Found problems: 1239
2011 Abels Math Contest (Norwegian MO), 1
Let $n$ be the number that is produced by concatenating the numbers $1, 2,... , 4022$,
that is, $n = 1234567891011...40214022$.
a. Show that $n$ is divisible by $3$.
b. Let $a_1 = n^{2011}$, and let $a_i$ be the sum of the digits of $a_{i-1}$ for $i > 1$. Find $a_4$
2023 Mongolian Mathematical Olympiad, 3
Let $m$ be a positive integer. We say that a sequence of positive integers written on a circle is [i] good [/i], if the sum of any $m$ consecutive numbers on this circle is a power of $m$.
1. Let $n \geq 2$ be a positive integer. Prove that for any [i] good [/i] sequence with $mn$ numbers, we can remove $m$ numbers such that the remaining $mn-m$ numbers form a [i] good [/i] sequence.
2. Prove that in any [i] good [/i] sequence with $m^2$ numbers, we can always find a number that was repeated at least $m$ times in the sequence.
1967 IMO Longlists, 57
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 $c_n = 0.$
1992 Putnam, B3
For any pair $(x,y)$ of real numbers, a sequence $(a_{n}(x,y))$ is defined as follows:
$$a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0$$
Find the area of the region $\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}$.
Mathley 2014-15, 2
Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$.
Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers.
Dang Hung Thang, University of Natural Sciences, Hanoi National University.
2013 IFYM, Sozopol, 1
Let $u_1=1,u_2=2,u_3=24,$ and
$u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$
Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.
1957 Moscow Mathematical Olympiad, 363
Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.
2022 Bosnia and Herzegovina BMO TST, 1
Let $a_1,a_2,a_3, \ldots$ be an infinite sequence of nonnegative real numbers such that for all positive integers $k$ the following conditions hold:
$i)$ $a_k-2a_{k+1}+a_{k+2} \geq 0$;
$ii)$ $\sum_{j=1}^{k} a_j \leq 1$.
Prove that for all positive integer $k$ holds: $0 \leq a_k - a_{k+1} < \frac{2}{k^2}$
2022 Brazil National Olympiad, 3
Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of integers numbers. Let $\Delta^1a_n=a_{n+1}-a_n$ for a non-negative integer $n$. Define $\Delta^Ma_n= \Delta^{M-1}a_{n+1}- \Delta^{M-1}a_n$. A sequence is [i]patriota[/i] if there are positive integers $k,l$ such that $a_{n+k}=\Delta^Ma_{n+l}$ for all non-negative integers $n$. Determine, with proof, whether exists a sequence that the last value of $M$ for which the sequence is [i]patriota[/i] is $2022$.
1977 IMO Shortlist, 13
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.
2017 Bundeswettbewerb Mathematik, 1
For which integers $n \geq 4$ is the following procedure possible? Remove one number of the integers $1,2,3,\dots,n+1$ and arrange them in a sequence $a_1,a_2,\dots,a_n$ such that of the $n$ numbers \[ |a_1-a_2|,|a_2-a_3|,\dots,|a_{n-1}-a_n|,|a_n-a_1| \] no two are equal.
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:
$\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$
2015 Harvard-MIT Mathematics Tournament, 4
Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold.
[list]
[*] $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$
[*] There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such that $a_i+a_j-N$ is divisible by $203$.
[/list]
2009 Brazil Team Selection Test, 4
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]