In a triangle $ ABC$, the angle bisector at $ B$ intersects $ AC$ at $ D$ and the circumcircle again at $ E$. The circumcircle of the triangle $ DAE$ meets the segment $ AB$ again at $ F$. Prove that the triangles $ DBC$ and $ DBF$ are congruent.

Mark has six boxes lined up in a straight line. Inside each of the first three boxes are a red ball, a blue ball, and a green ball. He randomly selects a ball from each of the three boxes and puts them into a fourth box. Then, he randomly selects a ball from each of the four boxes and puts them into a fifth box. Next, he randomly selects a ball from each of the five boxes and puts them into a sixth box, arriving at three boxes with $1, 3,$ and $5$ balls, respectively. The probability that the box with $3$ balls has each type of color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by [b]AOPS12142015[/b][/i]

Let $\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys "ab$\leftarrow$ cd $\leftarrow \leftarrow$ e $\leftarrow \leftarrow$ f", the result is "faecdb". We say that a string $B$ is [i]reachable[/i] from a string $A$ if it is possible to insert some amount of $\leftarrow$'s in $A$, such that typing the resulting characters produces $B$. So, our example shows that "faecdb" is reachable from "abcdef". Prove that for any two strings $A$ and $B$, $A$ is reachable from $B$ if and only if $B$ is reachable from $A$.

The sequence $ (x_n)_{n \in \mathbb{N}}$ is defined by $ x_1\equal{}2, x_2\equal{}3,$ and $ x_{2m\plus{}1}\equal{}x_{2m}\plus{}x_{2m\minus{}1}$ for $ m \ge 1;$ $ x_{2m}\equal{}x_{2m\minus{}1}\plus{}2x_{2m\minus{}2}$ for $ m \ge 2.$ Determine $ x_n$ as a function of $ n$.

For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.

Allen plays a game on a tree with $2n$ vertices, each of whose vertices can be red or blue. Initially, all of the vertices of the tree are colored red. In one move, Allen is allowed to take two vertices of the same color which are connected by an edge and change both of them to the opposite color. He wins if at any time, all of the verices of the tree are colored blue. (a) Show that Allen can win if and only if the vertices can be split up into two groups $V_1$ and $V_2$ to size $n$, such that each edge in the tree has one endpoint in $V_1$ and one endpoint in $V_2$. (b) Let $V_1 = \left\{ a_1, \ldots, a_n \right\}$ and $V_2 = \left\{ b_1, \ldots, b_n \right\}$ from part (a). Let $M$ be the minimum over all permutations $\sigma$ of $\left\{ 1, \ldots, n \right\}$ of the quantity \[ \sum\limits_{i = 1}^{n} d(a_i, b_{\sigma(i)}), \] where $d(v, w)$ denotes the number of edges along the shortest path between vertices $v$ and $w$ in the tree. Show that if Allen can win, then the minimum number of moves that it can take for Allen to win is equal to $M$.

A plane is divided into unit squares by two collections of parallel lines. For any $n\times n$ square with sides on the division lines, we define its frame as the set of those unit squares which internally touch the boundary of the $n\times n$ square. Prove that there exists only one way of covering a given $100\times 100$ square whose sides are on the division lines with frames of $50$ squares (not necessarily contained in the $100\times 100$ square). (A. Perlin)

$ABC$ is an arbitrary triangle. $A',B',C'$ are midpoints of arcs $BC, AC, AB$. Sides of triangle $ABC$, intersect sides of triangle $A'B'C'$ at points $P,Q,R,S,T,F$. Prove that \[\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}\]

For each $\alpha\in \mathbb{R}$ define $f_{\alpha}(x)=\lfloor{\alpha x}\rfloor$. Let $n\in \mathbb{N}$. Show there exists a real $\alpha$ such that for $1\le \ell \le n$ : \[ f_{\alpha}^{\ell}(n^2)=n^2-\ell=f_{\alpha^{\ell}}(n^2).\] Here $f^{\ell}_{\alpha}(x)=(f_{\alpha}\circ f_{\alpha}\circ \cdots \circ f_{\alpha})(x)$ where the composition is carried out $\ell$ times.

Determine all functions $f:\mathbb R\to\mathbb R$ such that for all $x, y \in \mathbb R$ $$f(xf(y) - yf(x)) = f(xy) - xy.$$

Let be given a parallelogram $ ABCD$ and two points $ A_1$, $ C_1$ on its sides $ AB$, $ BC$, respectively. Lines $ AC_1$ and $ CA_1$ meet at $ P$. Assume that the circumcircles of triangles $ AA_1P$ and $ CC_1P$ intersect at the second point $ Q$ inside triangle $ ACD$. Prove that $ \angle PDA \equal{} \angle QBA$.

A permutation $ (a_1,a_2,a_3,a_4,a_5)$ of $ (1,2,3,4,5)$ is heavy-tailed if $ a_1 \plus{} a_2 < a_4 \plus{} a_5$. What is the number of heavy-tailed permutations? $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 52$

Let $ a,b,c\in (0,1)$ and $ x,y,z\in (0, \plus{} \infty)$ be six real numbers such that \[ a^x \equal{} bc , \quad b^y \equal{} ca , \quad c^z \equal{} ab .\] Prove that \[ \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 .\] [i]Cezar Lupu[/i]

If $991+993+995+997+999=5000-N$, then $N=$ $\text{(A)}\ 5 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 25$

Let $n$ be a positive odd integer. $C$ is a set consists of integral points on a plane, which is defined by \[ C = \{(i, j): i, j = 0, 1, \dots, 2n-1\} \] and forms a $2n \times 2n$ array. On every point there is a Guinea pig, which is facing toward one of the following directions: [i]positive/negative $x$-axis[/i], or [i]positive/negative $y$-axis[/i]. Jeff wants to keep $n^2+1$ of the Guinea pigs on the plane and remove all the others. After that, the Guinea pigs on the plane will move as the following: 1. In every round, the Guinea pigs move toward by an unit, and keep facing the same direction. 2. If a Guinea pig move to a point $(i, j)$ which is [i]not[/i] in $C$, it will further move to another point $(p, q)$ in $C$, such that $p \equiv i \pmod {2n}$ and $q \equiv j \pmod {2n}$. [i](For example, if a Guinea pig move from $(2, 0)$ to $(2, -1)$, it will then further move to $(2, 2n-1)$.)[/i] The next round begins after all the Guinea pigs settle up. Jeff's goal is to keep the appropriate Guinea pigs on the plane, so that in every single round, any two Guinea pigs will never move to the same endpoint, and will never move to the startpoints[i](in that round)[/i] of each other simultaneously. Prove that Jeff can always succeed wherever the Guinea pigs initially face. [i]Proposed by Weijiun Kao[/i] Edit: By the way, it can be proven that the number $n^2+1$ is optimal, i.e. if the Guinea pigs face appropriately, Jeff can only keep at most $n^2+1$ of them on the plane to avoid any collision.

Find all pairs of integers $ (x,y)$, such that \[ x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0 \]

Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that \[ a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n \] and \[ a_1 + \dots + a_n = b_1 + \dots + b_n. \] Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.

Let $x$ and $y$ be relatively prime integers. Show that $x^2+xy+y^2$ and $x^2+3xy+y^2$ are relatively prime.

On the AMC 12, each correct answer is worth $ 6$ points, each incorrect answer is worth $ 0$ points, and each problem left unanswered is worth $ 2.5$ points. If Charlyn leaves $ 8$ of the $ 25$ problems unanswered, how many of the remaining problems must she answer correctly in order to score at least $ 100$? $ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

Let triangle $\vartriangle ABC$ be acute, and let point $M$ be the midpoint of $\overline{BC}$. Let $E$ be on line segment $\overline{AB}$ such that $\overline{AE} \perp \overline{EC}$. Then, suppose $T$ is a point on the other side of $\overleftrightarrow{BC}$ as $A$ is such that $\angle BTM = \angle ABC$ and $\angle TCA = \angle BMT$. If $AT = 14$, $AM = 9,$ and $\frac{AE}{AC} =\frac27$ , compute $BC$.

Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?

Left focal point and right focal point of a hyperbola are $F_1,F_2$, left focal point and right focal point of a hyperbola are $M,N$. If $P$ is a point on the hyperbola, then the tangent point of inscribed circle of $\triangle PF_1F_2$ on $F_1F_2$ is $\text{(A)}$a point on segment $MN$ $\text{(B)}$a point on segment $F_1M$ or $F_2N$ $\text{(C)}$point $M$ or $N$ $\text{(D)}$not sure

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

Let $O$ be the centre of the circle touching the side $AC$ of triangle $ABC$ and the continuations of the sides $BA$ and $BC$. $D$ is the centre of the circle passing through the points $A$, $B$ and $O$. Prove that the points $A$, $B$, $C$ and $D$ lie on a circle. (YF Akurlich)

Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.