Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)? \[ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )\] \[ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )\] (2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

You are given a word consisting of letters $a;b;c$ You can apply 3 operations on this word: [b]1)[/b] You can write any $3$ letter long $\text{subword}$ in reverse. ($\text{xyz}\rightarrow \text{zyx}$) [b]2)[/b] You can add same $2$ letters between any $2$ consecutive letters. ($\text{xyxy}\rightarrow \text{xy}$[b]zz[/b]$\text{xy}$) [b]3)[/b] You can remove any $\text{subword}$ in the given form $\text{xyyx}$ ($\text{x}$[b]yzzy[/b]$\text{xy}\rightarrow\text{xxy}$) Given these $3$ operations, can you go from $\text{abccab}$ to $\text{baccba}$? (Note: A $\text{subword}$ is a string of consecutive letters from the given word)

Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

Let $S$ be the set of all points in the plane whose coordinates are positive integers less than or equal to $100$ (so $S$ has $100^2$ elements), and let $L$ be the set of all lines $\ell$ such that $\ell$ passes through at least two points in $S$. Find, with proof, the largest integer $N \geq 2$ for which it is possible to choose $N$ distinct lines in $L$ such that every two of the chosen lines are parallel.

$20$ football teams take part in a tournament . On the first day all the teams play one match . On the second day all the teams play a further match . Prove that after the second day it is possible to select $10$ teams, so that no two of them have yet played each other. ( S . A . Genkin)

Determine all pairs of non-negative integers $(m, n)$ with m ≥n, such that $(m+n)^3$ divides $2n(3m^2+n^2)+8$

Let $ \{\xi_{k \ell} \}_{k,\ell=1}^{\infty}$ be a double sequence of random variables such that \[ \Bbb{E}( \xi_{ij} \xi_{k\ell})= \mathcal{O} \left(\frac{ \log(2|i-k|+2)}{ \log(2|j-\ell|+2)^{2}}\right) \;\;\;(i,j,k,\ell =1,2, \ldots ) \\\ .\] Prove that with probability one, \[ \frac{1}{mn} \sum_{k=1}^m \sum_{\ell=1}^n \xi_{k\ell} \rightarrow 0 \;\;\textrm{as} \; \max (m,n)\rightarrow \infty\ \\ .\] [i]F. Moricz[/i]

Find the maximum of $x+y$ given that $x$ and $y$ are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\]

Find all real values of a for which the equation $ (a \minus{} 3x^2 \plus{} \cos \frac {9\pi x}{2})\sqrt {3 \minus{} ax} \equal{} 0$ has an odd number of solutions in the interval $ [ \minus{} 1,5]$

Let $c>2$ and $a_0,a_1, \ldots$ be a sequence of real numbers such that \begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*} for any $n$ $\in$ $\mathbb{N}$. Prove, $a_1=0$

Dr. G has a bag of five marbles and enjoys drawing one marble from the bag, uniformly at random, and then putting it back in the bag. How many draws, on average, will it take Dr. G to reach a point where every marble has been drawn at least once?

Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.

An automobile went up a hill at a speed of $ 10$ miles an hour and down the same distance at a speed of $ 20$ miles an hour. The average speed for the round trip was: $ \textbf{(A)}\ 12\frac {1}{2} \text{ mph} \qquad\textbf{(B)}\ 13\frac {1}{3} \text{ mph} \qquad\textbf{(C)}\ 14\frac {1}{2} \text{ mph} \qquad\textbf{(D)}\ 15 \text{ mph}$ $ \textbf{(E)}\ \text{none of these}$

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Assume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$. Find $f(100)$.

Prove that among any ten consecutive integers at least one is relatively prime to each of the others.

For any two positive integers $n$ and $p$, prove that there are exactly ${{(p+1)}^{n+1}}-{{p}^{n+1}}$ functions $f:\left\{ 1,2,...,n \right\}\to \left\{ -p,-p+1,-p+2,....,p-1,p \right\}$ such that $\left| f(i)-f(j) \right|\le p$ for all $i,j\in \left\{ 1,2,...,n \right\}$.

Given an acute triangle $ABC$ with $AB<AC$, let $D$ be the foot of altitude from $A$ to $BC$ and let $M\neq D$ be a point on segment $BC$.$\,J$ and $K$ lie on $AC$ and $AB$ respectively such that $K,J,M$ lies on a common line perpendicular to $BC$. Let the circumcircles of $\triangle ABJ$ and $\triangle ACK$ intersect at $O$. Prove that $J,O,M$ are collinear if and only if $M$ is the midpoint of $BC$. [i]Proposed by Wong Jer Ren[/i]

Let $(a_n)_{n\ge1}$ and $(b_n)_{n\ge1}$ be positive real sequences such that $$\lim_{n\to\infty}\frac{a_{n+1}-a_n}n=a\in\mathbb R^*_+\enspace\text{and}\enspace\lim_{n\to\infty}\frac{b_{n+1}}{nb_n}=b\in\mathbb R^*_+$$ Compute $$\lim_{n\to\infty}\left(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}}\right)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Neculai Stanciu[/i]

Find the number of ordered triples of divisors $(d_1, d_2, d_3)$ of $360$ such that $d_1d_2d_3$ is also a divisor of $360$. In this section, the word [i]divisor [/i]is used to refer to a [i]positive divisor[/i] of an integer.

Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

Isabella the geologist discovers a diamond deep underground via an X-ray machine. The diamond has the shape of a convex cyclic pentagon $PABCD$ with $AD|| BC$. Soon after the discovery, her X-ray breaks, and she only recovers partial information about its dimensions. She knows that $AD = 70, BC = 55, PA : PD = 3 : 4$, and $PB : PC = 5 : 6$. Compute $PB$.

Given an $a\times b$ chessboard where $a,b\ge3$, alice wants to use only $L$-dominoes (as the figure shows) to cover this chessboard. How many grids, at least, are covered even times? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi82LzhmZDkwMGQzZjU3M2QxMzk4Y2NjNDg5ZTMwM2ZmYjJiMWU3MmUwLnBuZw==&rn=MjAxOC1DMy5wbmc=[/img]

Let $ABCD$ be a tetrahedron with $AB=CD=1300$, $BC=AD=1400$, and $CA=BD=1500$. Let $O$ and $I$ be the centers of the circumscribed sphere and inscribed sphere of $ABCD$, respectively. Compute the smallest integer greater than the length of $OI$. [i] Proposed by Michael Ren [/i]