[u]Down the Infinite Corridor[/u] Consider an isosceles triangle $T$ with base $10$ and height $12$. Define a sequence $\omega_1$, $\omega_2$,$...$of circles such that $\omega_1$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_i$ and both legs of the isosceles triangle for $i > 1$. [b]p1.[/b] Find the radius of $\omega_1$. [b]p2.[/b] Find the ratio of the radius of $\omega_{i+1}$ to the radius of $\omega_i$. [b]p3.[/b] Find the total area contained in all the circles.

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal area and equal perimeter.

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

Let $n$ be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves: (1) select a row and multiply each entry in this row by $n$; (2) select a column and subtract $n$ from each entry in this column. Find all possible values of $n$ for which the following statement is true: Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is $0$.

Let $ABC$ be a triangle such that $\angle A=30^\circ$ and $\angle B=80^\circ$. Let $D$ and $E$ be points on sides $AC$ and $BC$ respectively so that $\angle ABD=\angle DBC$ and $DE\parallel AB$. Determine the measure of $\angle EAC$.

Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

Let $ABC$ be a triangle, and let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to sides $AC$ and $AB$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with the tangents from $B$ and $C$ to $(ABC)$, respectively. If $M$ is the midpoint of $BC$, prove that $(PQM)$ is tangent to $BC$ at $M$.

In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. [i]Proposed by Michael Tang[/i]

A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.

Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute scalene triangle $ABC$. The perpendicular bisector of $\overline{AH}$ intersects $\overline{AB}$ and $\overline{AC}$ at $X_A$ and $Y_A$ respectively. Let $K_A$ denote the intersection of the circumcircles of triangles $OX_AY_A$ and $BOC$ other than $O$. Define $K_B$ and $K_C$ analogously by repeating this construction two more times. Prove that $K_A$, $K_B$, $K_C$, and $O$ are concyclic. [i]Hongzhou Lin[/i]

In triangle $ABC$, $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$. Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$. Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$. Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$. Prove that $MK,EF$ and $HL$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$. Determine the angles in the triangle for which the angle opposite to the side with the length $a$ is as big as possible.

Let $ABCD$ be a rectangle. Determine the set of all points $P$ from the region between the parallel lines $AB$ and $CD$ such that $\angle APB=\angle CPD$.

Two triangles $ABC, A’B’C’$ have the same orthocenter $H$ and the same circumcircle with center $O$. Letting $PQR$ be the triangle formed by $AA’, BB’, CC’$, prove that the circumcenter of $PQR$ lies on $OH$.

Exists a triangle whose three altitudes have lengths $1, 2$ and $3$ respectively?

Let $\Delta_0$ be an equilateral triangle with incircle $\omega$. A point on $\omega$ is reflected in the sides of $\Delta_0$ to obtain a new triangle $\Delta_1$. The same point is then reflected over the sides of $\Delta_1$ to obtain another triangle $\Delta_2$. Prove that the circumcircle of $\Delta_2$ is tangent to $\omega$. [i]Proposed by Siddharth Choppara[/i]

$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals: $\textbf{(A) }2\sqrt{3}\qquad\textbf{(B) }3\sqrt{2}\qquad\textbf{(C) }3\sqrt{3}\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }2$ [asy] draw((0,0)--(6.5,0)--(6.5,4.5)--(0,4.5)--cycle); draw((2.5,1.5)--(0,0)); draw((2.5,1.5)--(0,4.5)); draw((2.5,1.5)--(6.5,4.5)); draw((2.5,1.5)--(6.5,0),linetype("8 8")); label("$A$",(0,0),dir(-135)); label("$B$",(6.5,0),dir(-45)); label("$C$",(6.5,4.5),dir(45)); label("$D$",(0,4.5),dir(135)); label("$P$",(2.5,1.5),dir(-90)); label("$3$",(1.25,0.75),dir(120)); label("$4$",(1.25,3),dir(35)); label("$5$",(4.5,3),dir(120)); //Credit to bobthesmartypants for the diagram [/asy]

Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$. Suppose that $AM$ intersects the incircle at $K,L$. We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$. Prove that $BP=CQ$.

Each vertex of a convex hexagon is the center of a circle whose radius is equal to the shorter side of the hexagon that contains the vertex. Show that if the intersection of all six circles (including their boundaries) is not empty, then the hexagon is regular.

Let $ABC$ be a triangle inscribed in circle $C$. Circles $C_1, C_2, C_3$ are tangent internally with circle $C$ in $A_1, B_1, C_1$ and tangent to sides $[BC], [CA], [AB]$ in points $A_2, B_2, C_2$ respectively, so that $A, A_1$ are on one side of $BC$ and so on. Lines $A_1A_2, B_1B_2$ and $C_1C_2$ intersect the circle $C$ for second time at points $A’,B’$ and $C’$, respectively. If $ M = BB’ \cap CC’$, prove that $m (\angle MAA’) = 90^\circ$ .

A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$, $BE=4$, $AD=5$ and $AC=2\sqrt{7} $. Find the angle $ \angle BDC$.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Let $ABCD$ be circumscribed quadrilateral such that the midpoints of $AB$,$BC$,$CD$ and $DA$ are $K$, $L$, $M$, $N$ respectively. Let the reflections of the point $M$ wrt the lines $AD$ and $BC$ be $P$ and $Q$ respectively. Let the circumcenter of the triangle $KPQ$ be $R$. Prove that $RN=RL$

Let $ABCDE$ be a pentagon with right angles at vertices $B$ and $E$ and such that $AB = AE$ and $BC = CD = DE$. The diagonals $BD$ and $CE$ meet at point $F$. Prove that $FA = AB$.