The endpoints of a compass are at two lattice points of an infinite unit square grid. It is allowed to rotate the compass around one of its endpoints, not varying its radius, and thus move the other endpoint to another lattice point. Can the endpoints of the compass change places after several such steps?

Let $ABC$ be a right angled triangle at $A$. Denote $D$ the foot of the altitude through $A$ and $O_1, O_2$ the incentres of triangles $ADB$ and $ADC$. The circle with centre $A$ and radius $AD$ cuts $AB$ in $K$ and $AC$ in $L$. Show that $O_1, O_2, K$ and $L$ are on a line.

Two rooks on a chessboard are said to be attacking each other if they are placed in the same row or column of the board. (a) There are eight rooks on a chessboard, none of them attacks any other. Prove that there is an even number of rooks on black fields. (b) How many ways can eight mutually non-attacking rooks be placed on the 9 £ 9 chessboard so that all eight rooks are on squares of the same color.

How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the same color? Rotations and reflections are considered distinct.

Let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.

Let $\triangle ABC$ be an arbitary triangle, and construct $P,Q,R$ so that each of the angles marked is $30^\circ$. Prove that $\triangle PQR$ is an equilateral triangle. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair ext30(pair pt1, pair pt2) { pair r1 = pt1+rotate(-30)*(pt2-pt1), r2 = pt2+rotate(30)*(pt1-pt2); draw(anglemark(r1,pt1,pt2,25)); draw(anglemark(pt1,pt2,r2,25)); return intersectionpoints(pt1--r1, pt2--r2)[0]; } pair A = (0,0), B=(10,0), C=(3,7), P=ext30(B,C), Q=ext30(C,A), R=ext30(A,B); draw(A--B--C--A--R--B--P--C--Q--A); draw(P--Q--R--cycle, linetype("8 8")); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$P$", P, NE); label("$Q$", Q, NW); label("$R$", R, S);[/asy]

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 $

Consider the following statement: for any permutation $\pi_1\not=\mathbb{I}$ of $\{1,2,...,n\}$ there is a permutation $\pi_2$ such that any permutation on these numbers can be obtained by a finite compostion of $\pi_1$ and $\pi_2$. (a) Prove the statement for $n=3$ and $n=5$. (b) Disprove the statement for $n=4$.

10) Call a string of letters $S$ an [i]almost-palindrome[/i] if $S$ and the reverse of $S$ differ in exactly $2$ places. Find the number of ways to order the letters in $HMMTTHEMETEAM$ to get an almost-palindrome. 11) Find all integers $n$, not necessarily positive, for which there exist positive integers ${a,b,c}$ satisfying $a^n + b^n = c^n$. 12) Let $a$ and $b$ be positive real numbers. Determine the minimum possible value of $\sqrt{a^2 + b^2} + \sqrt{a^2 + (b-1)^2} + \sqrt{(a-1)^2 + b^2} + \sqrt{(a-1)^2 + (b-1)^2}$. 13) Consider a $4$ x $4$ grid of squares, each originally colored red. Every minute, Piet can jump on any of the squares, changing the color of it and any adjacent squares to blue (two squares are adjacent if they share a side). What is the minimum number of minutes it will take Piet to change the entire grid to blue? 14) Let $ABC$ be an acute triangle with orthocenter $H$. Let ${D,E}$ be the feet of the ${A,B}$-altitudes, respectively. Given that $\overline{AH} = 20$ and $\overline{HD} =16$ and $\overline{BE} = 56$, find the length of $\overline{BH}$. 15) Find the smallest positive integer $b$ such that $1111 _b$ ($1111$ in base $b$) is a perfect square. If no such $b$ exists, write "No Solution" 16) For how many triples $( {x,y,z} )$ of integers between $-10$ and $10$, inclusive, do there exist reals ${a,b,c}$ that satisfy $ab = x$ $ac = y$ $bc = z$? 17) Unit squares $ABCD$ and $EFGH$ have centers $O_1$ and $O_2$, respectively, and are originally oriented so that $B$ and $E$ are at the same position and $C$ and $H$ are at the same position. The squares then rotate clockwise around their centers at a rate of one revolution per hour. After $5$ minutes, what is the area of the intersection of the two squares? 18) A function $f$ satisfies, for all nonnegative integers $x$ and $y$, $f(x,0) = f(0,x) = x$ If $x \ge y \ge 0$, $f(x,y)=f(x-y,y)+1$ If $y \ge x \ge 0$, $f(x,y) = f(x,y-x)+1$ Find the maximum value of $f$ over $0 \le x,y \le 100$.

Given two (identical) polygonal domains in the Euclidean plane, it is not possible in general to superpose the two using only translations and rotations. Prove that this can however be achieved by splitting one of the domains into a finite number of polygonal subdomains which then fit together, via translations and rotations in the plane, to recover the other domain.

A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.

Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis. [i]1993 Hiroshima Un iversity entrance exam/Science[/i]

When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?

Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$. [b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$. [b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.

Lisa and Bart are playing a game. A round table has $n$ lights evenly spaced around its circumference. Some of the lights are on and some of them off; the initial configuration is random. Lisa wins if she can get all of the lights turned on; Bart wins if he can prevent this from happening. On each turn, Lisa chooses the positions at which to flip the lights, but before the lights are flipped, Bart, knowing Lisa’s choices, can rotate the table to any position that he chooses (or he can leave the table as is). Then the lights in the positions that Lisa chose are flipped: those that are off are turned on and those that are on are turned off. Here is an example turn for $n = 5$ (a white circle indicates a light that is on, and a black circle indicates a light that is off): [asy] size(250); defaultpen(linewidth(1)); picture p = new picture; real r = 0.2; pair s1=(0,-4), s2=(0,-8); int[][] filled = {{1,2,3},{1,2,5},{2,3,4,5}}; draw(p,circle((0,0),1)); for(int i = 0; i < 5; ++i) { pair P = dir(90-72*i); filldraw(p,circle(P,r),white); label(p,string(i+1),P,2*P,fontsize(10)); } add(p); add(shift(s1)*p); add(shift(s2)*p); for(int j = 0; j < 3; ++j) for(int i = 0; i < filled[j].length; ++i) filldraw(circle(dir(90-72*(filled[j][i]-1))+j*s1,r)); label("$\parbox{15em}{Initial Position.}$", (-4.5,0)); label("$\parbox{15em}{Lisa says ``1,3,4.'' \\ Bart rotates the table one \\ position counterclockwise. }$", (-4.5,0)+s1); label("$\parbox{15em}{Lights in positions 1,3,4 are \\ flipped.}$", (-4.5,0)+s2);[/asy] Lisa can take as many turns as she needs to win, or she can give up if it becomes clear to her that Bart can prevent her from winning. (a) Show that if $n = 7$ and initially at least one light is on and at least one light is off, then Bart can always prevent Lisa from winning. (b) Show that if $n = 8$, then Lisa can always win in at most 8 turns.

I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.

Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$. [b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$. [b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.

Let $M$ be a set of points in a plane with at least two elements. Prove that if $M$ has two axes of symmetry $g_1$ and $g_2$ intersecting at an angle $\alpha = q\pi$, where $q$ is irrational, then $M$ must be infinite.

Given a straight line $\ell$ and the point $O$ out of the line. Prove that it is possible to move an arbitrary point $A$ in the same plane to the $O$ point, using only rotations around $O$ and symmetry with respect to the $\ell$.

A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?

A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is $5/9$ that of water. At equilibrium, what portion of the rod is above water? $\textbf{(A) } 0.25\\ \textbf{(B) } 0.33\\ \textbf{(C) } 0.5\\ \textbf{(D) } 0.67\\ \textbf{(E) } 0.75$

Let $ABCD$ be a square, and $C$ the circle whose diameter is $AB.$ Let $Q$ be an arbitrary point on the segment $CD.$ We know that $QA$ meets $C$ on $E$ and $QB$ meets it on $F.$ Also $CF$ and $DE$ intersect in $M.$ show that $M$ belongs to $C.$

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}$

In an acute-angled triangle $ABC, CD$ is the altitude. A line through the midpoint $M$ of side $AB$ meets the rays $CA$ and $CB$ at $K$ and $L$ respectively such that $CK = CL.$ Point $S$ is the circumcenter of the triangle $CKL.$ Prove that $SD = SM.$

The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s = 6 \sqrt{2}$, what is the volume of the solid? [asy] import three; size(170); pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(30,-20,10); real s = 6 * 2^.5; triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6); draw(F--B--C--F--E--A--B); draw(A--D--E, dashed); draw(D--C, dashed); label("$2s$", (s/2, s/2, 6), N); label("$s$", (s/2, 0, 0), SW); [/asy]