$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

find all solutions, not necessarily positive integers for $(m^2+ n)(m+ n^2)= (m+ n)^3$

Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: $$\log_2\left({x \over yz}\right) = {1 \over 2}$$ $$\log_2\left({y \over xz}\right) = {1 \over 3}$$ $$\log_2\left({z \over xy}\right) = {1 \over 4}$$ Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$

The positive integer $n$ is the sum of two positive integers that divide $n+6$. Find all possible values of $n$

In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.

Laura has $2010$ lamps connected with $2010$ buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons simultaneously, so the lamps are lit simultaneously, too. a) If Richard chooses the buttons to be pressed, what is the maximum number of different combinations of buttons he can press until Laura can assign the buttons to the lamps correctly? b) Supposing that Laura will choose the combinations of buttons to be pressed, what is the minimum number of attempts she has to do until she is able to associate the buttons with the lamps in a correct way?

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

The points $A, B, C, D$ lie in that order on a circle. The segments $AC$ and $BD$ intersect at the point $P$. The point $B'$ lies on the line $AB$ such that $A$ is between $B$ and $B'$ and $|AB'| = |DP |$. The point $C'$ lies on the line $CD$ such that $D$ is between $C$ and $C'$ lies and $|DC' | = |AP|$. Prove that $\angle B'PC' = \angle ABD'$. [img]https://cdn.artofproblemsolving.com/attachments/2/2/7ec65246ff5ecfebc25ca13f3709d1791ceb6c.png[/img] =

A regular hexagon is cut up into $N$ parallelograms of equal area. Prove that $N$ is divisible by three. (V. Prasolov, I. Sharygin, Moscow)

Let $ P(x,y)$ be a polynomial in two variables $ x,y$ such that $ P(x,y)\equal{}P(y,x)$ for every $ x,y$ (for example, the polynomial $ x^2\minus{}2xy\plus{}y^2$ satisfies this condition). Given that $ (x\minus{}y)$ is a factor of $ P(x,y)$, show that $ (x\minus{}y)^2$ is a factor of $ P(x,y)$.

Find all pairs $(x, y)$ of real numbers that satisfy the system of equations $$\begin{cases} x^4 + 2z^3 - y =\sqrt3 - \dfrac14 \\ y^4 + 2y^3 - x = - \sqrt3 - \dfrac14 \end{cases}$$

Blue rolls a fair $n$-sided die that has sides its numbered with the integers from $1$ to $n$, and then he flips a coin. Blue knows that the coin is weighted to land heads either $\dfrac{1}{3}$ or $\dfrac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\dfrac{1}{15}$, find $n$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{10}$ The chance of getting any given number is $\dfrac{1}{n}$ , so the probability of getting $7$ and heads is either $\dfrac{1}{n} \cdot \dfrac{1}{3}=\dfrac{1}{3n}$ or $\dfrac{1}{n} \cdot \dfrac{2}{3}=\dfrac{2}{3n}$. We get that either $n = 5$ or $n = 10$, but since rolling a $7$ is possible, only $n = \boxed{10}$ is a solution.[/hide]

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

Two concentric circles have differing radii such that a chord of the outer circle which is tangent to the inner circle has length $18$. Compute the area inside the bigger circle which lies outside of the smaller circle.

Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre of $\triangle BCM$. Let $N$ be the second intersection pointof $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot |BN|$.

If $a,b,c,d$ be any four positive real numbers, then prove that \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4. \]

* A bus route has $14$ stops (counting the first and the last). A bus cannot carry more than $25$ passengers. We assume that a passenger takes a bus from $A$ to $B$ if (s)he enters the bus at $A$ and gets off at $B$. Prove that for any bus route: a) there are $8$ distinct stops $A_1, B_1, A_2, B_2, A_3, B_3, A_4, B_4$ such that no passenger rides from $A_k$ to $B_k$ for all $k = 1, 2, 3, 4$ (#) b) there might not exist $10$ distinct stops $A_1, B_1, . . . , A_5, B_5$ with property (#).

Given a triangle $ABC$, the internal bisectors through $A$ and $B$ meet the opposite sides in $D$ and $E$, respectively. Prove that \[DE \le (3 - 2\sqrt2)(AB + BC + CA)\] and determine the cases of equality.

Given $n \ge 2$ points on a circle, Alice and Bob play the following game. Initially, a tile is placed on one of the points and no segment is drawn. The players alternate in turns, with Alice to start. In a turn, a player moves the tile from its current position $P$ to one of the $n-1$ other points $Q$ and draws the segment $PQ$. This move is not allowed if the segment $PQ$ is already drawn. If a player cannot make a move, the game is over and the opponent wins. Determine, for each $n$, which of the two players has a winning strategy.

Show that if $b = \frac{a+c}{2}$ in the triangle $ABC$, then $\cos (A-C) + 4 \cos B = 3$.

For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions: [list] [*] no two consecutive digits are equal, and [*] the last digit is a prime. [/list]

For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that [b]for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. [/b] Find the minimal possible value of the maximum of all numbers.

For each real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer that does not exceed $x.$ For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_2 n\rfloor$ is a positive even integer.

In the figure, what is the ratio of the area of the gray squares to the area of the white squares? [asy] size((70)); draw((10,0)--(0,10)--(-10,0)--(0,-10)--(10,0)); draw((-2.5,-7.5)--(7.5,2.5)); draw((-5,-5)--(5,5)); draw((-7.5,-2.5)--(2.5,7.5)); draw((-7.5,2.5)--(2.5,-7.5)); draw((-5,5)--(5,-5)); draw((-2.5,7.5)--(7.5,-2.5)); fill((-10,0)--(-7.5,2.5)--(-5,0)--(-7.5,-2.5)--cycle, gray); fill((-5,0)--(0,5)--(5,0)--(0,-5)--cycle, gray); fill((5,0)--(7.5,2.5)--(10,0)--(7.5,-2.5)--cycle, gray); [/asy] $ \textbf{(A)}\ 3:10 \qquad\textbf{(B)}\ 3:8 \qquad\textbf{(C)}\ 3:7 \qquad\textbf{(D)}\ 3:5 \qquad\textbf{(E)}\ 1:1 $

Which of the following polygons has the largest area? [asy] size(330); int i,j,k; for(i=0;i<5; i=i+1) { for(j=0;j<5;j=j+1) { for(k=0;k<5;k=k+1) { dot((6i+j, k)); }}} draw((0,0)--(4,0)--(3,1)--(3,3)--(2,3)--(2,1)--(1,1)--cycle); draw(shift(6,0)*((0,0)--(4,0)--(4,1)--(3,1)--(3,2)--(2,1)--(1,1)--(0,2)--cycle)); draw(shift(12,0)*((0,1)--(1,0)--(3,2)--(3,3)--(1,1)--(1,3)--(0,4)--cycle)); draw(shift(18,0)*((0,1)--(2,1)--(3,0)--(3,3)--(2,2)--(1,3)--(1,2)--(0,2)--cycle)); draw(shift(24,0)*((1,0)--(2,1)--(2,3)--(3,2)--(3,4)--(0,4)--(1,3)--cycle)); label("$A$", (0*6+2, 0), S); label("$B$", (1*6+2, 0), S); label("$C$", (2*6+2, 0), S); label("$D$", (3*6+2, 0), S); label("$E$", (4*6+2, 0), S); [/asy] $ \textbf{(A)}\text{A}\qquad\textbf{(B)}\ \text{B}\qquad\textbf{(C)}\ \text{C}\qquad\textbf{(D)}\ \text{D}\qquad\textbf{(E)}\ \text{E} $