Two circles $\Gamma$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $P$ be a point on $\Gamma$. The tangent at $P$ to $\Gamma$ meets $\Omega$ at the points $C$ and $D$, where $D$ lies between $P$ and $C$, and $ABCD$ is a convex quadrilateral. The lines $CA$ and $CB$ meet $\Gamma$ again at $E$ and $F$ respectively. The lines $DA$ and $DB$ meet $\Gamma$ again at $S$ and $T$ respectively. Suppose the points $P,E,S,F,B,T,A$ lie on $\Gamma$ in this order. Prove that $PC,ET,SF$ are parallel.

Let $ x_1, x_2, \cdots, x_n$ be positive real numbers . Prove that: $$\sum_ {i = 1}^n x_i ^2\geq \frac {1} {n + 1} \left (\sum_ {i = 1}^n x_i \right)^2+\frac{12(\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}. $$

Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? $ \textbf{(A)}\ 2520\qquad\textbf{(B)}\ 2880\qquad\textbf{(C)}\ 3120\qquad\textbf{(D)}\ 3250\qquad\textbf{(E)}\ 3750 $

Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $. $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}$$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.

Show that exists a sequence of $ 100$ terms such that: 1)Every term is a perfect square 2) every term is greater than the one before it ( it is strictly increasing) 3)Every two terms of the sequence are relative prime 4) The average between two consecutive terms is also a perfect square Daniel

Consider the numbers $x_1, x_2,...,x_n$ that satisfy: $\bullet$ $x_i \in \{-1,1\}$, with $i = 1, 2,...,n$ $\bullet$ $x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0$ Prove that $n$ is a multiple of $4$.

In the arrow-shaped polygon [see figure], the angles at vertices $A$, $C$, $D$, $E$ and $F$ are right angles, $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. The area of the polygon is closest to [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,10), C=(10,5), D=(30,5), E=(30,-5), F=(10,-5), G=(10,-10); draw(A--B--C--D--E--F--G--A); label("$A$", A, W); label("$B$", B, NE); label("$C$", C, S); label("$D$", D, NE); label("$E$", E, SE); label("$F$", F, N); label("$G$", G, SE); label("$5$", (11,7.5)); label("$5$", (11,-7.5)); label("$20$", (C+D)/2, N); label("$20$", (F+E)/2, S); label("$10$", (31,0)); [/asy] $ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 291\qquad\textbf{(C)}\ 294\qquad\textbf{(D)}\ 297\qquad\textbf{(E)}\ 300 $

If $ (a,b)$ and $ (c,d)$ are two points on the line whose equation is $ y\equal{}mx\plus{}k$, then the distance between $ (a,b)$ and $ (c,d)$, in terms of $ a$, $ c$, and $ m$, is $ \textbf{(A)}\ |a\minus{}c|\sqrt{1\plus{}m^2} \qquad \textbf{(B)}\ |a\plus{}c|\sqrt{1\plus{}m^2} \qquad \textbf{(C)}\ \frac{|a\minus{}c|}{\sqrt{1\plus{}m^2}} \qquad$ $ \textbf{(D)}\ |a\minus{}c|(1\plus{}m^2) \qquad \textbf{(E)}\ |a\minus{}c|$ $ |m|$

Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations \begin{align*} \sqrt{2x - xy} + \sqrt{2y - xy} & = 1\\ \sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}\\ \sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}. \end{align*}Then $\big[ (1-x)(1-y)(1-z) \big] ^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Paul Revere is currently at $\left(x_0, y_0\right)$ in the Cartesian plane, which is inside a triangle-shaped ship with vertices at $\left(-\dfrac{7}{25},\dfrac{24}{25}\right),\left(-\dfrac{4}{5},\dfrac{3}{5}\right)$, and $\left(\dfrac{4}{5},-\dfrac{3}{5}\right)$. Revere has a tea crate in his hands, and there is a second tea crate at $(0,0)$. He must walk to a point on the boundary of the ship to dump the tea, then walk back to pick up the tea crate at the origin. He notices he can take 3 distinct paths to walk the shortest possible distance. Find the ordered pair $(x_0, y_0)$. [i]Proposed by Derek Zhao[/i] [hide=Solution][i]Solution.[/i] $\left(-\dfrac{7}{25},\dfrac{6}{25}\right)$ Let $L$, $M$, and $N$ be the midpoints of $BC$, $AC$, and $AB$, respectively. Let points $D$, $E$, and $F$ be the reflections of $O = (0,0)$ over $BC$, $AC$, and $AB$, respectively. Notice since $MN \parallel BC$, $BC \parallel EF$. Therefore, $O$ is the orthocenter of $DEF$. Notice that $(KMN)$ is the nine-point circle of $ABC$ because it passes through the midpoints and also the nine-point circle of $DEF$ because it passes through the midpoints of the segments connecting a vertex to the orthocenter. Since $O$ is both the circumcenter of $ABC$ and the orthocenter of $DEF$ and the triangles are $180^\circ$ rotations of each other, Revere is at the orthocenter of $ABC$. The answer results from adding the vectors $OA +OB +OC$, which gives the orthocenter of a triangle.[/hide]

For each pair of positive integers $(x, y)$ a nonnegative integer $x\Delta y$ is defined. It is known that for all positive integers $a$ and $b$ the following equalities hold: i. $(a + b)\Delta b = a\Delta b + 1$. ii. $(a\Delta b) \cdot (b\Delta a) = 0$. Find the values of the expressions $2016\Delta 121$ and $2016\Delta 144$.

Let $ABC$ be a triangle with $E$ being the centre of its Euler circle. Through $E$, construct the lines $PS, MQ, NR$ parallel to $BC,CA,AB$ ($R,Q$ are on the line $BC, N, P$ on the line $AC,M, S$ on the line $AB$). Prove that the four Euler lines of triangles $ABC,AMN,BSR,CPQ$ are concurrent. Nguyễn Văn Linh

Let $n{}$ be a positive integer. Show that there exists a polynomial $f{}$ of degree $n{}$ with integral coefficients such that \[f^2=(x^2-1)g^2+1,\] where $g{}$ is a polynomial with integral coefficients.

Given $\sum_{n=1}^{\infty} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$, find$$\sum_{j=1}^{\infty} \sum_{i=1}^j \frac{1}{ij(i+1)(j+1)}.$$

Let $ABC$ be a right triangle with hypotenuse $BC$. The tangent to the circumcircle of triangle $ABC$ at $A$ intersects the line $BC$ at $T$. The points $D$ and $E$ are chosen so that $AD = BD, AE = CE,$ and $\angle CBD = \angle BCE < 90^{\circ}$. Prove that $D, E,$ and $T$ are collinear. Proposed by [i]Nikola Velov, Macedonia[/i]

There are $n{}$ stones in a heap. Two players play the game by alternatively taking either 1 stone from the heap or a prime number of stones which divides the current number of stones in the heap. The player who takes the last stone wins. For which $n{}$ does the first player have a strategy so that he wins no matter how the other player plays? [i]Fedor Ivlev[/i]

Let $AD$, $BE$, $CF$ be the heights of triangle $ABC$ and let $K$, $L$, $M$ be the orthocenters of triangles $AEF$, $BFD$ and $CDE$, respectively. Let $G_1$ and $G_2$ denote the centroids of triangles $DEF$ and $KLM$, respectively. Show that $HG_1 = G_1G_2$, where $H$ is the orthocenter of triangle $ABC$.

Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$. (Yuri Biletsky)

Let $ABC$ be an equilateral triangle of side length $2$. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k<1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$. with $CB'=AC'=k$. Line segments are drawn from points $A',B',C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Prove that $\Delta PQR$ is an equilateral triangle with side length ${4(1-k) \over \sqrt{k^2-2k+4}}$.

Let $ABC$ be a scalene acute triangle whose orthocenter is $H$. $M$ is the midpoint of segment $BC$. $N$ is the point where the segment $AM$ intersects the circle determined by $B, C$, and $H$. Show that lines $HN$ and $AM$ are perpendicular.

Is there a circle which passes through five points with integer co-ordinates?

Let $n$ be a positive integer. Given a sequence of nonnegative real numbers $x_1,\ldots ,x_n$ we define the [i]transformed sequence[/i] $y_1,\ldots ,y_n$ as follows: the number $y_i$ is the greatest possible value of the average of consecutive terms of the sequence that contain $x_i$. For example, the transformed sequence of $2,4,1,4,1$ is $3,4,3,4,5/2$. Prove that a) For every positive real number $t$, the number of $y_i$ such that $y_i>t$ is less than or equal to $\frac{2}{t}(x_1+\cdots +x_n)$. b) The inequality $\frac{y_1+\cdots +y_n}{32n}\leq \sqrt{\frac{x_1^2+\cdots +x_n^2}{32n}}$ holds.

For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first? $ \textbf{(A)}\ \text{Andy} \qquad \textbf{(B)}\ \text{Beth} \qquad \textbf{(C)}\ \text{Carlos} \qquad \textbf{(D)}\ \text{Andy and Carlos tie for first.}$ $\textbf{(E)}\ \text{All three tie.}$

A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.