The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.

Real numbers $x, y, z$ satisfy $$x + xy + xyz = 1, y + yz + xyz = 2, z + xz + xyz = 4.$$ The largest possible value of $xyz$ is $\frac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, $d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d$.

[b]4.[/b] Let $P_1 P_2 P_3 P_4 P_5 P_6$ be a convex hexagon. Denote by $T$ its area and by $t$ the area of the triangle $Q_1 Q_2 Q_3$, where $Q_1,Q_2$ and $Q_3$ are the midpoints of $P_1P_4,P_2P_5,P_3P_6$ respectively. Prove that $t<\frac{1}{4}T$. [b](G. 3)[/b]

Consider an acute triangle $ABC$ with $|AB| > |CA| > |BC|$. The vertices $D, E$, and $F$ are the base points of the altitudes from $A, B$, and $C$, respectively. The line through F parallel to $DE$ intersects $BC$ in $M$. The angular bisector of $\angle MF E$ intersects $DE$ in $N$. Prove that $F$ is the circumcentre of $\vartriangle DMN$ if and only if $B$ is the circumcentre of $\vartriangle FMN$.

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Let $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?

Let $f\in C^3(\mathbb{R})$ nonnegative function with $f(0)=f'(0)=0, f''(0)>0$. Define $g(x)$ as follows: \[ \{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 \\ g(x)=0 &\text{for}& x=0\end{array} \] (a) Show that $g$ is bounded in some neighbourhood of $0$. (b) Is the above true for $f\in C^2(\mathbb{R})$?

Find the smallest integer $k$ for which the conditions $(1)$ $a_1, a_2, a_3, \ldots$ is a nondecreasing sequence of positive integers $(2)$ $a_n=a_{n-1}+a_{n-2}$ for all $n>2$ $(3)$ $a_9=k$ are satisfied by more than one sequence.

There are $27$ cardboard and $27$ plastic boxes. There are balls of certain colors inside the boxes. It is known that any two boxes of the same kind do not have a ball with the same color. Boxes of different kind have at least one ball of the same color. At each step we select two boxes that have a ball of same color and switch this common color into any other color we wish. Find the smallest number $n$ of moves required.

For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.

Sunaina and Malay play a game on the coordinate plane. Sunaina has two pawns on $(0,0)$ and $(x,0)$, and Malay has a pawn on $(y,w)$, where $x,y,w$ are all positive integers. They take turns alternately, starting with Sunaina. In their turn they can move one of their pawns one step vertically up or down. Sunaina wins if at any point in time all the three pawns are colinear. Find all values of $x,y$ for which Sunaina has a winning strategy irrespective of the value of $w$. Proposed by NV Tejaswi

There are $n$ airports which form a regular $n$-gon. In the beginnig, there is exactly one plane at only $k$ airports. Each of the planes flies to one of the nearest airport each day. For which of the following ordered pairs $(n,k)$, it is impossible to gather all planes at a airport on one day however the planes are arranged initially? $ \textbf{(A)}\ (10,6) \qquad\textbf{(B)}\ (10,4) \qquad\textbf{(C)}\ (11,3) \qquad\textbf{(D)}\ (11,5) \qquad\textbf{(E)}\ (13,8) $

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. [i]Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece[/i]

[hide=F stands for Fermat, E stands for Euler]they had two problem sets under those two names[/hide] [b]F1.[/b] A circle has area $\pi$. Find the circumference of the circle. [b]F2.[/b] In triangle $ABC$, $AB = 5$, $BC = 12$, and $\angle B = 90^o$. Compute $AC$. [b]F3 / E1.[/b] A square has area $2015$. Find the length of the square's diagonal. [b]F4.[/b] I have two cylindrical candles. The first candle has diameter $1$ and height $1$. The second candle has diameter $2$ and height $2$. Both candles are lit at $1:00$ PM and both burn at the same constant rate (volume per time period). The first candle burns out at $1:50$ PM. When does the second candle burn out? Specify AM or PM. [b]F5 / E2.[/b] In triangle $ABC$, $BC$ has length $12$, the altitude from $A$ to $BC$ has length $6$, and the altitude from $C$ to $AB$ has length $8$. Compute the length of $AB$. [b]F6 / E3.[/b] Let $ABC$ be an isosceles triangle with base $AC$. Suppose that there exists a point $D$ on side $AB$ such that $AC = CD = BD$. Find the degree measure of $\angle ABC$. [b]F7 / E6.[/b] In concave quadrilateral $ABCD$, $\angle ABC = 60^o$ and $\angle ADC = 240^o$. If $AD = CD = 4$, compute $BD$. [b]F8 / E7.[/b] A circle of radius $5$ is inscribed in an isosceles trapezoid with legs of length $14$. Compute the area of the trapezoid. [b]E4.[/b] The Egyptian goddess Isil has a staff consisting of a pole with a circle on top. The length of the pole is $32$ inches, and the tangent segment from the bottom of the pole to the circle is $40$ inches. Find the radius of the circle, in inches. [img]https://cdn.artofproblemsolving.com/attachments/5/b/01ea1819aa58c4bde105b9b885f658b3823494.png[/img] [b]E5.[/b] The two concentric circles shown below have radii $1$ and $2$. A chord of the larger circle that is tangent to the smaller circle is drawn. Find the area of the shaded region, bounded by the chord and the larger circle. [img]https://cdn.artofproblemsolving.com/attachments/e/5/425735d2717548552fda8363c201dc8043da13.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

Let $f(X, Y, Z)=X^5Y-XY^5+Y^5Z-YZ^5+Z^5X-ZX^5$. Find $\frac{f(2009, 2010, 2011)+f(2010, 2011, 2009)-f(2011, 2010, 2009)}{f(2009, 2010, 2011)}$

For three sequences $(x_n),(y_n),(z_n)$ with positive starting elements $x_1,y_1,z_1$ we have the following formulae: \[ x_{n+1} = y_n + \frac{1}{z_n} \quad y_{n+1} = z_n + \frac{1}{x_n} \quad z_{n+1} = x_n + \frac{1}{y_n} \quad (n = 1,2,3, \ldots)\] a.) Prove that none of the three sequences is bounded from above. b.) At least one of the numbers $x_{200},y_{200},z_{200}$ is greater than 20.

A particle $P$ moves in the plane in such a way that the angle between the two tangents drawn from $P$ to the curve $y^2=4ax$ is always $90^\circ$. The locus of $P$ is $\textbf{(A)}~\text{a parabola}$ $\textbf{(B)}~\text{a circle}$ $\textbf{(C)}~\text{an ellipse}$ $\textbf{(D)}~\text{a straight line}$

Prove that there is no polynomial $P$ with integer coefficients such that $P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$.

In a $N \times N$ array of numbers, all rows are different (two rows are said to be different even if they differ only in one entry). Prove that there is a column which can be deleted in such a way that the resulting rows will still be different. (A Andjans, Riga)

Circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ intersect at $ A$ and $ B$. Let $ M\in AB$. A line through $ M$ (different from $ AB$) cuts circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ at $ Z,D,E,C$ respectively such that $ D,E\in ZC$. Perpendiculars at $ B$ to the lines $ EB,ZB$ and $ AD$ respectively cut circle $ \mathcal{C}_2$ in $ F,K$ and $ N$. Prove that $ KF\equal{}NC$.

Vijay chooses three distinct integers $a,b,c$ from the set $\{1,2,3,4,5,6,7,8,9,10,11\}.$ If $k$ is the minimum value taken on by the polynomial $a(x - b)(x - c)$ over all real numbers $x,$ and $l$ is the minimum value taken on by the polynomial $a(x-b)(x+c)$ over all real numbers $x,$ compute the maximum possible value of $k -l.$

Solve the equation $$ tg 7x - \sin 6x=\cos 4x - ctg 7x.$$

Given is a positive integer $n \geq 2$ and three pairwise disjoint sets $A, B, C$, each of $n$ distinct real numbers. Denote by $a$ the number of triples $(x, y, z) \in A \times B \times C$ satisfying $x<y<z$ and let $b$ denote the number of triples $(x, y, z) \in A \times B \times C$ such that $x>y>z$. Prove that $n$ divides $a-b$.

For a positive integer $n$, let function $f(n)$ denote the number of positive integers $a\leq n$ such that $\gcd(a,n) = \gcd(a+1,n) = 1$. Find the sum of all $n$ such that $f(n)=15$. [i]Proposed by Harry Kim[/i]

Consider the sequence $(a^n + 1)_{n\geq 1}$, with $a>1$ a fixed integer. i) Prove there exist infinitely many primes, each dividing some term of the sequence. ii) Prove there exist infinitely many primes, none dividing any term of the sequence. [i](Dan Schwarz)[/i]