Found problems: 766
The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.
Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying
\[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\]
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]
How many ways are there to cover a $3\times 8$ rectangle with $12$ identical dominoes?
The sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,... $suffices for all positive integers $n$ of the following recursion:
$a_{n+1} = a_n - b_n$ and $b_{n+1} = 2b_n$, if $a_n \ge b_n$,
$a_{n+1} = 2a_n$ and $b_{n+1} = b_n - a_n$, if $a_n < b_n$.
For which pairs $(a_1, b_1)$ of positive real initial terms is there an index $k$ with $a_k = 0$?
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]
Periodic sequences $(a_n),(b_n),(c_n)$ and $(d_n)$ satisfy the following conditions:
$$a_{n+1}=a_n+b_n,\enspace\enspace b_{n+1}=b_n+c_n,$$
$$c_{n+1}=c_n+d_n,\enspace\enspace d_{n+1}=d_n+a_n,$$
for $n=1,2,\ldots$. Prove that $a_2=b_2=c_2=d_2=0$.
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that \[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\] for all $i,j$ in $\{1,2,\dots,n\}.$ Show that for $n\ge 2,$ the quantity \[P_{n+5}-P_{n+4}-P_{n+3}+P_n\] does not depend on $n,$ and find its value.
An integer $ m > 1$ is given. The infinite sequence $ (x_n)_{n\ge 0}$ is defined by $ x_i\equal{}2^i$ for $ i<m$ and $ x_i\equal{}x_{i\minus{}1}\plus{}x_{i\minus{}2}\plus{}\cdots \plus{}x_{i\minus{}m}$ for $ i\ge m$.
Find the greatest natural number $ k$ such that there exist $ k$ successive terms of this sequence which are divisible by $ m$.
Let $a_n$ be a sequence of positive numbers such that:
i) $\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}$, for every $n\in\mathbb{N}^{\star}$
ii) $\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1$, for every $ k,n\in\mathbb{N}^{\star}$ with $|k-n|\neq 1$.
(a) Prove that $(a_n)$ is a geometric progression.
(n) Prove that exists $t>0$, such that $\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t$
Let $f$ be a function defined on $\{(i,j): i,j \in \mathbb{N}\}$ satisfying
(i) $f(i,i+1)=\frac{1}{3}$ for all $i$
(ii) $f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j)$ for all $k$ such that $i <k<j$.
Find the value of $f(1,100)$.
The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and
$a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$.
Prove that no term in $a_i$ is in the range $[1612, 2012]$.
Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that
\[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\]
and
\[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\]
[i]Calvin Deng.[/i]
Let $P_1(x)=\frac{1}{x}$ and $P_n(x)=P_{n-1}(x)+P_{n-1}(x-1)$ for every natural $ n$ greater than $1$. Find the value of $P_{2008}(2008)$.
[i](Mathophile)[/i]
Let $m\ge2$ be an integer. The sequence $(a_n)_{n\in\mathbb N}$ is defined by $a_0=0$ and $a_n=\left\lfloor\frac nm\right\rfloor+a_{\left\lfloor\frac nm\right\rfloor}$ for all $n$. Determine $\lim_{n\to\infty}\frac{a_n}n$.
The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]
for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?
Find all possible $\{ x_1,x_2,...x_n \}$ permutations of $ \{1,2,...,n \}$ so that when $1\le i \le n-2 $ then we have $x_i < x_{i+2}$ and when $1 \le i \le n-3$ then we have $x_i < x_{i+3}$ . Here $n \ge 4$.
Let $p_n$ be a bounded sequence of integers which satisfies the recursion
$$p_n =\frac{p_{n-1} +p_{n-2} + p_{n-3}p _{n-4}}{p_{n-1} p_{n-2}+ p_{n-3} +p_{n-4}}.$$
Show that the sequence eventually becomes periodic.
Each term of an infinite sequence of natural numbers is obtained from the previous term by adding to it one of its nonzero digits. Prove that this sequence contains an even number.
Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation
\[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\]
For all $x,y\in\mathbb R.$
Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.
Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge 0$. Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.
The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$.
Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
Let $f(m,1)=f(1,n)=1$ for $m\geq 1, n\geq 1$ and let $f(m,n)=f(m-1, n)+ f(m, n-1) +f(m-1 ,n-1)$ for $m>1$ and $n>1$. Also let
$$ S(n)= \sum_{a+b=n} f(a,b) \,\,\;\; a\geq 1 \,\, \text{and} \,\; b\geq 1.$$
Prove that
$$S(n+2) =S(n) +2S(n+1) \,\, \; \text{for} \, \, n \geq 2.$$