In $ ABC$ we have $ AB \equal{} 25$, $ BC \equal{} 39$, and $ AC \equal{} 42$. Points $ D$ and $ E$ are on $ AB$ and $ AC$ respectively, with $ AD \equal{} 19$ and $ AE \equal{} 14$. What is the ratio of the area of triangle $ ADE$ to the area of quadrilateral $ BCED$? $ \textbf{(A)}\ \frac{266}{1521}\qquad \textbf{(B)}\ \frac{19}{75}\qquad \textbf{(C)}\ \frac{1}{3}\qquad \textbf{(D)}\ \frac{19}{56}\qquad \textbf{(E)}\ 1$

$AD$, $BE$, $CF$ are the heights of the acute—angled triangle $ABC$. A perpendicular is drawn to the segment $DE$ at point $E$. It intersects the height of $AD$ at point $G$. The point $J$ is chosen on the segment $BD$ in such a way that $BJ = CD$. The circumscribed circle of a triangle $BD$ intersects the segment $BE$ at point $Q$. Prove that the points $J$, $Q$, and $G$ are collinear.

Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle. Let this new rectangle be given. Restore the original rectangle using compass and ruler.

Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let $ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively. Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles. For which position of $ P$ will the triangle $ DEF$ become equilateral?

Let $ABC$ be an equilateral triangle. For a point $M$ inside $\vartriangle ABC$, let $D,E,F$ be the feet of the perpendiculars from $M$ onto $BC,CA,AB$, respectively. Find the locus of all such points $M$ for which $\angle FDE$ is a right angle.

Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.

We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that [b]i)[/b] For any box, all pockets in this box must include a ball with the same color or [b]ii)[/b] For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this box Find the smallest value of $k$ for which we can always do this placement whatever the number of balls in the pockets and whatever the colors of balls.

The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.

Let $ABCDEF$ be a convex hexagon, such that $FA = AB$, $BC = CD$, $DE = EF$, and $\angle FAB = 2\angle EAC$. Suppose that the area of $ABC$ is $25$, the area of $CDE$ is $10$, the area of $EF A$ is $25$, and the area of $ACE$ is $x$. Find, with proof, all possible values of $x$.

Vertices $A$ and $C$ of the quadrilateral $ABCD$ are fixed points of the circle $k$ and each of the vertices $B$ and $D$ is moving to one of the arcs of $k$ with ends $A$ and $C$ in such a way that $BC=CD$. Let $M$ be the intersection point of $AC$ and $BD$ and $F$ is the center of the circumscribed circle around $\triangle ABM$. Prove that the locus of $F$ is an arc of a circle. [i]J. Tabov[/i]

Let $S = \{P_1,P_2,...,P_{2000}\}$ be a set of $2000$ points in the interior of a circle of radius $1$, one of which at its center. For $i = 1,2,...,2000$ denote by $x_i$ the distance from $P_i$ to the closest point $P_j \ne P_i$. Prove that $x_1^2 +x_2^2 +...+x_{2000}^2<9$ .

Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.

Let points $B$ and $C$ lie on the circle with diameter $AD$ and center $O$ on the same side of $AD$. The circumcircles of triangles $ABO$ and $CDO$ meet $BC$ at points $F$ and $E$ respectively. Prove that $R^2 = AF.DE$, where $R$ is the radius of the given circle. [i]Proposed by N.Moskvitin[/i]

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?

Let $a_1>a_2>a_3>\cdots$ be a sequence of real numbers which converges to 0. We put circles of radius $a_1$ into a unit square until no more can fit. (A previously laid circle must not be moved.) Then we put circles of radius $a_2$ in the remaining space until no more can fit, continuing the process for $a_3$,... What can the area covered by the circles be? a similar problem involving circles in a square: [url]https://artofproblemsolving.com/community/c7h1979044[/url]

Let $ABCDEF$ be a convex equilateral hexagon such that lines $BC$, $AD$, and $EF$ are parallel. Let $H$ be the orthocenter of triangle $ABD$. If the smallest interior angle of the hexagon is $4$ degrees, determine the smallest angle of the triangle $HAD$ in degrees.

Through the vertices $A, B$ of the parallelogram $ABCD$ passes a circle that intersects for the second time diagonals $BD$ and $AC$ at points $X$ and $Y$, respectively. The circumsccribed circle of $\vartriangle ADX$ intersects diagonal $AC$ for the second time at the point $Z$. Prove that $AY = CZ$.

Let $ABC$ be a triangle such that $AB > BC$ and let $D$ be a variable point on the line segment $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$, lying on the opposite side of $BC$ from $A$ such that $\angle BAE = \angle DAC$. Let $I$ be the incenter of triangle $ABD$ and let $J$ be the incenter of triangle $ACE$. Prove that the line $IJ$ passes through a fixed point, that is independent of $D$. [i]Proposed by Merlijn Staps[/i]

Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

Let $A_0,A_1, \ldots , A_n$ be points in a plane such that (i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and (ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$ where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$

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]

Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$

Let $C$ be a semicircle with diameter $AB$. Circles $S$, $S_1$, $S_2$ with radii $r$, $r_1$, $r_2$, respectively, are tangent to $C$ and the segment $AB$, and moreover $S_1$ and $S_2$ are externally tangent to $S$. Prove that $\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}=\frac{2\sqrt{2}}{\sqrt{r}}$