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?

Let $A_{1}A_{2}\ldots A_{2n}$ be a convex polygon and let $P$ be a point in its interior such that it doesn't lie on any of the diagonals of the polygon. Prove that there is a side of the polygon such that none of the lines $PA_{1}$, $\ldots$, $PA_{2n}$ intersects it in its interior.

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]

We call a pair of polygons, $p$ and $q$, [i]nesting[/i] if we can draw one inside the other, possibly after rotation and/or reflection; otherwise we call them [i]non-nesting[/i]. Let $p$ and $q$ be polygons. Prove that if we can find a polygon $r$, which is similar to $q$, such that $r$ and $p$ are non-nesting if and only if $p$ and $q$ are not similar.

Let $ ABCD$ be a parallelogram with no angle equal to $ 60^{\textrm{o}}$. Find all pairs of points $ E, F$, in the plane of $ ABCD$, such that triangles $ AEB$ and $ BFC$ are isosceles, of basis $ AB$, respectively $ BC$, and triangle $ DEF$ is equilateral. [i]Valentin Vornicu[/i]

Let $ ABC$ be a triangle where $ \angle ACB\equal{}90^{\circ}$. Let $ D$ be the midpoint of $ BC$ and let $ E$, and $ F$ be points on $ AC$ such that $ CF\equal{}FE\equal{}EA$. The altitude from $ C$ to the hypotenuse $ AB$ is $ CG$, and the circumcentre of triangle $ AEG$ is $ H$. Prove that the triangles $ ABC$ and $ HDF$ are similar.

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.

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? $ \textbf{(A)}\ 60 \qquad\textbf{(B)}\ 170 \qquad\textbf{(C)}\ 290 \qquad\textbf{(D)}\ 320 \qquad\textbf{(E)}\ 660 $

A square $ R$ of sidelength $ 250$ lies inside a square $ Q$ of sidelength $ 500$. Prove that: One can always find two points $ A$ and $ B$ on the perimeter of $ Q$ such that the segment $ AB$ has no common point with the square $ R$, and the length of this segment $ AB$ is greater than $ 521$.

Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image? [asy] import cse5;pathpen=black;pointpen=black; size(2cm); D((0,-2)--MP("y",(0,7),N)); D((-3,0)--MP("x",(5,0),E)); D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5)); [/asy][asy] import cse5;pathpen=black;pointpen=black; unitsize(0.2cm); D((0,-2)--MP("y",(0,7),N)); D(MP("\textbf{(A) }",(-3,0),W)--MP("x",(5,0),E)); D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5)); // D((18,-2)--MP("y",(18,7),N)); D(MP("\textbf{(B) }",(13,0),W)--MP("x",(21,0),E)); D((17,0)--(17,2)--(16,2)--(16,3)--(17,3)--(17,4)--(15,4)--(15,5)--(18,5)); // D((36,-2)--MP("y",(36,7),N)); D(MP("\textbf{(C) }",(29,0),W)--MP("x",(38,0),E)); D((31,0)--(31,1)--(33,1)--(33,2)--(34,2)--(34,1)--(35,1)--(35,3)--(36,3)); // D((0,-17)--MP("y",(0,-8),N)); D(MP("\textbf{(D) }",(-3,-15),W)--MP("x",(5,-15),E)); D((3,-15)--(3,-14)--(1,-14)--(1,-13)--(2,-13)--(2,-12)--(1,-12)--(1,-10)--(0,-10)); // D((15,-17)--MP("y",(15,-8),N)); D(MP("\textbf{(E) }",(13,-15),W)--MP("x",(22,-15),E)); D((15,-14)--(17,-14)--(17,-13)--(18,-13)--(18,-14)--(19,-14)--(19,-12)--(20,-12)--(20,-15)); [/asy]

A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black? $ \textbf{(A)}\ \dfrac{49}{512} \qquad\textbf{(B)}\ \dfrac{7}{64} \qquad\textbf{(C)}\ \dfrac{121}{1024} \qquad\textbf{(D)}\ \dfrac{81}{512} \qquad\textbf{(E)}\ \dfrac{9}{32} $

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

An integer sequence $\{a_{n}\}_{n \ge 1}$ is given such that \[2^{n}=\sum^{}_{d \vert n}a_{d}\] for all $n \in \mathbb{N}$. Show that $a_{n}$ is divisible by $n$ for all $n \in \mathbb{N}$.

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 numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is [i]bad[/i] if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? $ \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $

A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called [i]symmetric[/i] if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes? $\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$? $\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$

Let $ABC$ be a triangle with $AB\neq AC$, and let $A_{1}B_{1}C_{1}$ be the image of triangle $ABC$ through a rotation $R$ centered at $C$. Let $M,E , F$ be the midpoints of the segments $BA_{1}, AC, BC_{1}$ respectively Given that $EM = FM$, compute $\angle EMF$.

A country has $n$ cities, labelled $1,2,3,\dots,n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly $1$, and it is also not permitted to put a road between cities $1$ and $n$. Let $T_n$ be the total number of possible ways to build these roads. (a) For all odd $n$, prove that $T_n$ is divisible by $n$. (b) For all even $n$, prove that $T_n$ is divisible by $n/2$.

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.

The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?

Define $\mathcal{A}=\{(x,y)|x=u+v,y=v, u^2+v^2\le 1\}$. Find the length of the longest segment that is contained in $\mathcal{A}$.

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$.