Found problems: 963
Oliforum Contest I 2008, 1
Consider the sequence of integer such that:
$ a_1 = 2$
$ a_2 = 5$
$ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$
Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.
2021 Azerbaijan IMO 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]
2023 Assara - South Russian Girl's MO, 7
Given an increasing sequence of different natural numbers $a_1 < a_2 < a_3 < ... < a_n$ such that for any two distinct numbers in this sequence their sum is not divisible by $10$. It is known that $a_n = 2023$.
a) Can $n$ be greater than $800$?
b) What is the largest possible value of $n$?
c) For the value $n$ found in question b), find the number of such sequences with $a_n = 2023$.
2012 Grand Duchy of Lithuania, 4
Let $m$ be a positive integer. Find all bounded sequences of integers $a_1, a_2, a_3,... $for which $a_n + a_{n+1} + a_{n+m }= 0$ for all $n \in N$.
2021 South East Mathematical Olympiad, 1
A sequence $\{a_n\}$ is defined recursively by $a_1=\frac{1}{2}, $ and for $n\ge 2,$ $0<a_n\leq a_{n-1}$ and
\[a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0.\]
$(1)$ Determine the general formula of the sequence $\{a_n\};$
$(2)$ Let $S_n=a_1+\cdots+a_n.$ Prove that for $n\ge 1,$ $\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).$
1977 All Soviet Union Mathematical Olympiad, 239
Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.
2010 Saudi Arabia Pre-TST, 2.3
Let $a_0$ be a positive integer and $a_{n + 1} =\sqrt{a_n^2 + 1}$, for all $n \ge 0$.
1) Prove that for all $a_0$ the sequence contains infinitely many integers and infinitely many irrational numbers.
2) Is there an $a_0$ for which $a_{2010}$ is an integer?
1980 Austrian-Polish Competition, 2
A sequence of integers $1 = x_1 < x_2 < x_3 <...$ satisfies $x_{n+1} \le 2n$ for all $n$. Show that every positive integer $k$ can be written as $x_j -x_i$ for some $i, j$.
2020 Francophone Mathematical Olympiad, 2
Let $a_1,a_2,\ldots,a_n$ be a finite sequence of non negative integers, its subsequences are the sequences of the form $a_i,a_{i+1},\ldots,a_j$ with $1\le i\le j \le n$. Two subsequences are said to be equal if they have the same length and have the same terms, that is, two subsequences $a_i,a_{i+1},\ldots,a_j$ and $a_u,a_{u+1},\ldots a_v$ are equal iff $j-i=u-v$ and $a_{i+k}=a_{u+k}$ forall integers $k$ such that $0\le k\le j-1$. Finally, we say that a subsequence $a_i,a_{i+1},\ldots,a_j$ is palindromic if $a_{i+k}=a_{j-k}$ forall integers $k$ such that $0\le k \le j-i$
What is the greatest number of different palindromic subsequences that can a palindromic sequence of length $n$ contain?
2019 Federal Competition For Advanced Students, P1, 1
We consider the two sequences $(a_n)_{n\ge 0}$ and $(b_n) _{n\ge 0}$ of integers, which are given by $a_0 = b_0 = 2$ and $a_1= b_1 = 14$ and for $n\ge 2$ they are defined as
$a_n = 14a_{n-1} + a_{n-2}$ ,
$b_n = 6b_{n-1}-b_{n-2}$.
Determine whether there are infinite numbers that occur in both sequences
1971 IMO Shortlist, 9
Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and
\[T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).\]
Show that for all $k$,
\[1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],\]
where $[x]$ denotes the greatest integer not exceeding $x.$
1982 Czech and Slovak Olympiad III A, 5
Given is a sequence of real numbers $\{a_n\}^{\infty}_{n=1}$ such that $a_n \ne a_m$ for $n\ne m,$ given is a natural number $k$. Construct an injective map $P:\{1,2,\ldots,20k\}\to\mathbb Z^+$ such that the following inequalities hold:
$$a_{p(1)}<a_{p(2)}<...<a_{p(10)}$$
$$ a_{p(10)}>a_{p(11)}>...>a_{p(20)}$$
$$a_{p(20)}<a_{p(21)}<...<a_{p(30)}$$
$$...$$
$$a_{p(20k-10)}>a_{p(20k-9)}>...>a_{p(20k)}$$
$$a_{p(10)}>a_{p(30)}>...>a_{p((20k-10))} $$
$$a_{p(1)}<a_{p(20)}<...<a_{p(20k)},$$
VMEO IV 2015, 10.1
Where $n$ is a positive integer, the sequence $a_n$ is determined by the formula $$a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2, \,a_1 = 1.$$ Find the limit of the sequence $S_n$ defined by $S_n=a_1 + a_2 +... + a_n$.
2001 China Team Selection Test, 2
Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds:
$\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$
2000 Estonia National Olympiad, 4
Let us define the sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,...$. with the following conditions
$a_1 = 3, b_1 = 1$ and $a_{n +1} =\frac{a_n^2+b_n^2}{2}$ and $b_{n + 1}= a_n \cdot b_n$ for each $n = 1, 2,...$.
Find all different prime factors οf the number $a_{2000} + b_{2000}$.
1998 Slovenia Team Selection Test, 6
Let $a_0 = 1998$ and $a_{n+1} =\frac{a_n^2}{a_n +1}$ for each nonnegative integer $n$.
Prove that $[a_n] = 1994- n$ for $0 \le n \le 1000$
2013 Irish Math Olympiad, 9
We say that a doubly infinite sequence
$. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .$
is subaveraging if $s_n = (s_{n−1} + s_{n+1})/4$ for all integers n.
(a) Find a subaveraging sequence in which all entries are different from each other. Prove that all
entries are indeed distinct.
(b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such
that $s_m = s_n$, then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .
Mathematical Minds 2023, P7
Does there exist an increasing sequence of positive integers for which any large enough integer can be expressed uniquely as the sum of two (possibly equal) terms of the sequence?
[i]Proposed by Vlad Spătaru and David Anghel[/i]
1967 IMO Shortlist, 1
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let
\[ c_n = \sum^8_{k=1} a^n_k\]
for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$
2018 Vietnam National Olympiad, 1
The sequence $(x_n)$ is defined as follows:
$$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$
for all $n\geq 1$.
a. Prove that $(x_n)$ has a finite limit and find that limit.
b. For every $n\geq 1$, prove that
$$n\leq x_1+x_2+\dots +x_n\leq n+1.$$
2009 Belarus Team Selection Test, 2
a) Prove that there is not an infinte sequence $(x_n)$, $n=1,2,...$ of positive real numbers satisfying the relation
$x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}$, $\forall n \in N$ (*)
b) Do there exist sequences satisfying (*) and containing arbitrary many terms?
I.Voronovich
2014 Balkan MO Shortlist, A5
$\boxed{A5}$Let $n\in{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$Prove that
\[(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3\]
2022 Macedonian Mathematical Olympiad, Problem 1
Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$
[i]Proposed by Nikola Velov[/i]
2010 Saudi Arabia IMO TST, 2
a) Prove that for each positive integer $n$ there is a unique positive integer $a_n$ such that $$(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . $$
b) Prove that $a_{2010}$ is divisible by $5\cdot 4^{2009}$ and find the quotient
2012 Brazil Team Selection Test, 3
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]