How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

Let $C$ be the part of the graph $y=\frac{1}{x}\ (x>0)$. Take a point $P\left(t,\ \frac{1}{t}\right)\ (t>0)$ on $C$. (i) Find the equation of the tangent $l$ at the point $A(1,\ 1)$ on the curve $C$. (ii) Let $m$ be the line passing through the point $P$ and parallel to $l$. Denote $Q$ be the intersection point of the line $m$ and the curve $C$ other than $P$. Find the coordinate of $Q$. (iii) Express the area $S$ of the part bounded by two line segments $OP,\ OQ$ and the curve $C$ for the origin $O$ in terms of $t$. (iv) Express the volume $V$ of the solid generated by a rotation of the part enclosed by two lines passing through the point $P$ and pararell to the $y$-axis and passing through the point $Q$ and pararell to $y$-axis, the curve $C$ and the $x$-axis in terms of $t$. (v) $\lim_{t\rightarrow 1-0} \frac{S}{V}.$

Let $ABC$ be an equilateral triangle. A point $T$ is chosen on $AC$ and on arcs $AB$ and $BC$ of the circumcircle of $ABC$, $M$ and $N$ are chosen respectively, so that $MT$ is parallel to $BC$ and $NT$ is parallel to $AB$. Segments $AN$ and $MT$ intersect at point $X$, while $CM$ and $NT$ intersect in point $Y$. Prove that the perimeters of the polygons $AXYC$ and $XMBNY$ are the same.

In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.

Let $ABC$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $ABC$ in its sides $BC,CA,AB$ are denoted by $G_1,G_2,G_3$ and $O_1,O_2,O_3$, respectively. Show that the circumcircles of triangles $G_1G_2C$, $G_1G_3B$, $G_2G_3A$, $O_1O_2C$, $O_1O_3B$, $O_2O_3A$ and $ABC$ have a common point. [i]The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.[/i]

Let $O$ be the centre of the incircle of $\triangle ABC$. Points $K,L$ are the intersection points of the circles circumscribed about triangles $BOC,AOC$ respectively with the bisectors of the angles at $A,B$ respectively $(K,L\not= O)$. Also $P$ is the midpoint of segment $KL$, $M$ is the reflection of $O$ with respect to $P$ and $N$ is the reflection of $O$ with respect to line $KL$. Prove that the points $K,L,M$ and $N$ lie on the same circle.

The circle $\Gamma$ and the line $\ell$ have no common points. Let $AB$ be the diameter of $\Gamma$ perpendicular to $\ell$, with $B$ closer to $\ell$ than $A$. An arbitrary point $C\not= A$, $B$ is chosen on $\Gamma$. The line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on the same side of $AC$. Let $BE$ intersect $\ell$ at $F$, and let $AF$ intersect $\Gamma$ at $G\not= A$. Let $H$ be the reflection of $G$ in $AB$. Show that $F,C$, and $H$ are collinear.

The parabola $\mathcal P$ given by equation $y=x^2$ is rotated some acute angle $\theta$ clockwise about the origin such that it hits both the $x$ and $y$ axes at two distinct points. Suppose the length of the segment $\mathcal P$ cuts the $x$-axis is $1$. What is the length of the segment $\mathcal P$ cuts the $y$-axis?

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation. [hide]translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here[/url][/hide]

A plane contains points $ A$ and $ B$ with $ AB \equal{} 1$. Let $ S$ be the union of all disks of radius $ 1$ in the plane that cover $ \overline{AB}$. What is the area of $ S$? $ \textbf{(A)}\ 2\pi \plus{} \sqrt3 \qquad \textbf{(B)}\ \frac {8\pi}{3} \qquad \textbf{(C)}\ 3\pi \minus{} \frac {\sqrt3}{2} \qquad \textbf{(D)}\ \frac {10\pi}{3} \minus{} \sqrt3 \qquad \textbf{(E)}\ 4\pi \minus{} 2\sqrt3$

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

Let $ABC$ be a triangle with $AB = AC$ and $\angle BAC = 30^{\circ}$, Let $A'$ be the reflection of $A$ in the line $BC$; $B'$ be the reflection of $B$ in the line $CA$; $C'$ be the reflection of $C$ in line $AB$, Show that $A'B'C'$ is an equilateral triangle.

Rectangle $ PQRS$ lies in a plane with $ PQ = RS = 2$ and $ QR = SP = 6$. The rectangle is rotated $ 90^\circ$ clockwise about $ R$, then rotated $ 90^\circ$ clockwise about the point that $ S$ moved to after the first rotation. What is the length of the path traveled by point $ P$? ${ \textbf{(A)}\ (2\sqrt3 + \sqrt5})\pi \qquad \textbf{(B)}\ 6\pi \qquad \textbf{(C)}\ (3 + \sqrt {10})\pi \qquad \textbf{(D)}\ (\sqrt3 + 2\sqrt5)\pi \\ \textbf{(E)}\ 2\sqrt {10}\pi$

In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$. [i]Proposed by Zuming Feng[/i]

Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.

Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ denote the reflection of $D$ over line $IO$. Find the integer closest to $1000 \cdot \frac{BE}{CE}$. [i]Proposed by Evan Chen[/i]

Let $ABC$ be a triangle. The incircle of $ABC$ touches $BC,AC,AB$ at $D,E,F$, respectively. Each pair of the incircles of triangles $AEF, BDF,CED$ has two pair of common external tangents, one of them being one of the sides of $ABC$. Show that the other three tangents divide triangle $DEF$ into three triangles and three parallelograms.

Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$. $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$, tangent to $(I),(O)$ in $D,K$ (for $\omega_a$), $E,M$ (for $\omega_b$), and $F,N$ (for $\omega_c$). [b]a)[/b] Show that $DK,EM,FN$ are concurrent in a point $P$; [b]b)[/b] Show that the orthocenter of $DEF$ lies on $OP$.

We call a path Valid if i. It only comprises of the following kind of steps: A. $(x, y) \rightarrow (x + 1, y + 1)$ B. $(x, y) \rightarrow (x + 1, y - 1)$ ii. It never goes below the x-axis. Let $M(n)$ = set of all valid paths from $(0,0) $, to $(2n,0)$, where $n$ is a natural number. Consider a Valid path $T \in M(n)$. Denote $\phi(T) = \prod_{i=1}^{2n} \mu_i$, where $\mu_i$= a) $1$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$ b) $y$, if the $i^{th} $ step is $(x, y) \rightarrow (x + 1, y - 1)$ Now Let $f(n) =\sum _{T \in M(n)} \phi(T)$. Evaluate the number of zeroes at the end in the decimal expansion of $f(2021)$

$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition: (1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$ (2) $d \mid (x_1+x_2+ \cdots x_n)$ Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.

Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.

In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.

Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.

Let a positive integer $n$.Consider square table $3*3$.One use $n$ colors to color all cell of table such that each cell is colored by exactly one color. Two colored table is same if we can receive them from other by a rotation through center of $3*3$ table How many way to color this square table satifies above conditions.