Found problems: 1239
Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and
\[
a_k = (n-k+1) \cdot c_{k-1}, \quad
b_k = \binom nk - c_k - a_k, \quad \text{and} \quad
c_k = \frac{b_{k-1}}{k}
\]
for each integer $1 \leq k \leq n$.
$ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$.
Proposed by [i]Jonathan Du[/i]
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$:
$p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.
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.
Let $n \ne 0$ be a natural number. A sequence of numbers is briefly called a sequence “$F_n$” if $n$ different numbers $z_1$, $z_2$, $...$, $z_n$ exist so that the following conditions are fulfilled:
(1) Each term of the sequence is one of the numbers $z_1$, $z_2$, $...$, $z_n$.
(2) Each of the numbers $z_1$, $z_2$, $...$, $z_n$ occurs at least once in the sequence.
(3) Any two immediately consecutive members of the sequence are different numbers.
(4) No subsequence of the sequence has the form $\{a, b, a, b\}$ with $a \ne b$.
Note: A subsequence of a given sequence $\{x_1, x_2, x_3, ...\}$ or $\{x_1, x_2, x_3, ..., x_s\}$ is called any sequence of the form $\{x_{m1}, x_{m2}, x_{m3}, ...\}$ or $\{x_{m1}, x_{m2}, x_{m3}, ..., x_{mt}\}$ with natural numbers $m_1 < m_2 < m_3 < ...$
Answer the following questions:
a) Given $n$, are there sequences $F_n$ of arbitrarily long length?
b) If question (a) is answered in the negative for an $n$:
What is the largest possible number of terms that a sequence $F_n$ can have (given $n$)?
Let $a_0$ be a positive integer and $a_{n + 1} =\sqrt{a_n^2 + 1}$, for all $n \ge 0$.
1) Prove that for all $a_0$ the sequence contains infinitely many integers and infinitely many irrational numbers.
2) Is there an $a_0$ for which $a_{2010}$ is an integer?
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:
\[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\]
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .
[i]Proposed by Marcin Kuczma, Poland[/i]
Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.
Let $a_1=1$ and $a_{n+1}=a_{n}+\frac{1}{2a_n}$ for $n \geq 1$. Prove that
$a)$ $n \leq a_n^2 < n + \sqrt[3]{n}$
$b)$ $\lim_{n\to\infty} (a_n-\sqrt{n})=0$
Let $(x_n), n\in\mathbb{N}$ be a sequence such that $x_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N}$
and $x_1=\frac{a}{b}$ where $a,b$ are positive integers such that $3\not|b$. If $x_m$ is a square of a rational number for some positive integer $m$, prove that $x_1$ is also a square of a rational number.
Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$
a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$
b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$
A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$
[i]Proposed by Gerhard Wöginger, Austria[/i]
Given a polynomial $P(x)$ with
a) natural coefficients;
b) integer coefficients;
Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
Let $a_1 = 1 +\sqrt2$ and for each $n \ge 1$ de ne $a_{n+1} = 2 -\frac{1}{a_n}$. Find the greatest integer less than or equal to the product $a_1a_2a_3 ... a_{200}$.
For $n \in \mathbb{N}$, let $a_n = \int _0 ^{1/\sqrt{n}} | 1 + e^{it} + e^{2it} + \dots + e^{nit} | \ dt$. Determine whether the sequence $(a_n) = a_1, a_2, \dots$ is bounded.
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 $n\geq 2$
Two positive integers $r$ and $k$ are given as is an infinite sequence of positive integers $a_1 \le a_2 \le a_3 \le ..$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a positive integer $t$ such that $\frac{t}{a_t}= k$.
For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$
(a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$
(b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$
[I]United Kingdom[/i]
(a) Find the numbers $a_0,. . . , a_{100}$, such that $a_0 = 0, a_{100} = 1$ and for all $k = 1,. . . , 99$ :
$$a_k = \frac12 a_{k- 1} + \frac12 a_{k+1 }$$
(b) Find the numbers $a_0,. . . , a_{100}$, such that $a_0 = 0, a_{100} = 1$ and for all $k = 1,. . . , 99$ :
$$a_k = 1+\frac12 a_{k- 1} + \frac12 a_{k+1 }$$.
The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.
Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\ (v_n)$ are defined as follows:
$$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$
a. Prove that
$${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$
b. Find all values of $a$ in the equality case.
$a_1,a_2,a_3,...$ is a monotone increasing sequence of natural numbers. It is known that for any $k, a_{a_k} = 3k$.
a) Find $a_{100}$.
b) Find $a_{1983}$.
(A Andjans, Riga)
PS. (a) for Juniors, (b) for Seniors
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$. From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are same as first $k$ elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element $1$.