Found problems: 1239
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/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]
Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
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]
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$.
Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.
Determine all pairs $(c, d) \in \mathbb{R}^2$ of real constants such that there is a sequence $(a_n)_{n\geq1}$ of positive real numbers such that, for all $n \geq 1$, $$a_n \geq c \cdot a_{n+1} + d \cdot \sum_{1 \leq j < n} a_j .$$
An infinite sequence is given by $x_1=2, x_2=7, x_{n+1} = 4x_n - x_{n-1}$ for all $n \geq 2$. Does there exist a perfect square in this sequence?
[hide="Remark"]During the test the initial value of $x_1$ was given as $1$, thus the problem was not graded[/hide]
Let $c_1, \ldots, c_n \in \mathbb{R}$ with $n \geq 2$ such that \[ 0 \leq \sum^n_{i=1} c_i \leq n. \] Show that we can find integers $k_1, \ldots, k_n$ such that \[ \sum^n_{i=1} k_i = 0 \] and \[ 1-n \leq c_i + n \cdot k_i \leq n \] for every $i = 1, \ldots, n.$
[hide="Another formulation:"]
Let $x_1, \ldots, x_n,$ with $n \geq 2$ be real numbers such that \[ |x_1 + \ldots + x_n| \leq n. \] Show that there exist integers $k_1, \ldots, k_n$ such that \[ |k_1 + \ldots + k_n| = 0. \] and \[ |x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1 \] for every $i = 1, \ldots, n.$ In order to prove this, denote $c_i = \frac{1+x_i}{2}$ for $i = 1, \ldots, n,$ etc.
[/hide]
Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $
[i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]
Prove that the sequence defined by:
$$ y_ {n + 1} = \frac {1} {2} (3y_ {n} + \sqrt {5y_ {n} ^ {2} -4}) , \,\, \forall n \ge 0$$ with $ y_ {0} = 1$ consists only of integers.
Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as:
$u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$
a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit
b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$
The teacher writes numbers $1$ at both ends of the blackboard. The first student adds a $2$ in the middle between them, each next student adds the sum of each two adjacent numbers already on the blackboard between them (hence there are numbers $1, 3, 2, 3, 1$ on the blackboard after the second student, $1, 4, 3, 5, 2, 5, 3, 4, 1$ after the third student etc.) Find the sum of all numbers on the blackboard after the $n$-th student.
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$.
Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.
There are $n=1681$ children, $a_1,a_2,...,a_{n}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.
Let $(x_n)_{n\ge2}$ be a sequence of real numbers such that $x_2>0$ and $x_{n+1}=-1+\sqrt[n]{1+nx_n}$ for $n\ge2$. Find
(a) $\lim_{n\to\infty}x_n$,
(b) $\lim_{n\to\infty}nx_n$.
There're two positive inegers $a_1<a_2$. For every positive integer $n \geq 3$ let $a_n$ be the smallest integer that bigger than $a_{n-1}$ and such that there's unique pair $1\leq i< j\leq n-1$ such that this number equals to $a_i+a_j$. Given that there're finitely many even numbers in this sequence. Prove that sequence $\{a_{n+1}-a_n \}$ is periodic starting from some element.
There are $2022$ numbers arranged in a circle $a_1, a_2, . . ,a_{2022}$. It turned out that for any three consecutive $a_i$, $a_{i+1}$, $a_{i+2}$ the equality $a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}$. Prove that $\sum^{2022}_{i=1} a_ia_{i+2} = 0$, if we know that $a_{2023} = a_1$, $a_{2024} = a_2$.
[b]a)[/b] Prove that there exists an unique sequence of real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying
$$ -\text{ctg} x_n=x_n\in\left( (2n+1)\pi /2,(n+1)\pi \right) , $$
for any nonnegative integer $ n. $
[b]b)[/b] Show that $ \lim_{n\to\infty } \left( \frac{x_n}{(n+1)\pi } \right)^{n^2} =e^{-1/\pi^2} . $
[i]Cătălin Zârnă[/i]
Consider the sequences $(a_n), (b_n)$ defined by
\[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \]
Find the smallest integer $m$ for which $b_m > a_{100}.$
Is there an infinite sequence of positive integers $\{a_n\}_{n = 1}^{\infty}$ which contains each positive integer exactly once and is such that the number $a_n + a_{n + 1} $ is a perfect square for each $n$?
The sequence $\{a_n\}$ satisfies $a_1 = \frac{1}{2}$, $a_2 = \frac{3}{8}$, and $a_{n + 1}^2 + 3 a_n a_{n + 2} = 2 a_{n + 1} (a_n + a_{n + 2}) (n \in \mathbb{N^*})$.
$(1)$ Determine the general formula of the sequence $\{a_n\}$;
$(2)$ Prove that for any positive integer $n$, there is $0 < a_n < \frac{1}{\sqrt{2n + 1}}$.
Let $(u_n)$ be a sequence of real numbers which satisfies
$$u_{n+2}=|u_{n+1}|-u_n\qquad\text{for all }n\in\mathbb N.$$Prove that there exists a positive integer $p$ such that $u_n=u_{n+p}$ holds for all $n\in\mathbb N$.
a) Prove that for each positive integer $n$ there is a unique positive integer $a_n$ such that $$(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . $$
b) Prove that $a_{2010}$ is divisible by $5\cdot 4^{2009}$ and find the quotient
Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows:
\[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\]
Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$