Found problems: 1239
2018 Saudi Arabia IMO TST, 1
Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that
i. All terms of sequences are pairwise coprime.
ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded.
Prove that this sequence contains infinitely many primes.
KoMaL A Problems 2017/2018, A. 713
We say that a sequence $a_1,a_2,\cdots$ is [i]expansive[/i] if for all positive integers $j,\; i<j$ implies $|a_i-a_j|\ge \tfrac 1j$. Find all positive real numbers $C$ for which one can find an expansive sequence in the interval $[0,C]$.
2008 Bosnia And Herzegovina - Regional Olympiad, 4
Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that:
$i)$ $a(0)=0$
$ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$.
If exists prove:
$a)$ $a(k)\geq a(k-1)$
$b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.
2024 Korea Summer Program Practice Test, 6
Does there exist a real sequence $\{a_n\}_{n=1}^\infty$ such that
$$a_na_{n+1}\ge a_{n+2}^2 +1$$
for all $n\ge 1$?
2017 IMO Shortlist, N1
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.
[i]Proposed by Stephan Wagner, South Africa[/i]
1998 Belarus Team Selection Test, 3
Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then
\[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\]
For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.
2013 Saudi Arabia Pre-TST, 4.1
Let $a_1,a_2, a_3,...$ be a sequence of real numbers which satisfy the relation $a_{n+1} =\sqrt{a_n^2 + 1}$
Suppose that there exists a positive integer $n_0$ such that $a_{2n_0} = 3a_{n_0}$ . Find the value of $a_{46}$.
2017 Serbia Team Selection Test, 5
Let $n \geq 2$ be a positive integer and $\{x_i\}_{i=0}^n$ a sequence such that not all of its elements are zero and there is a positive constant $C_n$ for which:
(i) $x_1+ \dots +x_n=0$, and
(ii) for each $i$ either $x_i\leq x_{i+1}$ or $x_i\leq x_{i+1} + C_n x_{i+2}$ (all indexes are assumed modulo $n$).
Prove that
a) $C_n\geq 2$, and
b) $C_n=2$ if and only $2 \mid n$.
1988 All Soviet Union Mathematical Olympiad, 466
Given a sequence of $19$ positive integers not exceeding $88$ and another sequence of $88$ positive integers not exceeding $19$. Show that we can find two subsequences of consecutive terms, one from each sequence, with the same sum.
2015 Silk Road, 3
Let $B_n$ be the set of all sequences of length $n$, consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$
\varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1).
$$. Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $.
.
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 Australian Maths Olympiad, 5
Each term of an infinite sequence $a_1 ,a_2 ,a_3 , \dots$ is equal to 0 or 1. For each positive integer $n$,
$$a_n + a_{n+1} \neq a_{n+2} + a_{n+3},\, \text{and}$$
$$a_n + a_{n+1} + a_{n+2} \neq a_{n+3} + a_{n+4} + a_{n+5}.$$
Prove that if $a_1 = 0$, then $a_{2020} = 1$.
2010 SEEMOUS, Problem 1
Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by
$$f_n(x)=\int^x_0f_{n-1}(t)dt$$
for all integers $n\ge1$.
a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$.
b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.
1994 IMO Shortlist, 4
Define the sequences $ a_n, b_n, c_n$ as follows. $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$.
If $ a_n$ is even then $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$, $ b_{n \plus{} 1} \equal{} 2b_n$, $ c_{n \plus{} 1} \equal{} c_n$.
If $ a_n$ is odd, then $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$, $ b_{n \plus{} 1} \equal{} b_n$, $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$.
Find the number of positive integers $ k < 1995$ such that some $ a_n \equal{} 0$.
2021 Science ON grade IX, 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]
1981 IMO Shortlist, 4
Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $
(a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence.
(b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.
1992 IMO Longlists, 49
Given real numbers $x_i \ (i = 1, 2, \cdots, 4k + 2)$ such that
\[\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ )\]
prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that
\[\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )\]
2019 Serbia National MO, 6
Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations :
$$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and
$$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$
Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$
1969 Bulgaria National Olympiad, Problem 2
Prove that
$$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.
1995 Belarus Team Selection Test, 3
Show that there is no infinite sequence an of natural numbers such that \[a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2\] for all $n\geq 2$
2020 Canadian Junior Mathematical Olympiad, 1
Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.
1999 Singapore Senior Math Olympiad, 3
Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence
$a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence
$ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.
2023 India EGMO TST, P5
Let $k$ be a positive integer. A sequence of integers $a_1, a_2, \cdots$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \cdots , a_{n+k} \}$. Determine, as a function of $k$, how many $k$-pop sequences there are.
[i]Proposed by Sutanay Bhattacharya[/i]
2021 Olimphíada, 1
The sequence of reals $a_1, a_2, a_3, \ldots$ is defined recursively by the recurrence:
$$\dfrac{a_{n+1}}{a_n} - 3 = a_n(a_n - 3)$$ Given that $a_{2021} = 2021$, find $a_1$.
1987 ITAMO, 5
Let $a_1,a_2,...$ and $b_1,b_2,..$. be two arbitrary infinite sequences of natural numbers.
Prove that there exist different indices $r$ and $s$ such that $a_r \ge a_s$ and $b_r \ge b_s$.