(a) Decide whether the fields of the $8 \times 8$ chessboard can be numbered by the numbers $1, 2, \dots , 64$ in such a way that the sum of the four numbers in each of its parts of one of the forms [list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28446[/img][/list] is divisible by four. (b) Solve the analogous problem for [list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28447[/img][/list]

Sequence $ \{ f_n(a) \}$ satisfies $ \displaystyle f_{n\plus{}1}(a) \equal{} 2 \minus{} \frac{a}{f_n(a)}$, $ f_1(a) \equal{} 2$, $ n\equal{}1,2, \cdots$. If there exists a natural number $ n$, such that $ f_{n\plus{}k}(a) \equal{} f_{k}(a), k\equal{}1,2, \cdots$, then we call the non-zero real $ a$ a $ \textbf{periodic point}$ of $ f_n(a)$. Prove that the sufficient and necessary condition for $ a$ being a $ \textbf{periodic point}$ of $ f_n(a)$ is $ p_n(a\minus{}1)\equal{}0$, where $ \displaystyle p_n(x)\equal{}\sum_{k\equal{}0}^{\left[ \frac{n\minus{}1}{2} \right]} (\minus{}1)^k C_n^{2k\plus{}1}x^k$, here we define $ \displaystyle \frac{a}{0}\equal{} \infty$ and $ \displaystyle \frac{a}{\infty} \equal{} 0$.

Let $\Delta ABC$ be a triangle with circumcenter $O$ and circumcircle $w$. Let $S$ be the center of the circle which is tangent with $AB$, $AC$, and $w$ (in the inside), and let the circle meet $w$ at point $K$. Let the circle with diameter $AS$ meet $w$ at $T$. If $M$ is the midpoint of $BC$, show that $K,T,M,O$ are concyclic.

Let $p=2017$ be a prime. Suppose that the number of ways to place $p$ indistinguishable red marbles, $p$ indistinguishable green marbles, and $p$ indistinguishable blue marbles around a circle such that no red marble is next to a green marble and no blue marble is next to a blue marble is $N$. (Rotations and reflections of the same configuration are considered distinct.) Given that $N=p^m\cdot n$, where $m$ is a nonnegative integer and $n$ is not divisible by $p$, and $r$ is the remainder of $n$ when divided by $p$, compute $pm+r$. [i]Proposed by Yannick Yao[/i]

A triangle is circumscribed about a circle of radius $r$ inches. If the perimeter of the triangle is $P$ inches and the area is $K$ square inches, then $\frac{P}{K}$ is: $ \text{(A)}\text{independent of the value of} \; r\qquad\text{(B)}\ \frac{\sqrt{2}}{r}\qquad\text{(C)}\ \frac{2}{\sqrt{r}}\qquad\text{(D)}\ \frac{2}{r}\qquad\text{(E)}\ \frac{r}{2} $

Let $ABC$ be a triangle such that $BC>AB$, $L$ be the internal angle bisector of $\angle ABC$. Let $P,Q$ be the feet from $A,C$ to $L$, respectively. Suppose $M,N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively. Let $O$ be the circumcenter of triangle $PQM$, and the circumcircle intersects $AC$ at point $H$. Prove that $O,M,N,H$ are concyclic.

Let $ p $ and $ q $ be given primes and the sequence $ \{ p_n \}_{n = 1}^{\infty} $ defined recursively as follows: $ p_1 = p $, $ p_2 = q $, and $ p_{n+2} $ is the largest prime divisor of the number $( p_n + p_{n + 1} + 2016) $ for all $ n \geq 1 $. Prove that this sequence is bounded. That is, there exists a positive real number $ M $ such that $ p_n < M $ for all positive integers $ n $.

Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened.

Solve the following equations in the set of natural numbers: a) $(5+11\sqrt2)^p=(11+5\sqrt2)^q$ b) $1005^x+2011^y=1006^z$

Prove or disprove the following statements. a) For every integer $ n \ge 3$, there exist $ n$ pairwise distinct positive integers such that the product of any two of them is divisible by the sum of the remaining $ n \minus{} 2$ numbers. b) For some integer $ n \ge 3$, there exist $ n$ pairwise distinct positive integers, such that the sum of any $ n \minus{} 2$ of them is divisible by the product of the remaining two numbers.

Suppose a circle $C$ of radius $\sqrt2$ touches the $Y$ -axis at the origin $(0, 0)$. A ray of light $L$, parallel to the $X$-axis, reflects on a point $P$ on the circumference of $C$, and after reflection, the reflected ray $L'$ becomes parallel to the $Y$ -axis. Find the distance between the ray $L$ and the $X$-axis.

Let $f : R^+ \to R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$ Prove that $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$.

Let be given a convex polygon $ M_0M_1\ldots M_{2n}$ ($ n\ge 1$), where $ 2n \plus{} 1$ points $ M_0$, $ M_1$, $ \ldots$, $ M_{2n}$ lie on a circle $ (C)$ with diameter $ R$ in an anticlockwise direction. Suppose that there is a point $ A$ inside this convex polygon such that $ \angle M_0AM_1$, $ \angle M_1AM_2$, $ \ldots$, $ \angle M_{2n \minus{} 1}AM_{2n}$, $ \angle M_{2n}AM_0$ are equal. Assume that $ A$ is not coincide with the center of the circle $ (C)$ and $ B$ be a point lies on $ (C)$ such that $ AB$ is perpendicular to the diameter of $ (C)$ passes through $ A$. Prove that \[ \frac {2n \plus{} 1}{\frac {1}{AM_0} \plus{} \frac {1}{AM_1} \plus{} \cdots \plus{} \frac {1}{AM_{2n}}} < AB < \frac {AM_0 \plus{} AM_1 \plus{} \cdots \plus{} AM_{2n}}{2n \plus{} 1} < R \]

Find all integers $a$, $b$, $c$, $d$, and $e$ such that \begin{align*}a^2&=a+b-2c+2d+e-8,\\b^2&=-a-2b-c+2d+2e-6,\\c^2&=3a+2b+c+2d+2e-31,\\d^2&=2a+b+c+2d+2e-2,\\e^2&=a+2b+3c+2d+e-8.\end{align*}

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

Find all positive integers $k$ such that there exists a positive integer $n$, for which $2^n + 11$ is divisible by $2^k - 1$.

Let $x$, $y$, and $z$ be positive real numbers with $x+y+z = 3$. Prove that at least one of the three numbers $$x(x+y-z)$$ $$y(y+z-x)$$ $$z(z+x-y)$$ is less or equal $1$. (Karl Czakler)

A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? $ \textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55 $

Find all positive integers $n$ which have exactly $\sqrt{n}$ positive divisors.

Let $(x_n)_{n\geq1}$ be a sequence that verifies: $$x_1=1, \quad x_2=7, \quad x_{n+1}=x_n+3x_{n-1}, \forall n \geq 2.$$ Prove that for every prime number $p$ the number $x_p-1$ is divisible by $3p.$

The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number. Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?

$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$ prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$

Suppose $a$, $b$, and $c$ are positive integers such that \[\dfrac ab+\dfrac bc+\dfrac ca-\dfrac{524}{abc}=\dfrac ba+\dfrac cb+\dfrac ac - \dfrac{518}{abc}=1.\] Find $a^2+b^2+c^2$. [i]Proposed by David Altizio[/i]

Let $ A_0 \equal{} (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k \equal{} (x_1,\dots,x_k)$ we construct a new sequence $ A_{k \plus{} 1}$ in the following way. 1. We choose a partition $ \{1,\dots,n\} \equal{} I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression \[ \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right| \] attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily. 2. We set $ A_{k \plus{} 1} \equal{} (y_1,\dots,y_n)$ where $ y_i \equal{} x_i \plus{} 1$ if $ i\in I$, and $ y_i \equal{} x_i \minus{} 1$ if $ i\in J$. Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$. [i]Author: Omid Hatami, Iran[/i]

In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$. Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively. (a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$. (b) Prove the converse of (a).