[b]carpeting[/b] suppose that $S$ is a figure in the plane such that it's border doesn't contain any lattice points. suppose that $x,y$ are two lattice points with the distance $1$ (we call a point lattice point if it's coordinates are integers). suppose that we can cover the plane with copies of $S$ such that $x,y$ always go on lattice points ( you can rotate or reverse copies of $S$). prove that the area of $S$ is equal to lattice points inside it. time allowed for this question was 1 hour.

A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.

Hexagon $ABCDEF$ is divided into four rhombuses, $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown. Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are congruent, and each has area $\sqrt{2006}$. Let $K$ be the area of rhombus $\mathcal{T}$. Given that $K$ is a positive integer, find the number of possible values for $K$. [asy] size(150);defaultpen(linewidth(0.7)+fontsize(10)); draw(rotate(45)*polygon(4)); pair F=(1+sqrt(2))*dir(180), C=(1+sqrt(2))*dir(0), A=F+sqrt(2)*dir(45), E=F+sqrt(2)*dir(-45), B=C+sqrt(2)*dir(180-45), D=C+sqrt(2)*dir(45-180); draw(F--(-1,0)^^C--(1,0)^^A--B--C--D--E--F--cycle); pair point=origin; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$\mathcal{P}$", intersectionpoint( A--(-1,0), F--(0,1) )); label("$\mathcal{S}$", intersectionpoint( E--(-1,0), F--(0,-1) )); label("$\mathcal{R}$", intersectionpoint( D--(1,0), C--(0,-1) )); label("$\mathcal{Q}$", intersectionpoint( B--(1,0), C--(0,1) )); label("$\mathcal{T}$", point); dot(A^^B^^C^^D^^E^^F);[/asy]

There are 8 members in a a bridge committee (committee for making bridges). Of these 8 members, 3 are chosen to be in special "approval" committee with 1 of 3 members being the "boss." In how many ways can this happen?

Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $ 20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $ 10$ vertical feet above the bottom? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7.5 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 15$

A semicircle with diameter $d$ is contained in a square whose sides have length $8$. Given the maximum value of $d$ is $m- \sqrt{n}$, find $m+n$.

Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$ $\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 30$

Ten chairs are evenly spaced around a round table and numbered clockwise from $ 1$ through $ 10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible? $ \textbf{(A)}\ 240\qquad \textbf{(B)}\ 360\qquad \textbf{(C)}\ 480\qquad \textbf{(D)}\ 540\qquad \textbf{(E)}\ 720$

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

In how many ways a cube can be painted using seven different colors in such a way that no two faces are in same color? $ \textbf{(A)}\ 154 \qquad\textbf{(B)}\ 203 \qquad\textbf{(C)}\ 210 \qquad\textbf{(D)}\ 240 \qquad\textbf{(E)}\ \text{None of the above} $

In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis. (1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section. (2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$. (3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis. [i]1987 Sophia University entrance exam/Science and Technology[/i]

In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$? [img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]

The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral. [i]Laurentiu Panaitopol[/i]

Determine all $n \in \mathbb{Z}^+$ such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many $n-$gons such that they can be composed into $n$ congruent regular hexagons in a non-overlapping way upon certain rotations and translations.

USAYNO: \url{http://goo.gl/wVR25} % USAYNO link: http://goo.gl/wVR25 [i]Proposed by Lewis Chen, Evan Chen, Eugene Chen[/i]

Isosceles $ \triangle ABC$ has a right angle at $ C$. Point $ P$ is inside $ \triangle ABC$, such that $ PA \equal{} 11, PB \equal{} 7,$ and $ PC \equal{} 6$. Legs $ \overline{AC}$ and $ \overline{BC}$ have length $ s \equal{} \sqrt {a \plus{} b\sqrt {2}}$, where $ a$ and $ b$ are positive integers. What is $ a \plus{} b$? [asy]pointpen = black; pathpen = linewidth(0.7); pen f = fontsize(10); size(5cm); pair B = (0,sqrt(85+42*sqrt(2))); pair A = (B.y,0); pair C = (0,0); pair P = IP(arc(B,7,180,360),arc(C,6,0,90)); D(A--B--C--cycle); D(P--A); D(P--B); D(P--C); MP("A",D(A),plain.E,f); MP("B",D(B),plain.N,f); MP("C",D(C),plain.SW,f); MP("P",D(P),plain.NE,f);[/asy] $ \textbf{(A) } 85 \qquad \textbf{(B) } 91 \qquad \textbf{(C) } 108 \qquad \textbf{(D) } 121 \qquad \textbf{(E) } 127$

Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy] size(200); defaultpen(fontsize(10)); real x=22.61986495; pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x); draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(13,13); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy]