The cross below is made up of five congruent squares. The perimeter of the cross is $72$. Find its area. [asy] import graph; size(3cm); pair A = (0,0); pair temp = (1,0); pair B = rotate(45,A)*temp; pair C = rotate(90,B)*A; pair D = rotate(270,C)*B; pair E = rotate(270,D)*C; pair F = rotate(90,E)*D; pair G = rotate(270,F)*E; pair H = rotate(270,G)*F; pair I = rotate(90,H)*G; pair J = rotate(270,I)*H; pair K = rotate(270,J)*I; pair L = rotate(90,K)*J; draw(A--B--C--D--E--F--G--H--I--J--K--L--cycle); [/asy]

Let $ABCD$ be a square and points $M$ $\in$ $BC$, $N \in CD$, $P$ $\in$ $DA$, such that $\angle BAM$ $=$ $x$, $\angle CMN$ $=$ $2x$, $\angle DNP$ $=$ $3x$ [list=i] [*] Show that, for any $x \in (0, \tfrac{\pi}{8} )$, such a configuration exists [*] Determine the number of angles $x \in ( 0, \tfrac{\pi}{8} )$ for which $\angle APB =4x$

In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.

For every nonnegative integer $ n$ and every real number $ x$ prove the inequality: $ |\cos x|\plus{}|\cos 2x|\plus{}\ldots\plus{}|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

Let $(a_n)_{n\ge 1}$ be a sequence of positive integers defined as $a_1,a_2>0$ and $a_{n+1}$ is the least prime divisor of $a_{n-1}+a_{n}$, for all $n\ge 2$. Prove that a real number $x$ whose decimals are digits of the numbers $a_1,a_2,\ldots a_n,\ldots $ written in order, is a rational number. [i]Laurentiu Panaitopol[/i]

In the sequence in the previous problem, how many of $u_1,u_2,u_3,\ldots, u_{2008}$ are pentagonal numbers?

Three real numbers $x$, $y$, and $z$ are such that $(x+4)/2=(y+9)/(z-3)=(x+5)/(z-5)$. Determine the value of $x/y$.

An equilateral triangle $ABC$ has sides of length $n$, a positive integer. Divide the triangle into equilateral triangles of length $ 1$, drawing parallel lines (at distance $ 1$) to all sides of the triangle. A path is a continuous path, starting at the triangle with vertex $A$ and always crossing from one small triangle to another on the side that both triangles share, in such a way that it never passes through a small triangle twice. Find the maximum number of triangles that can be visited.

For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied: (a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$; (b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$. Determine $N(n)$ for all $n\ge 2$.

In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.

For any three real numbers $ a$, $ b$, and $ c$, with $ b\neq c$, the operation $ \otimes$ is defined by: \[ \otimes(a,b,c) \equal{} \frac {a}{b \minus{} c} \]What is $ \otimes(\otimes(1,2,3), \otimes(2,3,1),\otimes(3,1,2))$? $ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad \textbf{(B)}\ \minus{}\!\frac {1}{4}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ \frac {1}{4}\qquad \textbf{(E)}\ \frac {1}{2}$

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

Let $ ABC$ be a triangle and let $ I$ be the incenter. $ Ia$ $ Ib$ and $ Ic$ are the excenters opposite to points $ A$ $ B$ and $ C$ respectively. Let $ La$ be the line joining the orthocenters of triangles $ IBC$ and $ IaBC$. Define $ Lb$ and $ Lc$ in the same way. Prove that $ La$ $ Lb$ and $ Lc$ are concurrent. Daniel

Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.

Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.

Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$, $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers.

Let $f$ be a real function defined in the interval $[0, +\infty )$ and suppose that there exist two functions $f', f''$ in the interval $[0, +\infty )$ such that \[f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.\] Let $g$ be a function for which \[g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.\] Prove that $g$ is bounded.

Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle. [i]M. Kungojin[/i]

Find all positive integers $n$ such that $27^n- 2^n$ is a perfect square.

Find all triples of natural $ (x,y,n)$ satisfying the condition: \[ \frac {x! \plus{} y!}{n!} \equal{} 3^n \] Define $ 0! \equal{} 1$

$A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are two squares with their vertices arranged clockwise.The perpendicular bisector of segment $A_1B_1,A_2B_2,A_3B_3,A_4B_4$ and the perpendicular bisector of segment $A_2B_2,A_3B_3,A_4B_4,A_1B_1$ intersect at point $P,Q,R,S$ respectively.Show that:$PR\perp QS$.

In a certain country there are more than $101$ towns. The capital of this country is connected by direct air routes with $100$ towns, and each town, except for the capital, is connected by direct air routes with $10$ towns (if $A$ is connected with $B, B$ is connected with $A$). It is known that from any town it is possible to travel by air to any other town (possibly via other towns). Prove that it is possible to close down half of the air routes connected with the capital, and preserve the capability of travelling from any town to any other town within the country. (IS Rubanov)

Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.