Found problems: 1239
Define a sequence $<x_n>$ by $x_0 = 0$ and $$\large x_n = \left\{
\begin{array}{ll}
x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\
& \\
x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\
\end{array}
\right. $$
where $k, r$ are integers. Prove that every integer occurs exactly once in the sequence.
Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.
Given three numbers $x, y, z$, and set $x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x|$.
From $x_1, y_1, z_1$, form in the same fashion the numbers $x_2, y_2, z_2$, and so on.
It is known that $x_n = x, y_n = y, z_n = z$ for some $n$. Find all possible values of $(x, y, z)$.
Let $a_n$ be the number of n-digit integers formed by $1, 2$ and $3$ which do not contain any consecutive $1$’s. Prove that $a_n$ is equal to $$\left( \frac12 + \frac{1}{\sqrt3}\right)(\sqrt{3} + 1)^n$$ rounded off to the nearest integer.
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.
Consider the following sequence
$$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$
Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$.
(Proposed by Tomas Barta, Charles University, Prague)
$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$.
$a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite.
$b)$ Find 3 different prime numbers that do not divide any terms of this sequence.
Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions:
\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n;
\]
\[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n.
\]
[i]Author: Dusan Dukic, Serbia[/i]
Let $a_0$ be a positive real number and consider the general term sequence $a_n$ defined by $$a_n =a_{n-1} + \frac{1}{a_{n-1}} \,\,\, n=1,2,3,...$$ Prove that $a_{1998} > 63$.
Given a real number $a$ and a sequence $(x_n)_{n=1}^\infty$ defined by:
$$\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right.$$
for all positive integers $n$.
1. For $a=0$, prove that $(x_n)$ converges.
2. Determine the largest possible value of $a$ such that $(x_n)$ converges.
The numbers $1$ to $12$ are arranged in a sequence. The number of ways this can be done equals $12 \times11 \times 10\times ...\times 1$. We impose the condition that in the sequence there should be exactly one number that is smaller than the number directly preceding it.
How many of the $12 \times11 \times 10\times ...\times 1$ sequences satisfy this condition?
Given are the sequences
\[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\]
of positive real numbers. For each integer $n$ the following inequalities hold:
\[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\]
\[b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})\]
\[c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})\]
Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.
Let $m \geq 2$ be a fixed positive integer, and $(a_n)_{n\geq 1}$ be a sequence of nonnegative real numbers such that, for all $n\geq 1$, we have that $a_{n+1} \leq a_n - a_{mn}$.
a) Prove that the sequence $b_n = \sum_{k=1}^{n} a_k$ is bounded above.
b) Prove that the sequence $c_n = \sum_{k=1}^{n} k^2 a_k$ is bounded above.
We say that a strictly increasing sequence of positive integers $n_1, n_2,\ldots$ is [i]non-decelerating[/i] if $n_{k+1}-n_k\le n_{k+2}-n_{k+1}$ holds for all positive integers $k$. We say that a strictly increasing sequence $n_1, n_2, \ldots$ is [i]convergence-inducing[/i], if the following statement is true for all real sequences $a_1, a_2, \ldots$: if subsequence $a_{m+n_1}, a_{m+n_2}, \ldots$ is convergent and tends to $0$ for all positive integers $m$, then sequence $a_1, a_2, \ldots$ is also convergent and tends to $0$. Prove that a non-decelerating sequence $n_1, n_2,\ldots$ is convergence-inducing if and only if sequence $n_2-n_1$, $n_3-n_2$, $\ldots$ is bounded from above.
[i]Proposed by András Imolay[/i]
Let $\alpha \leq -2$ be an integer. Prove that for every pair $(\beta_0, \beta_1)$ of integers there exists a uniquely determined sequence $0\leq q_0, \ldots, q_k<\alpha ^ 2 - \alpha$ of integers, such that $q_k\neq 0$ if $(\beta_0, \beta 1)\neq (0,0)$ and
$$\beta_i=\sum_{j=0}^k q_j(\alpha - i)^j,\text{ for }i=0,1$$
The sequence $ \{a_n\}$ of integers is defined by
\[ a_1 \equal{} 2, a_2 \equal{} 7
\]
and
\[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2.
\]
Prove that $ a_n$ is odd for all $ n > 1.$
For a positive integer $n$, an [i]$n$-sequence[/i] is a sequence $(a_0,\ldots,a_n)$ of non-negative integers satisfying the following condition: if $i$ and $j$ are non-negative integers with $i+j \leqslant n$, then $a_i+a_j \leqslant n$ and $a_{a_i+a_j}=a_{i+j}$.
Let $f(n)$ be the number of $n$-sequences. Prove that there exist positive real numbers $c_1$, $c_2$, and $\lambda$ such that \[c_1\lambda^n<f(n)<c_2\lambda^n\] for all positive integers $n$.
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$.
Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
Given an infinite positive integer sequence $\{x_i\}$ such that $$x_{n+2}=x_nx_{n+1}+1$$ Prove that for any positive integer $i$ there exists a positive integer $j$ such that $x_j^j$ is divisible by $x_i^i$.
[i]Remark: Unfortunately, there was a mistake in the problem statement during the contest itself. In the last sentence, it should say "for any positive integer $i>1$ ..."[/i]
Call an $n$-tuple $(a_1, . . . , a_n)$ [i]occasionally periodic [/i] if there exist a nonnegative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+p+j}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, . . . , a_n)$ with elements from set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.
$\{a_n\}$ is an infinite, strictly increasing sequence of positive integers and $a_{a_n}\leq a_n+a_{n+3}$ for all $n\geq 1$. Prove that, there are infinitely many triples $(k,l,m)$ of positive integers such that $k<l<m$ and $a_k+a_m=2a_l$
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 $a_{46}$.
Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two.
(N. Agakhanov)
let $(a_{n})$ be an arbitrary sequence of positive numbers. Show that
$$\limsup_{n\to \infty} \left(\frac{a_1 +a_{n+1}}{a_{n}}\right)^{n} \geq e.$$