Found problems: 1340
Let $A$ be the set $A = \{ 1,2, \ldots, n\}$. Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$, $x+y$ cannot be divisible by $x-y$.
[i]Mircea Lascu, Dorel Mihet[/i]
Consider a $3\times7$ grid of squares. Each square may be coloured green or white.
[list]
(a) Is it possible to find a colouring so that no subrectangle has all four corner squares of the same colour?
(b) Is it possible for a $4\times 6$ grid?
[/list]
[i]Subrectangles must have their corners at grid-points of the original diagram. The corner squares of a subrectangle must be different. The original diagram is a subrectangle of itself.[/i]
Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that
\[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0?
\]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.
$ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$
Define the sequence $a_n$ by the rule $$a_{n+1} =\left \lfloor \frac{a_n} 2 \right \rfloor + \left \lfloor \frac{a_n}3 \right \rfloor$$ for $n \in \{1, 2, 3, 4, 5, 6, 7\}$, where $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$. If $a_8=8$, how many possible values are there for $a_1$ given that it is a positive integer?
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 10)[/i]
Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$. Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\]
A set of (unit) squares of a $n\times n$ table is called [i]convenient[/i] if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a [i]convenient [/i] set made of $m$ squares, which becomes in[i]convenient [/i] when any of its squares is removed.
Let $a,b,c,d\in\mathbb{Z}_{\ge 0}$, $d\ne 0$ and the function $f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0}$ defined by
\[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]
Prove that the following are equivalent:
[list=1]
[*] $f$ is surjective;
[*] $c=0$, $b<d$ and $0<a\le d$.
[/list]
[i]Tiberiu Trif[/i]
Consider the numbers arranged in the following way:
\[\begin{array}{ccccccc} 1 & 3 & 6 & 10 & 15 & 21 & \cdots \\ 2 & 5 & 9 & 14 & 20 & \cdots & \cdots \\ 4 & 8 & 13 & 19 & \cdots & \cdots & \cdots \\ 7 & 12 & 18 & \cdots & \cdots & \cdots & \cdots \\ 11 & 17 & \cdots & \cdots & \cdots & \cdots & \cdots \\ 16 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\]
Find the row number and the column number in which the the number $20096$ occurs.
A $ 3 \times n$ grid is filled as follows. The first row consists of the numbers from $ 1$ to $ n$ arranged in ascending order. The second row is a cyclic shift of the top row: $ i,i\plus{}1,...,n,1,2,...,i\minus{}1$ for some $ i$. The third row has the numbers $ 1$ to $ n$ in some order so that in each of the $ n$ columns, the sum of the three numbers is the same. For which values of $ n$ is it possible to fill the grid in this way? For all such $ n$, determine the number of different ways of filling the grid.
For every positive integer $ n$ consider
\[ A_n\equal{}\sqrt{49n^2\plus{}0,35n}.
\]
(a) Find the first three digits after decimal point of $ A_1$.
(b) Prove that the first three digits after decimal point of $ A_n$ and $ A_1$ are the same, for every $ n$.
For the integer numbers $i,j,k$ satisfying the condtion $i^2+j^2+k^2=2011$, what is the largest value of $i+j+k$?
$\textbf{(A)}\ 71 \qquad\textbf{(B)}\ 73 \qquad\textbf{(C)}\ 74 \qquad\textbf{(D)}\ 76 \qquad\textbf{(E)}\ 77$
Let $a,b\in \mathbb{R}$. Evaluate:
\[\lim_{n\to \infty}\left(\sqrt{a^2n^2+bn}-an\right)\]
Consider the sequence $(x_n)_{n\ge 1}$, defined by $x_n=\sqrt{n}-\lfloor \sqrt{n}\rfloor$. Denote by $A$ the set of the points $x\in \mathbb{R}$, for which there is a subsequence of $(x_n)_{n\ge 1}$ tending to $x$.
a) Prove that $\mathbb{Q}\cap [0,1]\subset A$.
b) Find $A$.
Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$.
Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.
A rectangular array has $ n$ rows and $ 6$ columns, where $ n \geq 2$. In each cell there is written either $ 0$ or $ 1$. All rows in the array are different from each other. For each two rows $ (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$ and $ (y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})$, the row $ (x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6})$ can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.
Let $a_0$ be a positive integer. Define the sequence $\{a_n\}_{n \geq 0}$ as follows: if \[ a_n = \sum_{i = 0}^jc_i10^i \] where $c_i \in \{0,1,2,\cdots,9\}$, then \[ a_{n + 1} = c_0^{2005} + c_1^{2005} + \cdots + c_j^{2005}. \] Is it possible to choose $a_0$ such that all terms in the sequence are distinct?
There are $a$ straight lines in a plane, no two of which are parallel to each other and no three intersect in one point.
a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least
$\lfloor \frac{(a-1)(a-2)}{10} \rfloor$
intersection points.
b) Find all $a$ for which the evaluation in a) is the best possible.
Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.
There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$.
For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$.
Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible).
Sorry for my bad English.
Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer.
[list=a]
[*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number?
[*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]
Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[\left\lfloor\dfrac{2008}1\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^2}2\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^3}3\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^4}4\right\rfloor,\,\,\,\,\,\,\,\,\,\ldots.\]
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
For a given number $\alpha{}$ let $f_\alpha$ be a function defined as \[f_\alpha(x)=\left\lfloor\alpha x+\frac{1}{2}\right\rfloor.\]Let $\alpha>1$ and $\beta=1/\alpha$. Prove that for any natural $n{}$ the relation $f_\beta(f_\alpha(n))=n$ holds.
[i]Proposed by I. Dorofeev[/i]
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
For how many positive integers $n \le 1000$ is $$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$$ not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)
$\textbf{(A) } 22 \qquad\textbf{(B) } 23 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 26$