Lengths of two sides of a rectangle are $\sqrt2,1$. The rectangle rotates a round around one of its diagonal. Find the volume of the revolved body.

At lunch, the seven members of the Kubik family sit down to eat lunch together at a round table. In how many distinct ways can the family sit at the table if Alexis refuses to sit next to Joshua? (Two arrangements are not considered distinct if one is a rotation of the other.)

Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped. If it lands heads, we define $Q_i$ to be $P_i$; otherwise, we define $Q_i$ to be the reflection of $P_i$ over $O$. (So, it is possible for some of the $Q_i$ to coincide.) Let $D$ be the length of the vector $\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}$. Compute the expected value of $D^2$. [i]Ray Li[/i]

Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Prove that $AA'$, $BB'$ and $CC'$ are concurrent.

[b]points in plane[/b] set $A$ containing $n$ points in plane is given. a $copy$ of $A$ is a set of points that is made by using transformation, rotation, homogeneity or their combination on elements of $A$. we want to put $n$ $copies$ of $A$ in plane, such that every two copies have exactly one point in common and every three of them have no common elements. a) prove that if no $4$ points of $A$ make a parallelogram, you can do this only using transformation. ($A$ doesn't have a parallelogram with angle $0$ and a parallelogram that it's two non-adjacent vertices are one!) b) prove that you can always do this by using a combination of all these things. time allowed for this question was 1 hour and 30 minutes

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

Let $\overline{AH_1}, \overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$. The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$, respectively. For $k = 1, 2, 3$, let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$. Prove that the circumcircles of triangles $T_1P_1T_2$, $T_2P_2T_3$, $T_3P_3T_1$ pass through a common point.

Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.

In a quadrilateral $ABCD$ we have $AB||CD$ and $AB=2CD$. A line $\ell$ is perpendicular to $CD$ and contains the point $C$. The circle with centre $D$ and radius $DA$ intersects the line $\ell$ at points $P$ and $Q$. Prove that $AP\perp BQ$.

Squares $ABB_{1}A_{2}$ and $BCC_{1}B_{2}$ are externally drawn on the hypotenuse $AB$ and on the leg $BC$ of a right triangle $ABC$ . Show that the lines $CA_{2}$ and $AB_{2}$ meet on the perimeter of a square with the vertices on the perimeter of triangle $ABC .$

In acute triangle $ABC$, $AB > AC$. Let $M$ be the midpoint of side $BC$. The exterior angle bisector of $\widehat{BAC}$ meet ray $BC$ at $P$. Point $K$ and $F$ lie on line $PA$ such that $MF \perp BC$ and $MK \perp PA$. Prove that $BC^2 = 4 PF \cdot AK$. [asy] defaultpen(fontsize(10)); size(7cm); pair A = (4.6,4), B = (0,0), C = (5,0), M = midpoint(B--C), I = incenter(A,B,C), P = extension(A, A+dir(I--A)*dir(-90), B,C), K = foot(M,A,P), F = extension(M, (M.x, M.x+1), A,P); draw(K--M--F--P--B--A--C); pair point = I; pair[] p={A,B,C,M,P,F,K}; string s = "A,B,C,M,P,F,K"; int size = p.length; real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;} string[] k= split(s,","); for(int i = 0;i<p.length;++i) { label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i])); }[/asy]

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations: [list=] [*]$L,$ a rotation of $90^{\circ}$ counterclockwise around the origin; [*]$R,$ a rotation of $90^{\circ}$ clockwise around the origin; [*]$H,$ a reflection across the $x$-axis; and [*]$V,$ a reflection across the $y$-axis. [/list] Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.) $\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$

Let $(I_b)$, $(I_c)$ are excircles of a triangle $ABC$. Given a circle $ \omega $ passes through $A$ and externally tangents to the circles $(I_b)$ and $(I_c)$ such that it intersects with $BC$ at points $M$, $N$. Prove that $ \angle BAM=\angle CAN $. A. Smirnov

Given any set of $9$ points in the plane such that there is no $3$ of them collinear, show that for each point $P$ of the set, the number of triangles with its vertices on the other $8$ points and that contain $P$ on its interior is even.

Let $O$ be the circumcenter and $I$ be the incenter of an acute triangle $ABC$ with $m(\widehat{B}) \neq m(\widehat{C})$. Let $D$, $E$, $F$ be the midpoints of the sides $[BC]$, $[CA]$, $[AB]$, respectively. Let $T$ be the foot of perpendicular from $I$ to $[AB]$. Let $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of $[OI]$. If $A$, $P$, $Q$ are collinear, prove that \[\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.\]

The distinct points $X_1, X_2, X_3, X_4, X_5, X_6$ all lie on the same side of the line $AB$. The six triangles $ABX_i$ are all similar. Show that the $X_i$ lie on a circle.

The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.

Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$. Let $ \mathcal Q$ be the locus of the midpoint of $ AB$. It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2$ denote its vertex and focus, respectively. Determine the ratio $ F_1F_2/V_1V_2$.

$ M$ is the midpoint of base $ BC$ in a trapezoid $ ABCD$. A point $ P$ is chosen on the base $ AD$. The line $ PM$ meets the line $ CD$ at a point $ Q$ such that $ C$ lies between $ Q$ and $ D$. The perpendicular to the bases drawn through $ P$ meets the line $ BQ$ at $ K$. Prove that $ \angle QBC \equal{} \angle KDA$. [i]Proposed by S. Berlov[/i]

In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos ^2 \angle AEI = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, determine $m+n$. [i]Proposed by Ray Li[/i]

Isosceles right triangle $ ABC$ encloses a semicircle of area $ 2\pi$. The circle has its center $ O$ on hypotenuse $ \overline{AB}$ and is tangent to sides $ \overline{AC}$ and $ \overline{BC}$. What is the area of triangle $ ABC$? [asy]defaultpen(linewidth(0.8));pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(circle(o, 2)); clip(a--b--c--cycle); draw(a--b--c--cycle); dot(o); label("$C$", c, NW); label("$A$", a, NE); label("$O$", o, SE); label("$B$", b, SW);[/asy] $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi $

Let $ ABC$ be a triangle. Circle $ \omega$­ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$­ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of ­$ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.

We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly two units. Determine the number of regular size tables $n\times n$ ($n> 1$ - given a fixed positive integer). (We can assume that the rows and columns of the tables are numbered: the cases of coincidence in turn, reflect, and so considered different).

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

Let $ABC$ be a triangle in which $\angle A = 60^\circ$. Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$. Let $M$ be the reflection of $A$ in line $EF$. Prove that $M$ lies on $BC$.