In the lower left corner of the $8 \times 8$ chessboard there is a king. He can move one cell either to the right, or up, or diagonally - to the right and up. How many ways can the king go to the upper right corner of the board if his route does not contain cells located on opposite sides of the diagonal going from the lower left to the upper right corner of the board?

Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$

Compute the maximum area of a rectangle which can be inscribed in a triangle of area $M$.

Let S be the point of intersection of the two lines $ l_1 : 7x \minus{} 5y \plus{} 8 \equal{} 0$ and $ l_2 : 3x \plus{} 4y \minus{} 13 \equal{} 0.$ Let $ P \equal{} (3, 7), Q \equal{} (11, 13),$ and let $ A$ and $ B$ be points on the line $ PQ$ such that $ P$ is between $ A$ and $ Q,$ and $ B$ is between $ P$ and $ Q,$ and such that \[ \frac{PA}{AQ} \equal{} \frac{PB}{BQ} \equal{} \frac{2}{3}.\] Without finding the coordinates of $ B$ find the equations of the lines $ SA$ and $ SB.$

The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$: $(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$.

Let $a_1,a_2,a_3,\cdots $ be an infinite sequence of distinct integers. Prove that there are infinitely many primes $p$ that distinct positive integers $i,j,k$ can be found such that $p\mid a_ia_ja_k-1$. [i]Proposed by Mohsen Jamali[/i]

Let $p = 2017$ be a prime number. Let $E$ be the expected value of the expression \[3 \;\square\; 3 \;\square\; 3 \;\square\; \cdots \;\square\; 3 \;\square\; 3\] where there are $p+3$ threes and $p+2$ boxes, and one of the four arithmetic operations $\{+, -, \times, \div\}$ is uniformly chosen at random to replace each of the boxes. If $E = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $p$. [i]Proposed by Michael Tang[/i]

Let $n$ be a positive integer and $p_1, p_2, p_3, \ldots p_n$ be $n$ prime numbers all larger than $5$ such that $6$ divides $p_1 ^2 + p_2 ^2 + p_3 ^2 + \cdots p_n ^2$. prove that $6$ divides $n$.

How many positive divisors does number $20!$ have?

Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\ell$ in the plane which is horizontal, vertical, or has slope $+1$ or $-1$, she tells Bob the number of points of $S$ that lie on $\ell$. Bob wins if he can determine the set $S$. Prove that if Alice picks $S$ to be of the form \[S = \{(x, y) \in \mathbb{Z}^2 \mid m \le x^2 + y^2 \le n\}\] for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.) [i]Proposed by Mark Sellke[/i]

Let $\triangle ABC$ be a triangle with $\angle A$-acute. Let $P$ be a point inside $\triangle ABC$ such that $\angle BAP = \angle ACP$ and $\angle CAP =\angle ABP$. Let $M, N$ be the centers of the incircle of $\triangle ABP$ and $\triangle ACP$, and $R$ the radius of the circumscribed circle of $\triangle AMN$. Prove that $\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}. $

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

On the board all the 20-digit numbers which have 10 ones and 10 twos in their decimal form are written. It is allowed to choose two different digits in any number and to reverse the order of digits in the interval between them. What is the maximal quantity of equal numbers which is possible to get on the board using such operations?

Given a prime number $p\ge 3,$ and an integer $d \ge 1$. Prove that there exists an integer $n\ge 1,$ such that $\gcd(n,d) = 1,$ and the product \[P=\prod\limits_{1 \le i < j < p} {({i^{n + j}} - {j^{n + i}})} \text{ is not divisible by } p^n.\]

There is $n$ black points in the plane.We do the following algorithm: Start from any point from those $n$ points and colour it red. Then connect this point to the nearest black point available and colour this new point red. Then do the same with this point but at any step , but you are never allowed to draw a line which intersects on of the current drawn segments. If you reach an intersection , the algorithm is over. Is it true that for any $n$ and at any initial position , we can start from a point st in the algorithm , we reach all the points?

The sum of the digits of a natural number $n$ is denoted by $S(n)$. Prove that $S(8n) \ge \frac{1}{8} S(n)$ for each $n$.

An acute triangle $ABC$ is given. Let us denote by $D$ and $E$ the orthogonal projections, respectively of points $ B$ and $C$ on the bisector of the external angle $BAC$. Let $F$ be the point of intersection of the lines $BE$ and $CD$. Show that the lines $AF$ and $DE$ are perpendicular.

Indicate in which one of the following equations $ y$ is neither directly nor inversely proportional to $ x$: $ \textbf{(A)}\ x \plus{} y \equal{} 0 \qquad\textbf{(B)}\ 3xy \equal{} 10 \qquad\textbf{(C)}\ x \equal{} 5y \qquad\textbf{(D)}\ 3x \plus{} y \equal{} 10$ $ \textbf{(E)}\ \frac {x}{y} \equal{} \sqrt {3}$

Let $a,b,c,d x,y,z, w$ be real numbers such that $$\begin{matrix} ax -by-c z-dw =0\\ b x +a y -d z +cw=0\\ c x+ d y +a z -b w=0\\ dx-c y+bz+aw=0 \end{matrix}$$ prove that $$a=b=c=d=0, \ \ or \ \ x=y=z=w=0$$

Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.

Let \[S=\sqrt{1+\dfrac1{1^2}+\dfrac1{2^2}}+\sqrt{1+\dfrac1{2^2}+\dfrac1{3^2}}+\cdots+\sqrt{1+\dfrac1{2007^2}+\dfrac1{2008^2}}.\] Compute $\lfloor S^2\rfloor$.

Initially an urn contains 100 black marbles and 100 white marbles. Repeatedly, three marbles are removed from the urn and replaced from a pile outside the urn as follows: \[ \begin{tabular}{ccc} \textbf{\underline{MARBLES REMOVED}} & & \textbf{\underline{REPLACED WITH}} \\ 3 black & & 1 black \\ 2 black, 1 white & &1 black, 1 white\\ 1 black, 2 white & & 2 white \\ 3 white & & 1 black, 1 white \end{tabular} \] Which of the following sets of marbles could be the contents of the urn after repeated applications of this procedure? $ \textbf{(A)}\ \text{2 black marbles} $ $\textbf{(B)}\ \text{2 white marbles} $ $\textbf{(C)}\ \text{1 black marble} $ $\textbf{(D)}\ \text{1 black and 1 white marble} $ $\textbf{(E)}\ \text{1 white marble} $

Let $ f:\mathbb{D}\longrightarrow\mathbb{C} $ be a continuous function, where $ \mathbb{D} $ is the closed unit disk. Suppose that $ f $ is holomorphic on the open unit disk and that $ e^{i\theta } $ are roots, for any $ \theta\in\left[ 0,\pi /4 \right] . $ Show that $ f=0_{\mathbb{D}} . $

Elisenda has a piece of paper in the shape of a triangle with vertices $A, B,$ and $C$ such that $AB = 42.$ She chooses a point $D$ on segment $AC,$ and she folds the paper along line $BD$ so that $A$ lands at a point $E$ on segment $BC.$ Then, she folds the paper along line $DE.$ When she does this, $B$ lands at the midpoint of segment $DC.$ Compute the perimeter of the original unfolded triangle.

Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? [asy] size(6cm); draw(circle((0,1),1), linewidth(1.2)); draw((-1,0)--(1.25,0), linewidth(1.2)); draw(circle((1,1/4),1/4), linewidth(1.2)); [/asy] $\textbf{(A)}~\displaystyle\frac{1}{9} \qquad\textbf{(B)}~1 \qquad\textbf{(C)}~\displaystyle\frac{10}{9} \qquad\textbf{(D)}~\displaystyle\frac{11}{9} \qquad\textbf{(E)}~\displaystyle\frac{19}{9}$