Found problems: 34
PEN M Problems, 8
The Bernoulli sequence $\{B_{n}\}_{n \ge 0}$ is defined by \[B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)\] Show that for all $n \in \mathbb{N}$, \[(-1)^{n}B_{n}-\sum \frac{1}{p},\] is an integer where the summation is done over all primes $p$ such that $p| 2k-1$.
PEN M Problems, 28
Let $\{u_{n}\}_{n \ge 0}$ be a sequence of integers satisfying the recurrence relation $u_{n+2}=u_{n+1}^2 -u_{n}$ $(n \in \mathbb{N})$. Suppose that $u_{0}=39$ and $u_{1}=45$. Prove that $1986$ divides infinitely many terms of this sequence.
PEN M Problems, 21
In the sequence $1, 0, 1, 0, 1, 0, 3, 5, \cdots$, each member after the sixth one is equal to the last digit of the sum of the six members just preceeding it. Prove that in this sequence one cannot find the following group of six consecutive members: \[0, 1, 0, 1, 0, 1\]
PEN M Problems, 19
A sequence with first two terms equal $1$ and $24$ respectively is defined by the following rule: each subsequent term is equal to the smallest positive integer which has not yet occurred in the sequence and is not coprime with the previous term. Prove that all positive integers occur in this sequence.
PEN M Problems, 32
In an increasing infinite sequence of positive integers, every term starting from the $2002$-th term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
PEN M Problems, 29
The sequence $\{a_{n}\}_{n \ge 1}$ is defined by $a_{1}=1$ and \[a_{n+1}= \frac{a_{n}}{2}+\frac{1}{4a_{n}}\; (n \in \mathbb{N}).\] Prove that $\sqrt{\frac{2}{2a_{n}^{2}-1}}$ is a positive integer for $n>1$.
PEN M Problems, 30
Let $k$ be a positive integer. Prove that there exists an infinite monotone increasing sequence of integers $\{a_{n}\}_{n \ge 1}$ such that \[a_{n}\; \text{divides}\; a_{n+1}^{2}+k \;\; \text{and}\;\; a_{n+1}\; \text{divides}\; a_{n}^{2}+k\] for all $n \in \mathbb{N}$.
PEN M Problems, 25
Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive integers such that \[0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.\] Show that there are infinitely many pairs $(p, q)$ of positive integers such that $p>q$ and $a_{q}\; \vert \; a_{p}$.
PEN M Problems, 34
The sequence of integers $\{ x_{n}\}_{n\ge1}$ is defined as follows: \[x_{1}=1, \;\; x_{n+1}=1+{x_{1}}^{2}+\cdots+{x_{n}}^{2}\;(n=1,2,3 \cdots).\] Prove that there are no squares of natural numbers in this sequence except $x_{1}$.
PEN M Problems, 14
Let $x_{1}$ and $x_{2}$ be relatively prime positive integers. For $n \ge 2$, define $x_{n+1}=x_{n}x_{n-1}+1$.[list=a][*] Prove that for every $i>1$, there exists $j>i$ such that ${x_{i}}^{i}$ divides ${x_{j}}^{j}$. [*] Is it true that $x_{1}$ must divide ${x_{j}}^{j}$ for some $j>1$? [/list]
PEN M Problems, 9
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=2, \; a_{n+1}=\left\lfloor \frac{3}{2}a_{n}\right\rfloor.\] Show that it has infinitely many even and infinitely many odd integers.
PEN M Problems, 26
Let $p$ be an odd prime $p$ such that $2h \neq 1 \; \pmod{p}$ for all $h \in \mathbb{N}$ with $h< p-1$, and let $a$ be an even integer with $a \in] \tfrac{p}{2}, p [$. The sequence $\{a_n\}_{n \ge 0}$ is defined by $a_{0}=a$, $a_{n+1}=p -b_{n}$ \; $(n \ge 0)$, where $b_{n}$ is the greatest odd divisor of $a_n$. Show that the sequence $\{a_n\}_{n \ge 0}$ is periodic and find its minimal (positive) period.
PEN M Problems, 22
Let $\, a$, and $b \,$ be odd positive integers. Define the sequence $\{f_n\}_{n\ge 1}$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for sufficiently large $\, n \,$ and determine the eventual value as a function of $\, a \,$ and $\, b$.
PEN M Problems, 2
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.\] Show that $a_{n}$ is a square if and only if $n=2^{k}+k-2$ for some $k \in \mathbb{N}$.
PEN M Problems, 33
The sequence $ \{x_{n}\}_{n \ge 1}$ is defined by
\[ x_{1} \equal{} 2, x_{n \plus{} 1} \equal{} \frac {2 \plus{} x_{n}}{1 \minus{} 2x_{n}}\;\; (n \in \mathbb{N}).
\] Prove that
a) $ x_{n}\not \equal{} 0$ for all $ n \in \mathbb{N}$,
b) $ \{x_{n}\}_{n \ge 1}$ is not periodic.
PEN M Problems, 3
Let $f(n)=n+\lfloor \sqrt{n}\rfloor$. Prove that, for every positive integer $m$, the sequence \[m, f(m), f(f(m)), f(f(f(m))), \cdots\] contains at least one square of an integer.
PEN M Problems, 11
Let $a_{1}={11}^{11}$, $a_{2}={12}^{12}$, $a_{3}={13}^{13}$, and \[a_{n}= \vert a_{n-1}-a_{n-2}\vert+\vert a_{n-2}-a_{n-3}\vert, n \ge 4.\] Determine $a_{{14}^{14}}$.
PEN M Problems, 16
Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i\geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?
PEN M Problems, 20
Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?
PEN M Problems, 1
Let $P(x)$ be a nonzero polynomial with integer coefficients. Let $a_{0}=0$ and for $i \ge 0$ define $a_{i+1}=P(a_{i})$. Show that $\gcd ( a_{m}, a_{n})=a_{ \gcd (m, n)}$ for all $m, n \in \mathbb{N}$.
PEN M Problems, 15
For a given positive integer $k$ denote the square of the sum of its digits by $f_{1}(k)$ and let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of $f_{1991}(2^{1990})$.
PEN M Problems, 12
Let $k$ be a fixed positive integer. The sequence $\{a_{n}\}_{n\ge1}$ is defined by \[a_{1}=k+1, a_{n+1}=a_{n}^{2}-ka_{n}+k.\] Show that if $m \neq n$, then the numbers $a_{m}$ and $a_{n}$ are relatively prime.
PEN M Problems, 23
Define \[\begin{cases}d(n, 0)=d(n, n)=1&(n \ge 0),\\ md(n, m)=md(n-1, m)+(2n-m)d(n-1,m-1)&(0<m<n).\end{cases}\] Prove that $d(n, m)$ are integers for all $m, n \in \mathbb{N}$.
PEN M Problems, 4
The sequence $ \{a_{n}\}_{n \ge 1}$ is defined by \[ a_{1}=1, \; a_{2}=2, \; a_{3}=24, \; a_{n}=\frac{ 6a_{n-1}^{2}a_{n-3}-8a_{n-1}a_{n-2}^{2}}{a_{n-2}a_{n-3}}\ \ \ \ (n\ge4).\] Show that $ a_{n}$ is an integer for all $ n$, and show that $ n|a_{n}$ for every $ n\in\mathbb{N}$.
PEN M Problems, 5
Show that there is a unique sequence of integers $\{a_{n}\}_{n \ge 1}$ with \[a_{1}=1, \; a_{2}=2, \; a_{4}=12, \; a_{n+1}a_{n-1}=a_{n}^{2}\pm1 \;\; (n \ge 2).\]