Find all pairs of natural numbers $(m,n)$, $m\leq n$, such that there exists a table with $m$ rows and $n$ columns filled with the numbers 1 and 0, satisfying the following property: If in a cell there's a 0 (respectively a 1), then the number of zeros (respectively ones) in the row of this cell is equal to the number of zeros (respectively ones) in the column of this cell.

For the sequence \[S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},\]find the limit \[\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).\]

Nine journalists from different countries attend a press conference. None of these speaks more than three languages, and each pair of the journalists share a common language. Show that there are at least five journalists sharing a common language.

How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$?

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$ $\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$

Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Consider a point $P$ inside the triangle having $PA=1$, $PB=2$ and $PC=3$. Find the maximum possible area of the triangle $ABC$.

Let $n\ge 2$ be a positive integer. $S$ is a set of $2n$ airports. For two arbitrary airports $A,B$, if there is an airway from $A$ to $B$, then there is an airway from $B$ to $A$. Suppose that $S$ has only one independent set of $n$ airports. Let the independent set $X$. Prove that there exists an airport $P\in S\setminus X$ which satisfies following condition. [b]Condition[/b] : For two arbitrary distinct airports $A,B\in S\setminus \{P\}$, if there exists a path connecting $A$ and $B$, then there exists a path connecting $A$ and $B$ which does not pass $P$.

The altitude $CH$ of a right triangle $ABC$, with $\angle{C}=90$, cut the angles bisectors $AM$ and $BN$ at $P$ and $Q$, and let $R$ and $S$ be the midpoints of $PM$ and $QN$. Prove that $RS$ is parallel to the hypotenuse of $ABC$

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. The tangent line of the circumcircle of triangle $BHC$ at $H$ meets $AB$ and $AC$ at $E$ and $F$ respectively. If $O$ is the circumcenter of triangle $AEF$, prove that the circumcircle of triangle $EOF$ is tangent to $\Omega$.

The altitudes $AD$ and $BE$ of acute triangle $ABC$ intersect at $H$. Let $F$ be the intersection of $AB$ and a line that is parallel to the side $BC$ and goes through the circumcentre of $ABC$. Let $M$ be the midpoint of $AH$. Prove that $\angle CMF=90^\circ$

Suppose $[a \,\,\, b]$ denotes the average of $a$ and $b$, and $\{a\,\,\,b\,\,\,c\}$ denotes the average of $a$, $b$, and $c$. What is $\{\{1\,\,\, 1\,\,\, 0\}\,\,\, [0\,\,\, 1]\,\,\, 0\}$? $ \textbf{(A)}\ \frac{2}{9} \qquad\textbf{(B)}\ \frac{5}{18} \qquad\textbf{(C)}\ \frac{1}{3} \qquad\textbf{(D)}\ \frac{7}{18} \qquad\textbf{(E)}\ \frac{2}{3} $

Given is the polynomial $P(x)$ and the numbers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_1a_2a_3\not=0$. Suppose that for every $x$, we have \[P(a_1x+b_1)+P(a_2x+b_2)=P(a_3x+b_3)\] Prove that the polynomial $P(x)$ has at least one real root.

For integers $k$ and $n$ such that $1\le k<n$, let $C^n_k=\frac{n!}{k!(n-k)!}$. Then $\left(\frac{n-2k-1}{k+1}\right)C^n_k$ is an integer $\textbf{(A) }\text{for all }k\text{ and }n\qquad$ $\textbf{(B) }\text{for all even values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$ $\textbf{(C) }\text{for all odd values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$ $\textbf{(D) }\text{if }k=1\text{ or }n-1,\text{ but not for all odd values }k\text{ and }n\qquad$ $\textbf{(E) }\text{if }n\text{ is divisible by }k,\text{ but not for all even values }k\text{ and }n$

Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black) The figure can be rotated $90°, 180°$ or $270°$. Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.

Nonnegative reals $x_1$, $x_2$, $\dots$, $x_n$ satisfies $x_1+x_2+\dots+x_n=n$. Let $||x||$ be the distance from $x$ to the nearest integer of $x$ (e.g. $||3.8||=0.2$, $||4.3||=0.3$). Let $y_i = x_i ||x_i||$. Find the maximum value of $\sum_{i=1}^n y_i^2$.

Show that there are infinitely many primes.

Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?

A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$

Let $ABC$ be an acute triangle and $D$ be an intersection of the angle bisector of $A$ and side $BC$. Let $\Omega$ be a circle tangent to the circumcircle of triangle $ABC$ and side $BC$ at $A$ and $D$, respectively. $\Omega$ meets the sides $AB, AC$ again at $E, F$, respectively. The perpendicular line to $AD$, passing through $E, F$ meets $\Omega$ again at $G, H$, respectively. Suppose that $AE$ and $GD$ meet at $P$, $EH$ and $GF$ meet at $Q$, and $HD$ and $AF$ meet at $R$. Prove that $\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}$.

Suppose $a,b,c>0$ are integers such that \[abc-bc-ac-ab+a+b+c=2013.\] Find the number of possibilities for the ordered triple $(a,b,c)$.

The numbers one through eight are written, in that order, on a chalkboard. A mysterious higher power in possession of both an eraser and a piece of chalk chooses three distinct numbers $x$, $y$, and $z$ on the board, and does the following. First, $x$ is erased and replaced with $y$, after which $y$ is erased and replaced with $z$, and finally $z$ is erased and replaced with $x$. The higher power repeats this process some finite number of times. For example, if $(x,y,z)=(2,4,5)$ is chosen, followed by $(x,y,z)=(1,4,3)$, the board would change in the following manner: \[12345678 \rightarrow 14352678 \rightarrow 43152678\] Compute the number of possible final orderings of the eight numbers.

Let $f:\;\mathbb{R}\to\mathbb{R}$ be a continuously differentiable function that satisfies $f'(t)>f(f(t))$ for all $t\in\mathbb{R}$. Prove that $f(f(f(t)))\le0$ for all $t\ge0$. [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]

Let $ x_1, x_2, \cdots, x_n$ be positive real numbers . Prove that: $$\sum_ {i = 1}^n x_i ^2\geq \frac {1} {n + 1} \left (\sum_ {i = 1}^n x_i \right)^2+\frac{12(\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}. $$

Given a finite sequence $ S \equal{} (a_1,a_2,\ldots,a_n)$ of $ n$ real numbers, let $ A(S)$ be the sequence \[ \left(\frac {a_1 \plus{} a_2}2,\frac {a_2 \plus{} a_3}2,\ldots,\frac {a_{n \minus{} 1} \plus{} a_n}2\right) \]of $ n \minus{} 1$ real numbers. Define $ A^1(S) \equal{} A(S)$ and, for each integer $ m$, $ 2\le m\le n \minus{} 1$, define $ A^m(S) \equal{} A(A^{m \minus{} 1}(S)).$ Suppose $ x > 0$, and let $ S \equal{} (1,x,x^2,\ldots,x^{100})$. If $ A^{100}(S) \equal{} (1/2^{50})$, then what is $ x$? $ \textbf{(A) } 1 \minus{} \frac {\sqrt {2}}2\qquad \textbf{(B) } \sqrt {2} \minus{} 1\qquad \textbf{(C) } \frac 12\qquad \textbf{(D) } 2 \minus{} \sqrt {2}\qquad \textbf{(E) } \frac {\sqrt {2}}2$

Suppose $p < q < r < s$ are prime numbers such that $pqrs + 1 = 4^{p+q}$. Find $r + s$.