Compute the number of possible words $w=w_1w_2\dots w_{100}$ satisfying: $\bullet$ $w$ has exactly $50$ $A$'s and $50$ $B$'s (and no other letter). $\bullet$ For $i=1,2,\dots,100$, the number of $A$'s among $w_1, w_2, \dots, w_i$ is at most the number of $B$'s among $w_1, w_2, \dots, w_i$. $\bullet$ For all $i=44,45,\dots,57$, if $w_i$ is a $B$, then $w_{i+1}$ must be a $B$.

Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$

Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity $$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$ holds for all $x,y \in \mathbb N$

Let $n$ be a positive integer. Determine integers, $n+1 \leq r \leq 3n+2$ such that for all integers $a_1,a_2,\dots,a_m,b_1,b_2,\dots,b_m$ satisfying the equations \[ a_1b_1^k+a_2b_2^k+\dots + a_mb_m^k=0 \] for every $1 \leq k \leq n$, the condition \[ r \mid a_1b_1^r+a_2b_2^r+\dots + a_mb_m^r=0 \] also holds.

In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses. In each case, the sides of the rhombuses are the same length as the sides of the regular polygon. (a) Tile a regular decagon ($10$-gon) into rhombuses in this manner. (b) Tile a regular dodecagon ($12$-gon) into rhombuses in this manner. (c) How many rhombuses are in a tiling by rhombuses of a $2002$-gon? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png[/img]

In cyclic quadrilateral $ABCD$, $\angle DBC = 90^\circ$ and $\angle CAB = 30^\circ$. The diagonals of $ABCD$ meet at $E$. If $\frac{BE}{ED} = 2$ and $CD = 60$, compute $AD$. (Note: a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.)

Let $ABCD$ be an circumscribed quadrilateral such that $m(\widehat{A})=m(\widehat{B})=120^\circ$, $m(\widehat{C})=30^\circ$, and $|BC|=2$. What is $|AD|$? $ \textbf{(A)}\ \sqrt 3 - 1 \qquad\textbf{(B)}\ \sqrt 2 - 3 \qquad\textbf{(C)}\ \sqrt 6 - \sqrt 2 \qquad\textbf{(D)}\ 2 - \sqrt 2 \qquad\textbf{(E)}\ 3 - \sqrt 3 $

Rui Xuen has a circle $\omega$ with center $O$, and a square $ABCJ$ with vertices on $\omega$. Let $M$ be the midpoint of $AB$, and let $\Gamma$ be the circle passing through the points $J$, $O$, $M$. Suppose $\Gamma$ intersect line $AJ$ at a point $P \neq J$, and suppose $\Gamma$ intersect $\omega$ at a point $Q \neq J$. A point $R$ lies on side $BC$ so that $RC = 3RB$. Help Rui Xuen prove that the points $P$, $Q$, $R$ are collinear.

Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \$3, buy or sell bundles of three wheat for \$12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?

Let $p(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a complex polynomial. Suppose that $1=c_0\ge c_1\ge \cdots \ge c_n\ge 0$ is a sequence of real numbers which form a convex sequence. (That is $2c_k\le c_{k-1}+c_{k+1}$ for every $k=1,2,\cdots ,n-1$ ) and consider the polynomial \[ q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n \] Prove that : \[ \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z) \]

When flipped, coin A shows heads $\textstyle\frac{1}{3}$ of the time, coin B shows heads $\textstyle\frac{1}{2}$ of the time, and coin C shows heads $\textstyle\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as $\textstyle\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $p + q$. [i]Proposed by Eugene Chen[/i]

Using several white edge cubes of side $ 1$, Guille builds a large cube. Then he chooses $4$ faces of the big cube and paints them red. Finally, he takes apart the large cube and observe that the cubes with at least a face painted red is $431$. Find the number of cubes that he used to assemble the large cube. Analyze all the possibilities.

At least two of any three students of a school with $1001$ students are friends. How many of the numbers $334,412,450,499$ can be the number of friends of the one with the highest number of friends? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None} $

The incircle $ \omega $ of a quadrilateral $ ABCD $ touches $ AB, BC, CD, DA $ at $ E, F, G, H $, respectively. Choose an arbitrary point $ X$ on the segment $ AC $ inside $ \omega $. The segments $ XB, XD $ meet $ \omega $ at $ I, J $ respectively. Prove that $ FJ, IG, AC $ are concurrent.

Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure? [asy] defaultpen(linewidth(1)); real x=sqrt(3)/2; path p=rotate(30)*polygon(6); filldraw(p^^shift(0,3)*p^^shift(4x,0)*p^^shift(3x,1.5)*p^^shift(2x,3)*p^^shift(-4x,0)*p^^shift(-3x,1.5)*p^^shift(-2x,3)*p^^shift(3x,-1.5)*p^^shift(-3x,-1.5)*p^^shift(2x,-3)*p^^shift(-2x,-3)*p^^shift(0,-3)*p, black, black); draw(shift(2x,0)*p^^shift(-2x,0)*p^^shift(x,1.5)*p^^shift(-x,1.5)*p^^shift(x,-1.5)*p^^shift(-x,-1.5)*p); [/asy] $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 18 $

Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r^2_1+\dots+r^2_5$?

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.

Compute $$\frac{20+\frac{1}{25-\frac{1}{20}}}{25+\frac{1}{20-\frac{1}{25}}}.$$

Let $\triangle ABC$ be a triangle with $|AC|=2|AB|$ and let $O$ be its circumcenter. Let $D$ be the intersection of the bisector of $\angle A$ with $BC$. Let $E$ be the orthogonal projection of $O$ to $AD$ and let $F\ne D$ be the point on $AD$ satisfying $|CD|=|CF|$. Prove that $\angle EBF=\angle ECF$.

Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 \, , \,\, f(2n) = 2f(n) \, , \,\, nf(2n + 1) = (2n + 1)(f(n) + n) $$ for all $n \ge 1$. a) Prove that $f(n)$ is an integer for all $n$. b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$.

Find the sum of the non-repeated roots of the polynomial $P(x) = x^6-5x^5-4x^4-5x^3+8x^2+7x+7$.

In a quadrilateral $ABCD$ sides $AB$ and $CD$ are equal, $\angle A=150^\circ,$ $\angle B=44^\circ,$ $\angle C=72^\circ.$ Perpendicular bisector of the segment $AD$ meets the side $BC$ at point $P.$ Find $\angle APD.$ [i]Proposed by F. Bakharev[/i]

On a table near the sea, there are $N$ glass boxes where $N<2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let $S$ be the sum of all values of $N$ for which Sowdha has a winning strategy and let $R$ be the sum of all values of $N$ for which Rafi has a winning strategy. What is the value of $\frac{R-S}{10}$?

Let $ABC$ be a triangle, and $B_1,C_1$ be its excenters opposite $B,C$. $B_2,C_2$ are reflections of $B_1,C_1$ across midpoints of $AC,AB$. Let $D$ be the extouch at $BC$. Show that $AD$ is perpendicular to $B_2C_2$

Prove that $x^2+y^2 \ge 2\sqrt2 (x-y)$ if $xy = 1$