$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$

For $n \in \mathbb{N}$, define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$. Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$.

For each positive integer $n$, let $I_n$ denote the number of integers $p$ for which $50^n<7^p<50^{n+1}$. (a) Prove that, for each $n$, $I_n$ is either $2$ or $3$. (b) Prove that $I_n=3$ for infinitely many $n\in\mathbb N$, and find at least one such $n$.

Let there be a sequence $a_n$ such that $a_1 = 2,a_2 = 0, a_3 = 1, a_4 = 0$, and for $n \ge 1, a_{n+4}$ is the remainder when $a_n + 2a_{n+1} + 3a_{n+2} + 4a_{n+3}$ is divided by $9$. Prove that there are no positive integer $k$ such that $$a_k = 0, a_{k+1} = 1, a_{k+2} = 0,a_{k+3} = 2.$$

Let $\{a_n\}$ be a sequence of integers satisfying the following conditions. [list] [*] $a_1=2021^{2021}$ [*] $0 \le a_k < k$ for all integers $k \ge 2$ [*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$. [/list] Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.

Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.

Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\dots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2\dots,x_n\in \mathbb{R}$. Bob wins if he can pick a vector $(z_1,z_2,\dots,z_n)\in S$ such that $$\sum_{i=1}^n x_iy_i\le \sum_{i=1}^n x_iz_i,$$otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win. [i]Proposed by DVDthe1st[/i]

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold: (1) $a_i = a_{i+n}$ for any $i$, (2) $a_i$ is not divisible by $n$ for any $i$, (3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$. For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?

Let $1 + 1/2 + 1/3 +... + 1/n = a_n/b_n$, where $a_n$ and $b_n$ are relatively prime. Show that there exist infinitely many positive integers $n$, such that $b_{n+1} < b_n$. (8)

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

Prove that there are no positive integers $n$ and $k\le n$ such that the numbers $$\binom nk,\binom n{k+1},\binom n{k+2},\binom n{k+3}$$in this order form an arithmetic progression.

Consider a sequence $\{a_i\}^\infty_{i\ge1}$ of positive integers. For all positvie integers $n$ prove that there exists infinitely many positive integers $k$ such that there is no pair $(m,t)$ of positive integers where $m>n$ and $$kn+a_n=tm(m+1)+a_m$$

Let $f : N \to [0, \infty)$ be a function satisfying the following conditions: a) $f(4)=2$ b) $\frac{1}{f( 0 ) + f( 1)} + \frac{1}{f( 1 ) + f( 2 )} + ... + \frac{1}{f (n ) + f(n + 1) }= f ( n + 1)$ for all integers $n \ge 0$. Find $f(n)$ in closed form.

Let $n$ be a given positive integer. (a) Do there exist $2n+1$ consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in the ascending order such that $a_0+a_1+\ldots+a_n=a_{n+1}+\ldots+a_{2n}$? (b) Do there exist consecutive positive integers $a_0,a+1,\ldots,a_{2n}$ in ascending order such that $a_0^2+a_1^2+\ldots+a_n^2=a_{n+1}^2+\ldots+a_{2n}^2$? (c) Do there exist consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in ascending order such that $a_0^3+a_1^3+\ldots+a_n^3=a_{n+1}^3+\ldots+a_{2n}^3$? [hide=Official Hint]You may study the function $f(x)=(x-n)^3+\ldots+x^3-(x+1)^3-\ldots-(x+n)^3$ and prove that the equation $f(x)=0$ has a unique solution $x_n$ with $3n(n+1)<x_n<3n(n+1)+1$. You may use the identity $1^3+2^3+\ldots+n^3=\frac{n^2(n+1)^2}2$.[/hide]