Found problems: 1239
Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and
$$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$
Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$
Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions:
[list]
[*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square.
[*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$.
[/list]
Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$.
And determine the value of $\sum_{k=N}^{N+2021}a_k$.
Prove that for any integer $n \ge 2$, there exists a unique finite sequence $x_0, x_1,..., x_n$ of real numbers which satisfies $x_0 = x_n = 0$ and $x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0$ for all $i = 1,2,...,n - 1$. Prove moreover that $ |x_i| \le \frac12$ for all $i = 1,2,...,n - 1$.
Nguyễn Duy Thái Sơn
Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$.
Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.
Let $a_1, a_2, a_3, \dots$ be a sequence of integers such that
$\text{(i)}$ $a_1=0$
$\text{(ii)}$ for all $i\geq 1$, $a_{i+1}=a_i+1$ or $-a_i-1$.
Prove that $\frac{a_1+a_2+\cdots+a_n}{n}\geq-\frac{1}{2}$ for all $n\geq 1$.
Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.
Prove that the sequence $(a_n)$, where
$$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$
Let be a bounded sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying the recurrence:
$$ x_{n+3} =\sqrt[3]{3x_n-2} . $$
Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent.
[i]Cristinel Mortici[/i]
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.
For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $
where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$.
.
Given the sequence $(a_n)$, in which $a_n =\frac14 n^4 - 10n^2(n - 1)$, with $n = 0, 1, 2,...$ Determine the smallest term of the sequence.
Let $(s_n)_{n\geq 1}$ be a sequence given by $s_n=-2\sqrt{n}+\sum \limits_{k=1}^n\frac{1}{\sqrt{k}}$ with $\lim \limits_{n \to \infty}s_n=s=$Ioachimescu constant and $(a_n)_{n\geq 1}$ , $(b_n)_{n\geq 1}$ be a positive real sequences such that $$\lim \limits_{n\to \infty}\frac{a_{n+1}}{na_n}=a\in \mathbb{R}^*_+, \lim \limits_{n\to \infty}\frac{b_{n+1}}{b_n\sqrt{n}}=b\in \mathbb{R}^*_+$$Compute$$\lim \limits_{n\to \infty}\left(1+e^{s_n}-e^{s_{n+1}}\right)^{\sqrt[n]{a_nb_n}}$$
For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i\equal{}0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 \equal{} 1,$ $ a_i \leq a_{i\plus{}1} \plus{} a_{i\plus{}2}$ for $ i \equal{} 0, \ldots, n \minus{} 2.$
The sequences of natural numbers $p_n$ and $q_n$ are given such that
$$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$
Prove that $p_n$ and $q_m$ are coprime for any m and n.
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$.
[i]Remark:[/i] A sum over the elements of the empty set is assumed to be $0$.
Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.
Find the largest possible value of the positive integer $N$ given that there exist positive integers $a_1, a_2, \dots, a_N$ satisfying
$$ a_n = \sqrt{(a_{n-1})^2 + 2018 \, a_{n-2}}\:, \quad \text{for } n = 3,4,\dots,N. $$
In a sequence $a_1, a_2, ..$ of real numbers the product $a_1a_2$ is negative, and to define $a_n$ for $n > 2$ one pair $(i, j)$ is chosen among all the pairs $(i, j), 1 \le i < j < n$, not chosen before, so that $a_i +a_j$ has minimum absolute value, and then $a_n$ is set equal to $a_i + a_j$ . Prove that $|a_i| < 1$ for some $i$.
Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that
(a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence);
(b) if the sequence is periodic, then the first number is rational.
(G Shabat)
Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that
$$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$
[hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]