 Taiwan TST Round 1, N
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]
2021 Science ON all problems, 1
Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$.
$\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$.
$\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that
$$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$
Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$.
[i] (Stefan Obadă)[/i]
2024 Bundeswettbewerb Mathematik, 2
Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties:
(1) No number occurs more than once in the sequence.
(2) The sum of two different elements of the sequence is never a power of two.
(3) For all positive integers $n$, we have $a_n<r \cdot n$.
2017 IFYM, Sozopol, 6
The sequence $a_1,a_2…$ , is defined by the equations $a_1=1$ and $a_n=n.a_{[n/2]}$ for $n>1$. Prove that $a_n>n^2$ for $n>11$.
1978 All Soviet Union Mathematical Olympiad, 268
Consider a sequence $$x_n=(1+\sqrt2+\sqrt3)^n$$ Each member can be represented as $$x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$$ where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.
2021 Belarusian National Olympiad, 10.1
An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$
Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$
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 $a_{2019} < \frac12 <a_{2018}$.
2005 Morocco TST, 3
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.
1989 IMO Shortlist, 30
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
2022 Bosnia and Herzegovina BMO TST, 1
Let $a_1,a_2,a_3, \ldots$ be an infinite sequence of nonnegative real numbers such that for all positive integers $k$ the following conditions hold:
$i)$ $a_k-2a_{k+1}+a_{k+2} \geq 0$;
$ii)$ $\sum_{j=1}^{k} a_j \leq 1$.
Prove that for all positive integer $k$ holds: $0 \leq a_k - a_{k+1} < \frac{2}{k^2}$