Let $ \Omega$ be the circumcircle of triangle $ ABC$. Let $ D$ be the point at which the incircle of $ ABC$ touches its side $ BC$. Let $ M$ be the point on $ \Omega$ such that the line $ AM$ is parallel to $ BC$. Also, let $ P$ be the point at which the circle tangent to the segments $ AB$ and $ AC$ and to the circle $ \Omega$ touches $ \Omega$. Prove that the points $ P$, $ D$, $ M$ are collinear.

$32$ teams, ranked $1$ through $32$, enter a basketball tournament that works as follows: the teams are randomly paired and in each pair, the team that loses is out of the competition. The remaining $16$ teams are randomly paired, and so on, until there is a winner. A higher ranked team always wins against a lower-ranked team. If the probability that the team ranked $3$ (the third-best team) is one of the last four teams remaining can be written in simplest form as $\dfrac{m}{n}$, compute $m+n$.

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

Find all pairs of integer numbers $m$ and $n>2$ such that $((n-1)!-n)(n-2)!=m(m-2)$. [i]A. Kuznetsov[/i]

Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If $$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.

Denote $T$ the Toricelli point of the triangle $ABC$. Prove that $$AB^2 \times BC^2 \times CA^2 \geq 3(TA^2\times TB + TB^2 \times TC + TC^2 \times TA)(TA\times TB^2 + TB \times TC^2 + TC \times TA^2)$$

An elevator in a 100 floor building has only two buttons. The UP button makes the elevator go $13$ floors up, and the DOWN button makes the elevator go $8$ floors down. Is it possible to go from the $13^{\text{th}}$ floor to $8^{\text{th}}$ floor?

Suppose in a given collection of $2016$ integer, the sum of any $1008$ integers is positive. Show that sum of all $2016$ integers is positive.

Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.

$C$ is the unit circle in some plane. $R$ is a square with side $a$. $C$ is fixed and $R$ moves(without rotation) on the plane, in such a way that its center stays inside $C$(including boundaries). Find the maximum value of the area drawn by the trace of $R$.

For what real values of $ K$ does $ x \equal{} K^2 (x\minus{}1)(x\minus{}2)$ have real roots? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \minus{}2<K<1 \qquad \textbf{(C)}\ \minus{}2 \sqrt{2} < K < 2 \sqrt{2} \qquad \textbf{(D)}\ K>1 \text{ or } K<\minus{}2 \qquad \textbf{(E)}\ \text{all}$

A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power. [i](K. Kokhas)[/i]

In one side of a hall there are $2N$ rooms numbered from 1 to $2N$. In each room $i$ between 1 and $N$ there are $p_i$ beds. Is needed to move every one of this beds to the roms from $N+ 1$ to $2N$, in such a way that for every $j$ between $N+1$ and $2N$ the room $j$ will have $p_j$ beds. Supose that each bed can be move once and the price of moving a bed from room $i$ to room $j$ is $(i-j)^2$. Find a way to move every bed such that the total cost is minimize. Note: The numbers $p_i$ are given and satisfy that $p_1 + p_2 + \cdots + p_N = p_{N+1} + p_{N+2} + \cdots+ p_{2N}.$

Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.

Let $ABC$ be a triangle with $AB>BC>CA$. Let $D$ be a point on $AB$ such that $CD=BC$, and let $M$ be the midpoint of $AC$. Show that $BD=AC$ and that $\angle BAC=2\angle ABM.$

Problem 4. A chessboard with $2n \times 2n$ tiles is coloured such that every tile is coloured with one out of $n$ colours. Prove that there exists 2 tiles in either the same column or row such that if the colours of both tiles are swapped, then there exists a rectangle where all its four corner tiles have the same colour.

Let $x$,$y$,$z>0$ . Show that : $$\frac{x^3}{z^3+x^2y}+\frac{y^3}{x^3+y^2z}+\frac{z^3}{y^3+z^2x} \geq \frac{3}{2}$$

For how many positive real numbers $a$ has the equation $a^2x^2 + ax+1-7a^2 = 0$ two distinct integer roots? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of above} $

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that \[ A^2B+BA^2=2ABA \] Prove there exists $k\in \mathbb{N}$ such that \[ (AB-BA)^k=\mathbf{0}_n\] Here $\mathbf{0}_n$ is the null matrix of order $n$.

In the figure, $\angle A$, $\angle B$, and $\angle C$ are right angles. If $\angle AEB = 40^\circ $ and $\angle BED = \angle BDE$, then $\angle CDE = $ [asy] dot((0,0)); label("$E$",(0,0),SW); dot(dir(85)); label("$A$",dir(85),NW); dot((4,0)); label("$D$",(4,0),SE); dot((4.05677,0.648898)); label("$C$",(4.05677,0.648898),NE); draw((0,0)--dir(85)--(4.05677,0.648898)--(4,0)--cycle); dot((2,2)); label("$B$",(2,2),N); draw((0,0)--(2,2)--(4,0)); pair [] x = intersectionpoints((0,0)--(2,2)--(4,0),dir(85)--(4.05677,0.648898)); dot(x[0]); dot(x[1]); label("$F$",x[0],SE); label("$G$",x[1],SW); [/asy] $\text{(A)}\ 75^\circ \qquad \text{(B)}\ 80^\circ \qquad \text{(C)}\ 85^\circ \qquad \text{(D)}\ 90^\circ \qquad \text{(E)}\ 95^\circ$

We have two $20 \times 13$ rectangular grids with $260$ unit cells. each one. We insert in the boxes of each of the grids the numbers $1, 2, ..., 260$ as follows: $\bullet$ For the first grid, we start by inserting the numbers $1, 2, ..., 13$ in the boxes in the top row from left to right. We continue inserting numbers $14$, $ 15$, $...$, $26$ in the second row from left to right. We maintain the same procedure until in the last row, $20$, the numbers are placed $248$, $249$, $...$, $260$ from left to right. $\bullet$ For the second grid we start by inserting the numbers $1$, $2$,$ ..$., $20$ from top to bottom in the farthest column right. We continue inserting the numbers $21$, $22$,$ ...$, $40$ in the second column from the right also from top to bottom. We maintain that same procedure until we reach the column on the left where we place the numbers from top to bottom $241$, $242$, $ ...$, $260$. Determines the integers inserted in the boxes located in the same position in both grids.

Let $a$ and $b$ be two distinct natural numbers. It is known that $a^2+b|b^2+a$ and that $b^2+a$ is a power of a prime number. Determine the possible values of $a$ and $b$.

The incircle of a right-angled triangle $ABC$ ($\angle ABC =90^o$) touches $AB, BC, AC$ in points $C_1, A_1, B_1$, respectively. One of the excircles touches the side $BC$ in point $A_2$. Point $A_0$ is the circumcenter or triangle $A_1A_2B_1$, point $C_0$ is defined similarly. Find angle $A_0BC_0$.

[b]p1.[/b] Anne purchased yesterday at WalMart in Puerto Rico $6$ identical notebooks, $8$ identical pens and $7$ identical erasers. Anne remembers that each eraser costs $73$ cents. She did not buy anything else. Anne told her mother that she spent $12$ dollars and $76$ cents at Walmart. Can she be right? Note that in Puerto Rico there is no sales tax. [b]p2.[/b] Two men ski one after the other first in a flat field and then uphill. In the field the men run with the same velocity $12$ kilometers/hour. Uphill their velocity drops to $8$ kilometers/hour. When both skiers enter the uphill trail segment the distance between them is $300$ meters less than the initial distance in the field. What was the initial distance between skiers? (There are $1000$ meters in 1 kilometer.) [b]p3.[/b] In the equality $** + **** = ****$ all the digits are replaced by $*$. Restore the equality if it is known that any numbers in the equality does not change if we write all its digits in the opposite order. [b]p4.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started: ”-None of you has $8$ legs. Only I have 8 legs!” Which polyleg has exactly $8$ legs? [b]p5.[/b] Cut the figure shown below in two equal pieces. (Both the area and the form of the pieces must be the same.) [img]https://cdn.artofproblemsolving.com/attachments/e/4/778678c1e8748e213ffc94ba71b1f3cc26c028.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a convex quadrilateral. Let $E$ be the intersection of line $AB$ with the line $CD$, and $F$ is the intersection of line $BC$ with the line $AD$. Let $P$ and $Q$ be the foots of the perpendicular of $E$ to the lines $AD$ and $BC$ respectively, and let $R$ and $S$ be the foots of the perpendicular of $F$ to the lines $AB$ and $CD$, respectively.The point $T$ is the intersection of the line $ER$ with the line $FS$. a) Show that, there exists a circle that passes in the points $E, F, P, Q, R$ and $S$. b)Show that, the circumcircle of triangle $RST$ is tangent with the circumcircle of triangle $QRB$.