The $12$-sided polygon below was created by placing three $3$ × $3$ squares with their sides parallel so that vertices of two of the squares are at the center of the third square. Find the perimeter of this $12$-sided polygon.

Xavier takes a permutation of the numbers $1$ through $2011$ at random, where each permutation has an equal probability of being selected. He then cuts the permutation into increasing contiguous subsequences, such that each subsequence is as long as possible. Compute the expected number of such subsequences. [i]Author: Alex Zhu[/i] [hide="Clarification"]An increasing contiguous subsequence is an increasing subsequence all of whose terms are adjacent in the original sequence. For example, 1,3,4,5,2 has two maximal increasing contiguous subsequences: (1,3,4,5) and (2). [/hide]

Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\},$ \[\sum_{k=0}^{p-1}k!n^k \quad \text{is not divisible by }p.\]

For real numbers $x$, let \[P(x)=1+\cos (x)+i \sin (x)-\cos (2 x)-i \sin (2 x)+\cos (3 x)+i \sin (3 x)\] where $i=\sqrt{-1}$. For how many values of $x$ with $0 \leq x<2 \pi$ does $P(x)=0 ?$ $\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 4$

Prove that infinitely many positive integers can be represented as $(a-1)/b + (b-1)/c + (c-1)/a$, where $a$, $b$ and $c$ are pairwise distinct positive integers greater than 1.

Nine numbers $a, b, c, \dots$ are arranged around a circle. All numbers of the form $a+b^c, \dots$ are prime. What is the largest possible number of different numbers among $a, b, c, \dots$?

Prove that for any integers $a,b$, the equation $2abx^4 - a^2x^2 - b^2 - 1 = 0$ has no integer roots. [i](Dan Schwarz)[/i]

In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, $CH$ is the height of the triangle, and the point $T$ is the foot of the perpendicular dropped from the vertex $C$ on the line $AO$. Prove that the line $TH$ passes through the midpoint of the side $BC$ .

Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$, \[a^n + a^{n-1} +
\cdots
+ a^1 + 1 \mid a^{n!} + a^{(n-1)!} +
\cdots

+ a^{1!} + 1.\] [i]Proposed by Milos Milosavljevic[/i]

We say a number is irie if it can be written in the form $1+\dfrac{1}{k}$ for some positive integer $k$. Prove that every integer $n \geq 2$ can be written as the product of $r$ distinct irie numbers for every integer $r \geq n-1$.

Let $x,y,z,w $ be real numbers such that $x+2y+3z+4w=1$. Find the minimum of $x^2+y^2+z^2+w^2+(x+y+z+w)^2$.

Working alone, the expert can paint a car in one day, the amateur can paint a car in two days, and the beginner can paint a car in three days. If the three painters work together at these speeds to paint three cars, it will take them $\frac{m}{n}$ days where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers.

Find all polynomials $P(X)$ of fourth degree with real coefficients that verify the properties: [b]a)[/b] $P(-x)=P(x), \forall x\in\mathbb{R};$ [b]b)[/b] $P(x)\geq0, \forall x\in\mathbb{R};$ [b]c)[/b] $P(0)=1;$ [b]d)[/b] $P(X)$ has exactly two local minimums $x_1$ and $x_2$ such that $|x_1-x_2|=2.$

A [i]cross-pentomino[/i] is a shape that consists of a unit square and four other unit squares each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R,$ what is $100R^2$? [i]Author: Ray Li[/i]

Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\]

Prove that there exists infinitely many positive integers $n$ such that equation $(x+y+z)^3=n^2xyz$ has solution $(x,y,z)$ in set $\mathbb{N}^3$

Let $ABC$ be an isosceles triangle $(AB=AC)$ so that $\angle A< 2 \angle B$ . Let $D,Z $ points on the extension of height $AM$ so that $\angle CBD = \angle A$ and $\angle ZBA = 90^\circ$. Let $E$ the orthogonal projection of $M$ on height $BF$, and let $K$ the orthogonal projection of $Z$ on $AE$. Prove that $ \angle KDZ = \angle KDB = \angle KZB$.

What is the units digit of $13^{2012}$ ? $\textbf{(A)}\hspace{.05in}1 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}5 \qquad \textbf{(D)}\hspace{.05in}7 \qquad \textbf{(E)}\hspace{.05in}9 $

There are $N$ monsters, each with a positive weight. On each step, two of the monsters are merged into one, whose weight is the sum of weights for the two original monsters. At the end, all monsters will be merged into one giant monster. During this process, if at any mergence, one of the two monsters has a weight greater than $2.020$ times the other monster's weight, we will call this mergence [b]dangerous[/b]. The dangerous level of a sequence of mergences is the number of dangerous mergence throughout its process. Prove that, no matter how the weights being distributed among the monsters, "for every step, merge the lightest two monsters" is always one of the merging sequences that obtain the minimum possible dangerous level. [i]Proposed by houkai[/i]

Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.

In how many ways can you choose several numbers from the numbers $1,2,3,..., 11$ so that among the selected numbers there are not three consecutive numbers?

In the picture below, a white square is surrounded by four black squares and three white squares. They are surrounded by seven black squares. [img]https://i.stack.imgur.com/Dalmm.png[/img] What is the maximum number of white squares that can be surrounded by $ n $ black squares?

A sequence $\{a_i\}$ is given such that $a_1 = \frac13$ and for all positive integers $n$ $$a_{n+1} =\frac{a^2_n}{a^2_n - a_n + 1}.$$ Prove that $$\frac12 - \frac{1}{3^{2^{n-1}}} < a_1 + a_2 +... + a_n <\frac12 - \frac{1}{3^{2^n}} ,$$ for all positive integers $n$.