Found problems: 1239
To commemorate the $43rd$ anniversary of the restoration of mathematics competitions, a mathematics enthusiast
arranges the first $2021$ integers $1,2,\dots,2021$ into a sequence $\{a_n\}$ in a certain order, so that the sum of any consecutive $43$ items in the sequence is a multiple of $43$.
(1) If the sequence of numbers is connected end to end into a circle, prove that the sum of any consecutive $43$ items on the circle is also a multiple of $43$;
(2) Determine the number of sequences $\{a_n\}$ that meets the conditions of the question.
The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.
A sequence $\{y_i\}$ is given, where $y_0=-\frac{1}{4},y_1=0$. For every positive integer $n$ the following equality holds:
$$y_{n-1}+y_{n+1}=4y_n+1$$
Prove that for every positive integer $n$ the number $2y_{2n}+\frac{3}{2}$
a) is a positive integer
b) is a square of a positive integer
[i]D. Zmiaikou[/i]
The sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,... $suffices for all positive integers $n$ of the following recursion:
$a_{n+1} = a_n - b_n$ and $b_{n+1} = 2b_n$, if $a_n \ge b_n$,
$a_{n+1} = 2a_n$ and $b_{n+1} = b_n - a_n$, if $a_n < b_n$.
For which pairs $(a_1, b_1)$ of positive real initial terms is there an index $k$ with $a_k = 0$?
Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:
\[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\]
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .
[i]Proposed by Marcin Kuczma, Poland[/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]
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
[b]a)[/b] Provide an example of a sequence $ \left( a_n \right)_{n\ge 1} $ of positive real numbers whose series converges, and has the property that each member (sequence) of the family of sequences $ \left(\left( n^{\alpha } a_n \right)_{n\ge 1}\right)_{\alpha >0} $ is unbounded.
[b]b)[/b] Let $ \left( b_n \right)_{n\ge 1} $ be a sequence of positive real numbers, having the property that
$$ nb_{n+1}\leqslant b_1+b_2+\cdots +b_n, $$
for any natural number $ n. $ Prove that the following relations are equivalent:
$\text{(i)} $ there exists a convergent member (series) of the family of series $ \left( \sum_{i=1}^{\infty } b_i^{\beta } \right)_{\beta >0} $
$ \text{(ii)} $ there exists a member (sequence) of the family of sequences $ \left(\left( n^{\beta } b_n \right)_{n\ge 1}\right)_{\beta >0} $ that is convergent to $ 0. $
[i]Eugen Păltănea[/i]
Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $
Periodic sequences $(a_n),(b_n),(c_n)$ and $(d_n)$ satisfy the following conditions:
$$a_{n+1}=a_n+b_n,\enspace\enspace b_{n+1}=b_n+c_n,$$
$$c_{n+1}=c_n+d_n,\enspace\enspace d_{n+1}=d_n+a_n,$$
for $n=1,2,\ldots$. Prove that $a_2=b_2=c_2=d_2=0$.
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
The natural numbers $x_1$ and $x_2$ are less than $1000$. We construct a sequence:
$$x_3 = |x_1 - x_2|$$
$$x_4 = min \{ |x_1 - x_2|, |x_1 - x_3|, |x_2 - x_3|\}$$
$$...$$
$$x_k = min \{ |x_i - x_j|, 0 <i < j < k\}$$
$$...$$
Prove that $x_{21} = 0$.
Let $a_n$ be a sequence of positive numbers such that:
i) $\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}$, for every $n\in\mathbb{N}^{\star}$
ii) $\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1$, for every $ k,n\in\mathbb{N}^{\star}$ with $|k-n|\neq 1$.
(a) Prove that $(a_n)$ is a geometric progression.
(n) Prove that exists $t>0$, such that $\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t$
Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/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?
The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and
$a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$.
Prove that no term in $a_i$ is in the range $[1612, 2012]$.
All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$-th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$.) Is there an integer $n$ such that all the numbers
\[f(n),f(2n),f(3n),\dots,f(10n^2)\]
are perfect squares?
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.
Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?
[i]Proposed by Mihai Bălună, Romania[/i]
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.
[i]Proposed by Ilija Jovčeski[/i]
Let $m\ge2$ be an integer. The sequence $(a_n)_{n\in\mathbb N}$ is defined by $a_0=0$ and $a_n=\left\lfloor\frac nm\right\rfloor+a_{\left\lfloor\frac nm\right\rfloor}$ for all $n$. Determine $\lim_{n\to\infty}\frac{a_n}n$.
Let $0 < \frac{1}{q} \leq \frac{1}{p} < 1$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u^2_k > a^p_k$ and $v_k > b^q_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1\leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$ , then $$\left(\sum \limits_{k=1}^n\left(l^p\left(u_k+v_k\right)^2-\left(a_k+b_k\right)^p\right)\right)^{\frac{1}{p}}\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^p\right)\right)^{\frac{1}{p}}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^p\right)\right)^{\frac{1}{p}}$$where $$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$
Let $S$ be a sequence $n_1, n_2,..., n_{1995}$ of positive integers such that $n_1 +...+ n_{1995 }=m < 3990$. Prove that for each integer $q$ with $1 \le q \le m$, there is a sequence $n_{i_1} , n_{i_2} , ... , n_{i_k}$ , where $1 \le i_1 < i_2 < ...< i_k \le 1995$, $n_{i_1} + ...+ n_{i_k} = q$ and $k$ depends on $q$.
Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.