Found problems: 766
The sequence $(Q_{n}(x))$ of polynomials is defined by
$$Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x,$$
and for $m \geq 1 $ by
$$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$
$$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$
Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$
Let $n$ be a positive integer, and let $S_n$ be the set of all permutations of $1,2,...,n$. let $k$ be a non-negative integer, let $a_{n,k}$ be the number of even permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$ and $b_{n,k}$ be the number of odd permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$. Evaluate $a_{n,k}-b_{n,k}$.
[i]* * *[/i]
Determine all sequences $a_1,a_2,a_3,\dots$ of positive integers that satisfy the equation
$$(n^2+1)a_{n+1} - a_n = n^3+n^2+1$$
for all positive integers $n$.
A drunken horse moves on an infinite board whose squares are numbered in pairs $(a, b) \in \mathbb{Z}^2$. In each movement, the 8 possibilities $$(a, b) \rightarrow (a \pm 1, b \pm 2),$$ $$(a, b) \rightarrow (a \pm 2, b \pm 1)$$ are equally likely. Knowing that the knight starts at $(0, 0)$, calculate the probability that, after $2023$ moves, it is in a square $(a, b)$ with $a \equiv 4 \pmod 8$ and $b \equiv 5 \pmod 8$.
The sequence $(a_n)$ is defined by $a_0 = 0$ and $a_{n+1} = [\sqrt[3]{a_n +n}]^3$ for $n \ge 0$.
(a) Find $a_n$ in terms of $n$.
(b) Find all $n$ for which $a_n = n$.
Let the sequence $\{a_n\}_{n \geq 1}$ be defined by
\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]
Prove that
\[
a_n^{2025} >n^{2024}
\]
for all positive integers $n \geq 2$.
$\textbf{Proposed by Prajit Adhikari, Nepal.}$
Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define
\[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\]
for $n\ge 0.$ Evaluate
\[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]
Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula
$$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit.
(b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous.
Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals:
$ \textbf{(A)}\ \frac{13}{27}\qquad
\textbf{(B)}\ 33\qquad
\textbf{(C)}\ 21\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ \minus{}17$
The [i]Fibonacci numbers[/i] are defined recursively by the equation
\[ F_n \equal{} F_{n \minus{} 1} \plus{} F_{n \minus{} 2}\]
for every integer $ n \ge 2$, with initial values $ F_0 \equal{} 0$ and $ F_1 \equal{} 1$. Let $ G_n \equal{} F_{3n}$ be every third Fibonacci number. There are constants $ a$ and $ b$ such that every integer $ n \ge 2$ satisfies
\[ G_n \equal{} a G_{n \minus{} 1} \plus{} b G_{n \minus{} 2}.\]
Compute the ordered pair $ (a, b)$.
An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
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 $f_{1991}(2^{1990})$.
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
Let $c$ be a positive integer. Consider the sequence $x_1,x_2,\ldots$ which satisfies $x_1=c$ and, for $n\ge 2$,
\[x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1\]
where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. Determine an expression for $x_n$ in terms of $n$ and $c$.
[i]Huang Yumin[/i]
Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula
\[u_{n+2 }= u_n - u_{n+1}.\]
[b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined.
[b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.
Starting with four given integers $a_1, b_1, c_1, d_1$ is defined recursively for all positive integers $n$:
$$a_{n+1} := |a_n - b_n|, b_{n+1} := |b_n - c_n|, c_{n+1} := |c_n - d_n|, d_{n+1} := |d_n - a_n|.$$
Prove that there is a natural number $k$ such that all terms $a_k, b_k, c_k, d_k$ take the value zero.
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?
A sequence $(x_n)$ satisfies $x_{n+1}=\frac{x_n^2+a}{x_{n-1}}$ for all $n\in\mathbb N$. Prove that if $x_0,x_1$, and $\frac{x_0^2+x_1^2+a}{x_0x_1}$ are integers, then all the terms of sequence $(x_n)$ are integers.
Polynomial $ f\left(x\right)$ satisfies $ \left(x \minus{} 1\right)f\left(x \plus{} 1\right) \minus{} \left(x \plus{} 2\right)f\left(x\right) \equal{} 0$ for every $ x\in \Re$. If $ f\left(2\right) \equal{} 6$, $ f\left({\tfrac{3}{2}} \right) \equal{} ?$
$\textbf{(A)}\ -6 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ \frac {3}{2} \qquad\textbf{(D)}\ \frac {15}{8} \qquad\textbf{(E)}\ \text{None}$
15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ?
Consider the sequence of numbers defined by $a_1 = 7$, $a_2 = 7^7$ , $ ...$ , $a_n = 7^{a_{n-1}}$ for $n \ge 2$. Determine the last digit of the decimal representation of $a_{2021}$.
We have an infinite sequence of real numbers $x_0,x_1, x_2, ... $ such that $x_{n+1} = \sqrt{x_n -\frac14}$ holds for all natural $n$ and moreover $x_0 \in \frac12$.
(a) Prove that for every natural $n$ holds: $x_n > \frac12$
(b) Prove that $\lim_{n \to \infty} x_n$ exists. Calculate this limit.
Consider the sequence $a_1 = 3$ and $a_{n + 1} =\frac{3a_n^2+1}{2}-a_n$ for $n = 1 ,2 ,...$.
Prove that if $n$ is a power of $3$ then $n$ divides $a_n$ .