Found problems: 89
2015 Indonesia MO Shortlist, C6
Let $k$ be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance $k$ is the original distance. Formally, we may choose $2$ tails $A, B$ that are spaced $n$ and move $A$ to the different side of $B$ so the current distance is $kn$. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of $k$ so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.
2015 EGMO, 4
Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers
which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.
2023 Indonesia TST, N
Given an integer $a>1$. Prove that there exists a sequence of positive integers
\[ n_1, n_2, n_3, \ldots \]
Such that
\[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.
2023 Switzerland Team Selection Test, 1
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2013 Spain Mathematical Olympiad, 5
Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions
$i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct).
$ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$
1940 Moscow Mathematical Olympiad, 059
Consider all positive integers written in a row: $123456789101112131415...$ Find the $206788$-th digit from the left.
1946 Moscow Mathematical Olympiad, 121
Given the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, ... ,$ ascertain whether among its first $(10^8+1)$ terms there is a number that ends with four zeros.
1994 IMO Shortlist, 6
Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?
1991 All Soviet Union Mathematical Olympiad, 551
A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?
2023 Thailand Online MO, 9
Find all sequences of positive integers $a_1,a_2,\dots$ such that $$(n^2+1)a_n = n(a_{n^2}+1)$$ for all positive integers $n$.
2011 Tuymaada Olympiad, 4
Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.
2023 Brazil Team Selection Test, 1
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2018 ELMO Shortlist, 3
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$
[i]Proposed by Krit Boonsiriseth[/i]
2020 Bulgaria National Olympiad, P3
Let $a_1\in\mathbb{Z}$, $a_2=a_1^2-a_1-1$, $\dots$ ,$a_{n+1}=a_n^2-a_n-1$. Prove that $a_{n+1}$ and $2n+1$ are coprime.