Found problems: 963
2024 Romania National Olympiad, 4
Let $a$ be a given positive integer. We consider the sequence $(x_n)_{n \ge 1}$ defined by $x_n=\frac{1}{1+na},$ for every positive integer $n.$
Prove that for any integer $k \ge 3,$ there exist positive integers $n_1<n_2<\ldots<n_k$ such that the numbers $x_{n_1},x_{n_2},\ldots,x_{n_k}$ are consecutive terms in an arithmetic progression.
2013 Nordic, 3
Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$
2010 Bundeswettbewerb Mathematik, 2
The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.
2017 Korea National Olympiad, problem 4
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function as
\[ f(x) = \begin{cases} \frac{1}{x-1}& (x > 1)\\ 1& (x=1)\\ \frac{x}{1-x} & (x<1) \end{cases} \]
Let $x_1$ be a positive irrational number which is a zero of a quadratic polynomial with integer coefficients. For every positive integer $n$, let $x_{n+1} = f(x_n)$. Prove that there exists different positive integers $k$ and $\ell$ such that $x_k = x_\ell$.
2024-IMOC, A3
Find all infinite integer sequences $a_1,a_2,\ldots$ satisfying
\[a_{n+2}^{a_{n+1}}=a_{n+1}+a_n\] holds for all $n\geq 1$. Define $0^0=1$
2013 Junior Balkan Team Selection Tests - Romania, 4
For any sequence ($a_1,a_2,...,a_{2013}$) of integers, we call a triple ($i,j, k$) satisfying $1 \le i < j < k \le 2013$ to be [i]progressive [/i] if $a_k-a_j = a_j -a_i = 1$. Determine the maximum number of progressive triples that a sequence of $2013$ integers could have.
2012 Balkan MO Shortlist, N2
Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$.
Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.
1980 IMO Longlists, 13
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.
2020 LIMIT Category 2, 4
Define the sequence $\{a_n\}_{n\geq 1}$ as $a_n=n-1$, $n\leq 2$ and $a_n=$ remainder left by $a_{n-1}+a_{n-2}$ when divided by $3$ $\forall n\geq 2$. Then $\sum_{i=2018}^{2025}a_i=$?
(A)$6$
(B)$7$
(C)$8$
(D)$9$
1989 IMO Longlists, 93
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
2020 Australian Maths Olympiad, 4
Define the sequence $A_1, A_2, A_3, \dots$ by $A_1 = 1$ and for $n=1,2,3,\dots$
$$A_{n+1}=\frac{A_n+2}{A_n +1}.$$
Define the sequences $B_1, B_2, B_3,\dots$ by $B_1=1$ and for $n=1,2,3,\dots$
$$B_{n+1}=\frac{B_n^2 +2}{2B_n}.$$
Prove that $B_{n+1}=A_{2^n}$ for all non-negative integers $n$.
2020 Paraguay Mathematical Olympiad, 5
The general term of a sequence of numbers is defined as $a_n =\frac{1}{n^2 - n}$, for every integer $n \ge 3$.
That is, $a_3 =\frac16$, $a_4 =\frac{1}{12}$, $a_5 =\frac{1}{20}$, and so on.
Find a general expression for the sum $S_n$, which is the sum of all terms from $a_3$ until $a_n$.
1999 Tournament Of Towns, 5
For every non-negative integer $i$, define the number $M(i)$ as follows:
write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ )
(a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $.
Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ .
(b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ .
Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$.
(A Kanel)
2018 India IMO Training Camp, 3
Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$.
Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.
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 $a_{2022}\leq 1$.
2004 Federal Competition For Advanced Students, P2, 4
Show that there is an infinite sequence $a_1,a_2,...$ of natural numbers such that $a^2_1+a^2_2+ ...+a^2_N$ is a perfect square for all $N$. Give a recurrent formula for one such sequence.
2014 IMO, 1
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
[i]Proposed by Gerhard Wöginger, Austria.[/i]
1972 Czech and Slovak Olympiad III A, 3
Consider a sequence of polynomials such that $P_0(x)=2,P_1(x)=x$ and for all $n\ge1$ \[P_{n+1}(x)+P_{n-1}(x)=xP_n(x).\]
a) Determine the polynomial \[Q_n(x)=P^2_n(x)-xP_n(x)P_{n-1}(x)+P^2_{n-1}(x)\] for $n=1972.$
b) Express the polynomial \[\bigl(P_{n+1}(x)-P_{n-1}(x)\bigr)^2\] in terms of $P_n(x),Q_n(x).$
2014 German National Olympiad, 2
For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$
1983 IMO Longlists, 9
Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.
2002 Switzerland Team Selection Test, 6
A sequence $x_1,x_2,x_3,...$ has the following properties:
(a) $1 = x_1 < x_2 < x_3 < ...$
(b) $x_{n+1} \le 2n$ for all $n \in N$.
Prove that for each positive integer $k$ there exist indices $i$ and $j$ such that $k =x_i -x_j$.
1991 IMO Shortlist, 13
Given any integer $ n \geq 2,$ assume that the integers $ a_1, a_2, \ldots, a_n$ are not divisible by $ n$ and, moreover, that $ n$ does not divide $ \sum^n_{i\equal{}1} a_i.$ Prove that there exist at least $ n$ different sequences $ (e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $ \sum^n_{i\equal{}1} e_i \cdot a_i$ is divisible by $ n.$
1973 Dutch Mathematical Olympiad, 5
An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ .
(a) Prove that there are infinitely many numbers in the sequence equal to $0$.
(b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.
2018 Vietnam National Olympiad, 6
The sequence $(x_n)$ is defined as follows:
$$x_0=2,\, x_1=1,\, x_{n+2}=x_{n+1}+x_n$$
for every non-negative integer $n$.
a. For each $n\geq 1$, prove that $x_n$ is a prime number only if $n$ is a prime number or $n$ has no odd prime divisors
b. Find all non-negative pairs of integers $(m,n)$ such that $x_m|x_n$.
1983 IMO Shortlist, 5
Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.