Found problems: 963
1969 IMO Longlists, 28
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
2000 China Second Round Olympiad, 2
Define the sequence $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ as $a_0=1,a_1=4,a_2=49$ and for $n \geq 0$
$$
\begin{cases}
a_{n+1}=7a_n+6b_n-3, \\
b_{n+1}=8a_n+7b_n-4.
\end{cases}
$$
Prove that for any non-negative integer $n,$ $a_n$ is a perfect square.
1971 Dutch Mathematical Olympiad, 2
A sequence of real numbers is called a [i]Fibonacci [/i] sequence if $$t_{n+2} = t_{n+1} + t_n$$ for $n= 1,2,3,. .$ .
Two Fibonacci sequences are said to be [i]essentially different[/i] if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences $1,2,3,5,8,...$ and $1,3,4,7,11,...$ are essentially different, but the sequences $1,2,3,5,8,...$ and $2,4,6,10,16,...$ are not.
(a) Prove that there exist real numbers $p$ and $q$ such that the sequences $1,p,p^2,p^3,...$ and $1,q,q^2,q^3,...$ are essentially different Fibonacci sequences.
(b) Let $a_1,a_2,a_3,...$ and $b_1,b_2,b_3,...$ be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence $t_1,t_2,t_3,...$, there exists exactly one number $\alpha$ and exactly one number $\beta$, such that: $$t_n = \alpha a_n + \beta b_n$$ for $n = 1,2,3,...$
(c) $t_1,t_2,t_3,...$, is the Fibonacci sequence with $t_1 = 1$ and $t_2= 2$. Express $t_n$ in terms of $n$.
2012 German National Olympiad, 1
Define a sequence $(a_n)$ by $a_0 =-4 , a_1 =-7$ and $a_{n+2}= 5a_{n+1} -6a_n$ for $n\geq 0.$ Prove that there are infinitely many positive integers $n$ such that $a_n$ is composite.
2017 Purple Comet Problems, 16
Let $a_1 = 1 +\sqrt2$ and for each $n \ge 1$ de ne $a_{n+1} = 2 -\frac{1}{a_n}$. Find the greatest integer less than or equal to the product $a_1a_2a_3 ... a_{200}$.
1995 North Macedonia National Olympiad, 1
Let $ a_0 $ be a real number. The sequence $ \{a_n \} $ is given by $ a_ {n + 1} = 3 ^ n-5a_n $, $ n = 0,1,2, \ldots $.
a) Express the general member $ a_n $ through $ a_0 $ and $ n. $
b) Find such $ a_0, $ that $ a_ {n + 1}> a_n, $ for every $ n. $
1996 Akdeniz University MO, 2
Let $u_1=1,u_2=1$ and for all $k \geq 1$'s
$$u_{k+2}=u_{k+1}+u_{k}$$
Prove that for all $m \geq 1$'s $5$ divides $u_{5m}$
1980 IMO Longlists, 2
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way:
\[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \]
Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]
1963 Putnam, A4
Let $(a_n)$ be a sequence of positive real numbers. Show that
$$ \limsup_{n \to \infty} n \left(\frac{1 +a_{n+1}}{a_n } -1 \right) \geq 1$$
and prove that $1$ cannot be replaced by any larger number.
VII Soros Olympiad 2000 - 01, 11.7
Consider all possible functions defined for $x = 1, 2, ..., M$ and taking values $y = 1, 2, ..., n$. We denote the set of such functions by $T.$ By $T_0$ we denote the subset of $T$ consisting of functions whose value changes exactly by $ 1$ (in one direction or another) when the argument changes by $1$. Prove that if $M\ge 2n-4$, then among the functions from of the set $T$, there is a function that coincides at least at one point with any function from $T_0$. Specify at least one such function. Prove that if $M <2n-4$, then there is no such function.
2002 Estonia National Olympiad, 5
The teacher writes numbers $1$ at both ends of the blackboard. The first student adds a $2$ in the middle between them, each next student adds the sum of each two adjacent numbers already on the blackboard between them (hence there are numbers $1, 3, 2, 3, 1$ on the blackboard after the second student, $1, 4, 3, 5, 2, 5, 3, 4, 1$ after the third student etc.) Find the sum of all numbers on the blackboard after the $n$-th student.
1980 IMO, 1
Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.
2016 Thailand TSTST, 1
Let $a_1, a_2, a_3, \dots$ be a sequence of integers such that
$\text{(i)}$ $a_1=0$
$\text{(ii)}$ for all $i\geq 1$, $a_{i+1}=a_i+1$ or $-a_i-1$.
Prove that $\frac{a_1+a_2+\cdots+a_n}{n}\geq-\frac{1}{2}$ for all $n\geq 1$.
1983 Tournament Of Towns, (047) 4
$a_1,a_2,a_3,...$ is a monotone increasing sequence of natural numbers. It is known that for any $k, a_{a_k} = 3k$.
a) Find $a_{100}$.
b) Find $a_{1983}$.
(A Andjans, Riga)
PS. (a) for Juniors, (b) for Seniors
2020 Kyiv Mathematical Festival, 1.1
(a) Find the numbers $a_0,. . . , a_{100}$, such that $a_0 = 0, a_{100} = 1$ and for all $k = 1,. . . , 99$ :
$$a_k = \frac12 a_{k- 1} + \frac12 a_{k+1 }$$
(b) Find the numbers $a_0,. . . , a_{100}$, such that $a_0 = 0, a_{100} = 1$ and for all $k = 1,. . . , 99$ :
$$a_k = 1+\frac12 a_{k- 1} + \frac12 a_{k+1 }$$.
1989 IMO Longlists, 55
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 India IMO Training Camp, 3
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
2017 Bosnia and Herzegovina EGMO TST, 1
It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$. From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are same as first $k$ elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element $1$.
2024 Brazil EGMO TST, 4
The infinite sequence \( a_1, a_2, \ldots \) is defined by \( a_1 = 1 \) and, for each \( n \geq 1 \), the number \( a_{n+1} \) is the smallest positive integer greater than \( a_n \) that has the following property: for each \( k \in \{1, 2, \ldots, n\} \), the number \( a_{n+1} + a_k \) is not a perfect square. Prove that, for all \( n \), it holds that \( a_n \leq (n - 1)^2 + 1 \).
2005 Taiwan TST Round 3, 1
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.
Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?
[i]Proposed by Mihai Bălună, Romania[/i]
2024 IFYM, Sozopol, 3
Let $(a_n)_{n\geq 1}$ be a (not necessarily strictly) increasing sequence of positive integers, such that $a_n \leq 1000n^{0.999}$ for every positive integer $n$. Prove that there exist infinitely many positive integers $n$ for which $a_n$ divides $n$.
2014 Contests, A3
$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$
2016 Dutch Mathematical Olympiad, 2
For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$. The [i]sum product value[/i] of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together.
For example, if we take $n = 2$ and the sequence $0,1, 1, -1$, then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$. Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$. The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$, which equals $0$.
Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have.
[i]Attention: you are required to prove that a smaller sum product value is impossible.[/i]
1998 Moldova Team Selection Test, 10
Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.
2025 Bulgarian Winter Tournament, 11.4
Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties:
1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$.
2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct.
Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.