Found problems: 766
Let $a_1, a_2, ..., a_n, a_{n+1}$ be a finite sequence of real numbers satisfying $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1$ for $k = 1, 2, ..., n$
Prove that for $k=0, 1, ..., n+1,$ $|a_k| \leq \frac{k(n+1-k)}{2}$
For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$.
(a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$.
(b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.
How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one?
[i]Proposed by Germany, FR.[/i]
Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$.
Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i]
[i]Proposed by Laurentiu Panaitopol, Romania[/i]
How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ :
(A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.
The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and will paint each of the six sections a solid color. Find the number of ways you can choose to paint each of the six sections if no two adjacent section can be painted with the same color.
[asy]
size(3cm);
draw(unitcircle);
draw(scale(0.6)*unitcircle);
for(int i = 0; i < 6; ++i){
draw(dir(60*i)--0.6*dir(60*i));
}
[/asy]
The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are defined by $u_{0} =u_{1} =1$ ,$u_{n}=2u_{n-1}-3u_{n-2}$ $(n\geq2)$ , $v_{0} =a, v_{1} =b , v_{2}=c$ ,$v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}$ $(n\geq3)$. There exists a positive integer $N$ such that when $n> N$, we have $u_{n}\mid v_{n}$ . Prove that $3a=2b+c$.
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.
Let $c>2$ and $a_0,a_1, \ldots$ be a sequence of real numbers such that
\begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*}
for any $n$ $\in$ $\mathbb{N}$. Prove, $a_1=0$
Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers for which
\[ a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. \]
Prove that there exists a positive integer $m$ such that $a_m=4$ and $a_{m+1} \in\{3,4\}$.
[b]Note.[/b] $[x]$ is the greatest integer not exceeding $x$.
Let $b$ be an odd positive integer. The sequence $a_1, a_2, a_3, a_4$, is definedin the next way: $a_1$ and $a_2$ are positive integers and for all $k \ge 2$,
$$a_{k+1}= \begin{cases} \frac{a_k + a_{k-1}}{2} \,\,\, if \,\,\, a_k + a_{k-1} \,\,\, is \,\,\, even \\ \frac{a_k + a_{k-1+b}}{2}\,\,\, if \,\,\, a_k + a_{k-1}\,\,\, is \,\,\,odd\end{cases}$$
a) Prove that if $b = 1$, then after a certain term, the sequence will become constant.
b) For each $b \ge 3$ (odd), prove that there exist values of $a_1$ and $a_2$ for which the sequence will become constant after a certain term.
Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$.
[i]Proposed by Okan Tekman, Turkey[/i]
Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
The sequence $\{y_{n}\}_{n \ge 1}$ is defined by \[y_{1}=y_{2}=1,\;\; y_{n+2}= (4k-5)y_{n+1}-y_{n}+4-2k.\] Determine all integers $k$ such that each term of this sequence is a perfect square.
each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P.
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
Define the sequence $a_1, a_2,...$ as follows: $a_1 = 1$, and for every $n \ge 2$, $a_n = n - 2$ if $a_{n-1} = 0$ and $a_n = a_{n-1} - 1$, otherwise. Find the number of $1 \le k \le 2016$ such that there are non-negative integers $r, s$ and a positive integer $n$ satisfying $k = r + s$ and $a_{n+r} = a_n + s$.
Let $Q_0(x)=1$, $Q_1(x)=x,$ and
\[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\]
for all $n\ge 2.$ Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
Denote by $a_n$ the greatest number that is not divisible by $3$ and that divides $n$. Consider the sequence $s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N$. Denote by $A(n)$ the number of all sums $s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)$ that are divisible by $3$. Prove the formula
\[A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.\]
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be?
b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$:
(a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$, (b) $f(2n) \ge f(n)$
where $Z_{\ge 0}$ is the set of nonnegative integers. Solve the equation $f(n) = 1000$
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:
[list]
[*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and
[*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$.
[/list]
Show that $a_n$ is always a strictly positive integer.
A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at least once. For example, a $1 \times 1$ red tile followed by a $2 \times 1$ green tile, a $1 \times 1$ green tile, a $2 \times 1$ blue tile, and a $1 \times 1$ green tile is a valid tiling. Note that if the $2 \times 1$ blue tile is replaced by two $1 \times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000$.