City of Mar del Plata is a square shaped $WSEN$ land with $2(n + 1)$ streets that divides it into $n \times n$ blocks, where $n$ is an even number (the leading streets form the perimeter of the square). Each block has a dimension of $100 \times 100$ meters. All streets in Mar del Plata are one-way. The streets which are parallel and adjacent to each other are directed in opposite direction. Street $WS$ is driven in the direction from $W$ to $S$ and the street $WN$ travels from $W$ to $N$. A street cleaning car starts from point $W$. The driver wants to go to the point $E$ and in doing so, he must cross as much as possible roads. What is the length of the longest route he can go, if any $100$-meter stretch cannot be crossed more than once? (The figure shows a plan of the city for $n=6$ and one of the possible - but not the longest - routes of the street cleaning car. See http://goo.gl/maps/JAzD too.) [img]http://s14.postimg.org/avfg7ygb5/CPS_2012_P5.jpg[/img]

line $l$ through the point $A$ from triangle $ABC$ . Point $X$ is on line $l$.${\omega}_b$ and ${\omega}_c$ are circles that through points $X,A$ and respectively tanget to $AB$ adn $AC$. tangets from $B,C$ respectively to ${\omega}_b$ and ${\omega}_c$ meet them in $Y,Z$. Prove that by changing $X$, the circumcircle of the circle $XYZ$ passes through two fixed points. [i]Proposed by Ali Zamani [/i]

A tetrahedron $ABCD$ satisfies $AB=6$, $CD=8$, and $BC=DA=5$. Let $V$ be the maximum value of $ABCD$ possible. If we can write $V^4=2^n3^m$ for some integers $m$ and $n$, find $mn$.

The points $A$, $B$, $C$, $D$, and $E$ lie in one plane and have the following properties: $AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$. Find the length of the segment $ED$.

Let $ABC$ be a triangle such that the coordinates of the points $A$ and $B$ are rational numbers. Prove that the coordinates of $C$ are rational if, and only if, $\tan A$, $\tan B$, and $\tan C$, when defined, are all rational numbers.

Let $A_1A_2 \dots A_{13}$ and $B_1B_2 \dots B_{13}$ be two regular $13$-gons in the plane such that the points $B_1$ and $A_{13}$ coincide and lie on the segment $A_1B_{13}$, and both polygons lie in the same semiplane with respect to this segment. Prove that the lines $A_1A_9, B_{13}B_8$ and $A_8B_9$ are concurrent.

In triangle $ABC$, let $D$ be the midpoint of $BC$, and $BE$, $CF$ are the altitudes. Prove that $DE$ and $DF$ are both tangents to the circumcircle of triangle $AEF$

Consider a container of a hollow cube $ABGCDEPF$ (where $ABGC$, $DEPF$ are squares and $AD\parallel BE\parallel GP\parallel CF$). The cube is placed on a table in a way that the space diagonal $AP=1$ is perpendicular to the table. Then, water is poured into the cube. Denote $x$ the length of part of $AP$ submerged in water. Determine the volume of water $y$ in terms of $x$ when a) $0 < x \leq\frac13$, b) $\frac13 < x \leq\frac12$.

$O$ is the point of intersection of the diagonals of the convex quadrilateral $ABCD$. It is known that the areas of triangles $AOB, BOC, COD$ and $DOA$ are expressed in natural numbers. Prove that the product of these areas cannot end in $1985$.

Given is the triangle $ABC$ such that $BC=13, CA=14, AB=15$ Prove that $B$, the incenter $J$ and the midpoints of $AB$ and $BC$ all lie on a circle

A line tangent to the incircle of the equilateral triangle ABC intersects the sides AB and BC at points D and E respectively. Prove that $\frac{AD}{DB}+\frac{AE}{EC} = 1$.

Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

Consider an equilateral triangle with side length $9$. Each side is divided into $3$ equal segments by $2$ points, for a total of $6$ points. Compute the area of the circle passing through these$ 6$ points. [img]https://cdn.artofproblemsolving.com/attachments/7/b/1860a3ff86a0e4b93a4891861300dcb09adad4.png[/img]

Let $ABCD$ be a square such that $A=(0,0)$ and $B=(1,1)$. $P(\frac{2}{7},\frac{1}{4})$ is a point inside the square. An ant starts walking from $P$, touches $3$ sides of the square and comes back to the point $P$. The least possible distance traveled by the ant can be expressed as $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers and $a$ not divisible by any square number other than $1$. What is the value of $(a+b)$?

In right $ \triangle ABC$ with legs $ 5$ and $ 12$, arcs of circles are drawn, one with center $ A$ and radius $ 12$, the other with center $ B$ and radius $ 5$. They intersect the hypotenuse at $ M$ and $ N$. Then, $ MN$ has length: [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A); real r=degrees(B); draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270)); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N)); label("$12$", (6,0), S); label("$5$", (12,3.5), E);[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac {13}{5} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \frac {24}{5}$

Five points are given in the plane that each of $10$ triangles they define has area greater than $2$. Prove that there exists a triangle of area greater than $3$.

Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.

A $5\times5$ grid of squares is filled with integers. Call a rectangle [i]corner-odd[/i] if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid? Note: A rectangles must have four distinct corners to be considered [i]corner-odd[/i]; i.e. no $1\times k$ rectangle can be [i]corner-odd[/i] for any positive integer $k$.

Let $ A_1,B_1,C_1 $ be points on the sides (excluding their endpoints) $ BC,CA,AB, $ respectively, of a triangle $ ABC, $ such that $ \angle A_1AB =\angle B_1BC=\angle C_1CA. $ Let $ A^* $ be the intersection of $ BB_1 $ with $ CC_1,B^* $ be the intersection of $ CC_1 $ with $ AA_1, $ and $ C^* $ be the intersection of $ AA_1 $ with $ BB_1. $ Denote with $ r_A,r_B,r_C $ the inradii of $ A^*BC,AB^*C,ABC^*, $ respectively. Prove that $$ \frac{r_A}{BC}=\frac{r_B}{CA}=\frac{r_C}{AB} $$ if and only if $ ABC $ is equilateral. [i]Daniel Văcărețu[/i]

Let $\triangle ABC$ have side lengths $AB = 5, BC = 9,$ and $CA = 10.$ The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ intersect at point $D,$ and $\overline{AD}$ intersects the circumcircle at $P \ne A.$ The length of $\overline{AP}$ is equal to $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

In $\triangle ABC$, the corresponding sides of angle $A,B,C$ are $a,b,c$ respectively. If $a\cos B-b\cos A=\frac{3}{5}c$, find the value of $\frac{\tan A}{\tan B}$.

Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that \[\frac{TP}{TQ}\ge \frac{b}{a}.\] [i]Proposed by Morteza Saghafian[/i]

Triangle $ABC$ has a right angle at $B$. The perpendicular bisector of $\overline{AC}$ meets segment $\overline{BC}$ at $D$, while the perpendicular bisector of segment $\overline{AD}$ meets $\overline{AB}$ at $E$. Suppose $CE$ bisects acute $\angle ACB$. What is the measure of angle $ACB$?

$A_1B_1A_2$, $B_1A_2B_2$, $A_2B_2A_3$,...,$B_{13}A_{14}B_{14}$, $A_{14}B_{14}A_1$ and $B_{14}A_1B_1$ are equilateral rigid plates that can be folded along the edges $A_1B_1$,$B_1A_2$, ..., $A_{14}B_{14}$ and $B_{14}A_1$ respectively. Can they be folded so that all $28$ plates lie in the same plane?

Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.