Found problems: 963
2017 International Zhautykov Olympiad, 1
Let $(a_n)$ be sequnce of positive integers such that first $k$ members $a_1,a_2,...,a_k$ are distinct positive integers, and for each $n>k$, number $a_n$ is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers $a_1,a_2,...,a_{n-1}$. Prove that $a_n=2a_{n-1}$ for all sufficently large $n$.
2024 Brazil National Olympiad, 1
Let \( a_1 \) be an integer greater than or equal to 2. Consider the sequence such that its first term is \( a_1 \), and for \( a_n \), the \( n \)-th term of the sequence, we have
\[
a_{n+1} = \frac{a_n}{p_k^{e_k - 1}} + 1,
\]
where \( p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is the prime factorization of \( a_n \), with \( 1 < p_1 < p_2 < \cdots < p_k \), and \( e_1, e_2, \dots, e_k \) positive integers.
For example, if \( a_1 = 2024 = 2^3 \cdot 11 \cdot 23 \), the next two terms of the sequence are
\[
a_2 = \frac{a_1}{23^{1-1}} + 1 = \frac{2024}{1} + 1 = 2025 = 3^4 \cdot 5^2;
\]
\[
a_3 = \frac{a_2}{5^{2-1}} + 1 = \frac{2025}{5} + 1 = 406.
\]
Determine for which values of \( a_1 \) the sequence is eventually periodic and what all the possible periods are.
[b]Note:[/b] Let \( p \) be a positive integer. A sequence \( x_1, x_2, \dots \) is eventually periodic with period \( p \) if \( p \) is the smallest positive integer such that there exists an \( N \geq 0 \) satisfying \( x_{n+p} = x_n \) for all \( n > N \).
2010 Dutch IMO TST, 1
Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if
(i) $a_n < a_{n+1}$ for all $n\ge 1$,
(ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.
2014 Balkan MO Shortlist, 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 $5$
2021 Baltic Way, 3
Determine all infinite sequences $(a_1,a_2,\dots)$ of positive integers satisfying
\[a_{n+1}^2=1+(n+2021)a_n\]
for all $n \ge 1$.
1980 IMO Shortlist, 18
Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]
1998 Belarus Team Selection Test, 2
For any sequence of real numbers $(a_n), n \in N$, define a new sequence $(b_n)$ as $b_n =a_{n+2}+sa_{n+1}+ta_{n}$, where $s,t$ are given real numbers.
Find all ordered pairs $(s,t)$ satisfying the following property: any sequence $(a_n)$ converges as soon as the sequence $(b_n)$ converges.
2015 Peru MO (ONEM), 4
Let $b$ be an odd positive integer. The sequence $a_1, a_2, a_3, a_4$, is definedin the next way: $a_1$ and $a_2$ are positive integers and for all $k \ge 2$,
$$a_{k+1}= \begin{cases} \frac{a_k + a_{k-1}}{2} \,\,\, if \,\,\, a_k + a_{k-1} \,\,\, is \,\,\, even \\ \frac{a_k + a_{k-1+b}}{2}\,\,\, if \,\,\, a_k + a_{k-1}\,\,\, is \,\,\,odd\end{cases}$$
a) Prove that if $b = 1$, then after a certain term, the sequence will become constant.
b) For each $b \ge 3$ (odd), prove that there exist values of $a_1$ and $a_2$ for which the sequence will become constant after a certain term.
2015 Singapore Junior Math Olympiad, 3
There are $30$ children, $a_1,a_2,...,a_{30}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.
1977 IMO, 2
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.
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)} $.
.
1981 Czech and Slovak Olympiad III A, 4
Let $n$ be a positive integer. Show that there is a prime $p$ and a sequence $\left(a_k\right)_{k\ge1}$ of positive integers such that the sequence $\left(p+na_k\right)_{k\ge1}$ consists of distinct primes.
1975 Kurschak Competition, 3
Let $$x_0 = 5\,\, ,\, \,\,x_{n+1} = x_n +\frac{1}{x_n}.$$
Prove that $45 < x_{1000} < 45.1$.
2018 Peru Cono Sur TST, 9
Find the largest possible value of the positive integer $N$ given that there exist positive integers $a_1, a_2, \dots, a_N$ satisfying
$$ a_n = \sqrt{(a_{n-1})^2 + 2018 \, a_{n-2}}\:, \quad \text{for } n = 3,4,\dots,N. $$
2024 Brazil National Olympiad, 1
Consider a sequence whose first term is a given positive integer \( N > 1 \). Consider the prime factorization of \( N \). If \( N \) is a power of 2, the sequence consists of a single term: \( N \). Otherwise, the second term of the sequence is obtained by replacing the largest prime factor \( p \) of \( N \) with \( p + 1 \) in the prime factorization. If the new number is not a power of 2, we repeat the same procedure with it, remembering to factor it again into primes. If it is a power of 2, the numerical sequence ends. And so on.
For example, if the first term of the sequence is \( N = 300 = 2^2 \cdot 3 \cdot 5^2 \), since its largest prime factor is \( p = 5 \), the second term is \( 2^2 \cdot 3 \cdot (5 + 1)^2 = 2^4 \cdot 3^3 \). Repeating the procedure, the largest prime factor of the second term is \( p = 3 \), so the third term is \( 2^4 \cdot (3 + 1)^3 = 2^{10} \). Since we obtained a power of 2, the sequence has 3 terms: \( 2^2 \cdot 3 \cdot 5^2 \), \( 2^4 \cdot 3^3 \), and \( 2^{10} \).
a) How many terms does the sequence have if the first term is \( N = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \)?
b) Show that if a prime factor \( p \) leaves a remainder of 1 when divided by 3, then \( \frac{p+1}{2} \) is an integer that also leaves a remainder of 1 when divided by 3.
c) Present an initial term \( N \) less than 1,000,000 (one million) such that the sequence starting from \( N \) has exactly 11 terms.
1977 IMO Longlists, 34
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
2020 Canadian Mathematical Olympiad Qualification, 5
We define the following sequences:
• Sequence $A$ has $a_n = n$.
• Sequence $B$ has $b_n = a_n$ when $a_n \not\equiv 0$ (mod 3) and $b_n = 0$ otherwise.
• Sequence $C$ has $c_n =\sum_{i=1}^{n} b_i$
.• Sequence $D$ has $d_n = c_n$ when $c_n \not\equiv 0$ (mod 3) and $d_n = 0$ otherwise.
• Sequence $E$ has $e_n =\sum_{i=1}^{n}d_i$
Prove that the terms of sequence E are exactly the perfect cubes.
2021 Simon Marais Mathematical Competition, A2
Define the sequence of integers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$, and
\[ a_{n+1} = \left(n+1-\gcd(a_n,n) \right) \times a_n \]
for all integers $n \ge 1$.
Prove that $\frac{a_{n+1}}{a_n}=n$ if and only if $n$ is prime or $n=1$.
[i]Here $\gcd(s,t)$ denotes the greatest common divisor of $s$ and $t$.[/i]
2018 Istmo Centroamericano MO, 3
Determine all sequences of integers $a_1, a_2,. . .,$ such that:
(i) $1 \le a_i \le n$ for all $1 \le i \le n$.
(ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$
2008 Dutch IMO TST, 3
Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le i\le m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$,
so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j \ge i$.
Similarly, we define, for $1\le j \le n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$.
E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$.
(a) Prove that $a_j = c_j $ for $1 \le j \le n$.
(b) Prove that for $1\le k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.
2022 USAJMO, 1
For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties?
[list]
[*] $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$;
[*] $a_2 - a_1$ is not divisible by $m$.
[/list]
[i]Holden Mui[/i]
2014 IMO Shortlist, N7
Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ .
[i]Proposed by Austria[/i]
1994 Tournament Of Towns, (414) 2
Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that
(a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence);
(b) if the sequence is periodic, then the first number is rational.
(G Shabat)
2024 Abelkonkurransen Finale, 2a
Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$.
2010 Germany Team Selection Test, 3
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]