A board of $2n$ x $2n$ is colored chess style, a movement is the changing of colors of a $2$ x $2$ square. For what integers $n$ is possible to complete the board with one color using a finite number of movements?

The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC.

The number of teeth in three meshed gears $A$, $B$, and $C$ are $x$, $y$, and $z$, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of $A$, $B$, and $C$ are in the proportion $\text{(A)} \ x: y: z ~~\text{(B)} \ z: y: x ~~ \text{(C)} \ y: z: x~~ \text{(D)} \ yz: xz: xy ~~ \text{(E)} \ xz: yx: zy$

Triangle $ABC$ is given. The circle $\omega$ with center $I$ is tangent at points $D,E,F$ to segments $BC,AC,AB$ respectively. When $ABC$ is rotated $180$ degrees about point $I$, triangle $A'B'C'$ results. Lines $AD, B'C'$ meet at $U$, lines $BE, A'C'$ meet at $V$, and lines $CF, A'B'$ meet at $W$. Line $BC$ meets $A'C', A'B'$ at points $D_1, D_2$ respectively. Line $AC$ meets $A'B', B'C'$ at $E_1, E_2$ respectively. Line $AB$ meets $B'C', A'C'$ at $F_1,F_2$ respectively. Six (not necessarily convex) quadrilaterals were colored orange: \[AUIF_2 , C'FIF_2 , BVID_1 , A'DID_2 , CWIE_1 , B'EIE_2\] Six other quadrilaterals were colored green: \[AUIE_2 , C'FIF_1 , BVIF_2 , A'DID_1 , CWID_2 , B'EIE_1\] Prove that the sum of the green areas equals the sum of the orange areas.

A block of mass $m$ on a frictionless inclined plane of angle $\theta$ is connected by a cord over a small frictionless, massless pulley to a second block of mass $M$ hanging vertically, as shown. If $M=1.5m$, and the acceleration of the system is $\frac{g}{3}$, where $g$ is the acceleration of gravity, what is $\theta$, in degrees, rounded to the nearest integer? [asy]size(12cm); pen p=linewidth(1), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75); pair B = (-1,-1); pair C = (-1,-7); pair A = (-13,-7); path inclined_plane = A--B--C--cycle; draw(inclined_plane, p); real r = 1; // for marking angles draw(arc(A, r, 0, degrees(B-A))); // mark angle label("$\theta$", A + r/1.337*(dir(C-A)+dir(B-A)), (0,0), fontsize(16pt)); // label angle as theta draw((C+(-r/2,0))--(C+(-r/2,r/2))--(C+(0,r/2))); // draw right angle real h = 1.2; // height of box real w = 1.9; // width of box path box = (0,0)--(0,h)--(w,h)--(w,0)--cycle; // the box // box on slope with label picture box_on_slope; filldraw(box_on_slope, box, light_grey, black); label(box_on_slope, "$m$", (w/2,h/2)); pair V = A + rotate(90) * (h/2 * dir(B-A)); // point with distance l/2 from AB pair T1 = dir(125); // point of tangency with pulley pair X1 = intersectionpoint(T1--(T1 - rotate(-90)*(2013*dir(T1))), V--(V+B-A)); // construct midpoint of right side of box draw(T1--X1); // string add(shift(X1-(w,h/2))*rotate(degrees(B-A), (w,h/2)) * box_on_slope); // picture for the hanging box picture hanging_box; filldraw(hanging_box, box, light_grey, black); label(hanging_box, "$M$", (w/2,h/2)); pair T2 = (1,0); pair X2 = (1,-3); draw(T2--X2); // string add(shift(X2-(w/2,h)) * hanging_box); // Draws the actual pulley filldraw(unitcircle, grey, p); // outer boundary of pulley wheel filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley[/asy][i](Proposed by Ahaan Rungta)[/i]

A token starts at the point $(0,0)$ of an $xy$-coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Two transformations are said to [i]commute[/i] if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane: - A translation $2$ units to the right - A $90^\circ$- rotation counterclockwise about the origin. - A reflection across the $x$-axis, and - A dilation centered at the origin with scale factor $2$. Of the $6$ pairs of distinct transformations from this list, how many commute? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5 \qquad $

The solutions of the equation $ z^4 \plus{} 4z^3i \minus{} 6z^2 \minus{} 4zi \minus{} i \equal{} 0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon? $ \textbf{(A)}\ 2^{5/8} \qquad \textbf{(B)}\ 2^{3/4} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2^{5/4} \qquad \textbf{(E)}\ 2^{3/2}$

Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.

Let $X,Y,Z,T$ be four points in the plane. The segments $[XY]$ and $[ZT]$ are said to be [i]connected[/i], if there is some point $O$ in the plane such that the triangles $OXY$ and $OZT$ are right-angled at $O$ and isosceles. Let $ABCDEF$ be a convex hexagon such that the pairs of segments $[AB],[CE],$ and $[BD],[EF]$ are [i]connected[/i]. Show that the points $A,C,D$ and $F$ are the vertices of a parallelogram and $[BC]$ and $[AE]$ are [i]connected[/i].

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that \[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle? $ \textbf{(A)}\ \frac{1}{36} \qquad \textbf{(B)}\ \frac{1}{24} \qquad \textbf{(C)}\ \frac{1}{18} \qquad \textbf{(D)}\ \frac{1}{12} \qquad \textbf{(E)}\ \frac{1}{9}$

Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, of a square $ABCD$. If $|BM|=21$, $|DN|=4$, and $|NC|=24$, what is $m(\widehat{MAN})$? $ \textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 30^\circ \qquad\textbf{(C)}\ 37^\circ \qquad\textbf{(D)}\ 45^\circ \qquad\textbf{(E)}\ 60^\circ $

Show that for every convex polygon whose area is less than or equal to $1$, there exists a parallelogram with area $2$ containing the polygon.

Answer the following questions: (1) Let $a$ be non-zero constant. Find $\int x^2 \cos (a\ln x)dx.$ (2) Find the volume of the solid generated by a rotation of the figures enclosed by the curve $y=x\cos (\ln x)$, the $x$-axis and the lines $x=1,\ x=e^{\frac{\pi}{4}}$ about the $x$-axis.

Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance $x$ from where it starts. The distance $x$ can be expressed in the form $a\pi+b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c.$

We want to paint some identically-sized cubes so that each face of each cube is painted a solid color and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce?

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

For a real number $ a$, let $ \lfloor a \rfloor$ denominate the greatest integer less than or equal to $ a$. Let $ \mathcal{R}$ denote the region in the coordinate plane consisting of points $ (x,y)$ such that \[\lfloor x \rfloor ^2 \plus{} \lfloor y \rfloor ^2 \equal{} 25.\] The region $ \mathcal{R}$ is completely contained in a disk of radius $ r$ (a disk is the union of a circle and its interior). The minimum value of $ r$ can be written as $ \tfrac {\sqrt {m}}{n}$, where $ m$ and $ n$ are integers and $ m$ is not divisible by the square of any prime. Find $ m \plus{} n$.

Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.

An $n$-mino is a connected figure made by connecting $n$ $1 \times 1 $ squares. Two polyminos are the same if moving the first we can reach the second. For a polymino $P$ ,let $|P|$ be the number of $1 \times 1$ squares in it and $\partial P$ be number of squares out of $P$ such that each of the squares have at least on edge in common with a square from $P$. (a) Prove that for every $x \in (0,1)$:\[\sum_P x^{|P|}(1-x)^{\partial P}=1\] The sum is on all different polyminos. (b) Prove that for every polymino $P$, $\partial P \leq 2|P|+2$ (c) Prove that the number of $n$-minos is less than $6.75^n$. [i]Proposed by Kasra Alishahi[/i]

Let $ABCD$ be a square and let $\ell$ be a line. Let $M$ be the centre of the square. The diagonals of the square have length 2 and the distance from $M$ to $\ell$ exceeds 1. Let $A',B',C',D'$ be the orthogonal projections of $A,B,C,D$ onto $\ell$. Suppose that one rotates the square, such that $M$ is invariant. The positions of $A,B,C,D,A',B',C',D'$ change. Prove that the value of $AA'^2 + BB'^2 + CC'^2 + DD'^2$ does not change.

A polygon is called concave if it has at least one angle strictly greater than $180^{\circ}$. What is the maximum number of symmetries that an 11-sided concave polygon can have? (A [i]symmetry[/i] of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.)

For which positive integers $n$ is it possible to cover a $(2n+1) \times (2n+1)$ chessboard which has one of its corner squares cut out with tiles shown in the figure (each tile covers exactly $4$ squares, tiles can be rotated and turned around)? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]

A right circular cone has base radius $ r$ and height $ h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $ 17$ complete rotations. The value of $ h/r$ can be written in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.