Points $P,Q,R,S$ are taken on respective edges $AC$, $AB$, $BD$, and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$. The line $MN$ meets the plane $ABC$ at point $L$. Prove that the lines $AL$, $BP$, and $CQ$ are concurrent.

Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas? ${ \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D}}\ 840\qquad\textbf{(E)}\ 870 $

Let $H$ be the orthocentre and $G$ be the centroid of acute-angled triangle $ABC$ with $AB\ne AC$. The line $AG$ intersects the circumcircle of $ABC$ at $A$ and $P$. Let $P'$ be the reflection of $P$ in the line $BC$. Prove that $\angle CAB = 60$ if and only if $HG = GP'$

Let $ABC$ be an acute triangle with $AB<AC<BC$, inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Let $O_1$ be the symmetric point of $O$ wrt $AC$. Circle $c_1(O_1,R)$ intersects $BC$ at $Z$. If the extension of the altitude $AD$ intersects the cicrumscribed circle $c(O,R)$ at point $E$, prove that $EC$ is perpendicular on $AZ$.

The convex pentagon $ ABCDE $ has the following properties: $ \text{(i)} AB=BC $ $ \text{(ii)} \angle ABE+\angle CBD =\angle DBE $ $ \text{(iii)} \angle AEB +\angle BDC=180^{\circ} $ Prove that the orthocenter of $ BDE $ lies on $ AC. $

Find the point $ p $ in the first quadrant on the line $ y = 2x $ such that the distance between $ p $ and $ p' $, the point reflected across the line $ y = x $, is equal to $ \sqrt{32} $.

In the triangle $ABC$ we have $\angle A = 2\angle C$ and $2\angle B = \angle A + \angle C$. The angle bisector of $\angle C$ intersects the segment $AB$ in $E$, let $F$ be the midpoint of $AE$, let $AD$ be the altitude of the triangle $ABC$. The perpendicular bisector of $DF$ intersects $AC$ in $M$. Prove that $AM = CM$.

A circle and a point $P$ higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from $P$ to a point $Q$ on the circle. Given that the distance travelled from $P$ in time $t$ is equal to $\dfrac{1}{2}gt^2 \sin{\alpha}$, where $\alpha$ is the angle of inclination of the line $PQ$ to the horizontal, give a geometrical characterization of the point $Q$ for which the time taken from $P$ to $Q$ is a minimum.

Let $w$ be the circumcircle of non-isosceles acute triangle $ABC$. Tangent lines to $w$ in $A$ and $B$ intersect at point $S$. Let M be the midpoint of $AB$, and $H$ be the orthocenter of triangle $ABC$. The line $HA$ intersects lines $CM$ and $CS$ at points $M_a$ and $S_a$, respectively. The points $M_b$ and $S_b$ are defined analogously. Prove that $M_aS_b$ and $M_bS_a$ are the altitudes of triangle $M_aM_bH$.

The circle $ \omega $ inscribed in triangle $ABC$ touches its sides $AB, BC, CA$ at points $K, L, M$ respectively. In the arc $KL$ of the circle $ \omega $ that does not contain the point $M$, we select point $S$. Denote by $P, Q, R, T$ the intersection points of straight $AS$ and $KM, ML$ and $SC, LP$ and $KQ, AQ$ and $PC$ respectively. It turned out that the points $R, S$ and $M$ are collinear. Prove that the point $T$ also lies on the line $SM$.

Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that \[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\] where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.

Cut a square with three straight lines into three triangles and four quadrilaterals.

In the triangle $ABC$, $\angle A$ is biggest. On the circumcircle of $\triangle ABC$, let $D$ be the midpoint of $\widehat{ABC}$ and $E$ be the midpoint of $\widehat{ACB}$. The circle $c_1$ passes through $A,B$ and is tangent to $AC$ at $A$, the circle $c_2$ passes through $A,E$ and is tangent $AD$ at $A$. $c_1$ and $c_2$ intersect at $A$ and $P$. Prove that $AP$ bisects $\angle BAC$. [hide="Diagram"][asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(14.4cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -5.23, xmax = 9.18, ymin = -2.97, ymax = 4.82; /* image dimensions */ /* draw figures */ draw(circle((-1.32,1.36), 2.98)); draw(circle((3.56,1.53), 3.18)); draw((0.92,3.31)--(-2.72,-1.27)); draw(circle((0.08,0.25), 3.18)); draw((-2.72,-1.27)--(3.13,-0.65)); draw((3.13,-0.65)--(0.92,3.31)); draw((0.92,3.31)--(2.71,-1.54)); draw((-2.41,-1.74)--(0.92,3.31)); draw((0.92,3.31)--(1.05,-0.43)); /* dots and labels */ dot((-1.32,1.36),dotstyle); dot((0.92,3.31),dotstyle); label("$A$", (0.81,3.72), NE * labelscalefactor); label("$c_1$", (-2.81,3.53), NE * labelscalefactor); dot((3.56,1.53),dotstyle); label("$c_2$", (3.43,3.98), NE * labelscalefactor); dot((1.05,-0.43),dotstyle); label("$P$", (0.5,-0.43), NE * labelscalefactor); dot((-2.72,-1.27),dotstyle); label("$B$", (-3.02,-1.57), NE * labelscalefactor); dot((2.71,-1.54),dotstyle); label("$E$", (2.71,-1.86), NE * labelscalefactor); dot((3.13,-0.65),dotstyle); label("$C$", (3.39,-0.9), NE * labelscalefactor); dot((-2.41,-1.74),dotstyle); label("$D$", (-2.78,-2.07), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

Let $\{A_n | n = 1, 2, \cdots \} $ be a set of points in the plane such that for each $n$, the disk with center $A_n$ and radius $2^n$ contains no other point $A_j$ . For any given positive real numbers $a < b$ and $R$, show that there is a subset $G$ of the plane satisfying: [b](i)[/b] the area of $G$ is greater than or equal to $R$; [b](ii) [/b]for each point $P$ in $G$, $a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.$

Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively. a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$. b) Does the converse implication also always hold?

Let $ABC$ be acute-angled triangle . A circle $\omega_1(O_1,R_1)$ passes through points $B$ and $C$ and meets the sides $AB$ and $AC$ at points $D$ and $E$ ,respectively . Let $\omega_2(O_2,R_2)$ be the circumcircle of triangle $ADE$ . Prove that $O_1O_2$ is equal to the circumradius of triangle $ABC$ .

A hiking club wants to hike around a lake along an exactly circular route. On the shoreline they determine two points, which are the most distant from each other, and start to walk along the circle, which has these two points as the endpoints of its diameter. Can they be sure that, independent of the shape of the lake, they do not have to swim across the lake on any part of their route?

Let $P_1,P_2,P_3,P_4$ be the graphs of four quadratic polynomials drawn in the coordinate plane. Suppose that $P_1$ is tangent to $P_2$ at the point $q_2,P_2$ is tangent to $P_3$ at the point $q_3,P_3$ is tangent to $P_4$ at the point $q_4$, and $P_4$ is tangent to $P_1$ at the point $q_1$. Assume that all the points $q_1,q_2,q_3,q_4$ have distinct $x$-coordinates. Prove that $q_1,q_2,q_3,q_4$ lie on a graph of an at most quadratic polynomial.

The segments $BF$ and $CN$ are the altitudes in the acute-angled triangle $ABC$. The line $OI$, which connects the centers of the circumscribed and inscribed circles of triangle $ABC$, is parallel to the line $FN$. Find the length of the altitude $AK$ in the triangle $ABC$ if the radii of its circumscribed and inscribed circles are $R$ and $r$, respectively. (Grigory Filippovsky)

Let $ABC$ be a triangle, and let $M$ be a point on the side $(AC)$ .The line through $M$ and parallel to $BC$ crosses $AB$ at $N$. Segments $BM$ and $CN$ cross at $P$, and the circles $BNP$ and $CMP$ cross again at $Q$. Show that angles $BAP$ and $CAQ$ are equal.

In the triangle $ABC$ we have $\angle ABC=100^\circ$, $\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and $\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$. [i]St.Petersburg folklore[/i]

Let $ABC$ be a triangle. For a point $P$ in its interior, we draw the threee lines through $P$ parallel to the sides of the triangle. This partitions $ABC$ in three triangles and three quadrilaterals. Let $V_A$ be the area of the quadrilateral which has $A$ as one vertex. Let $D_A$ be the area of the triangle which has a part of $BC$ as one of its sides. Define $V_B, D_B$ and $V_C, D_C$ similarly. Determine all possible values of $\frac{D_A}{V_A}+\frac{D_B}{V_B}+\frac{D_C}{V_C}$, as $P$ varies in the interior of the triangle.

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$. *The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$. [i]Proposed by Zack Chroman[/i]

Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

In a matrix $2n \times 2n$, $n \in N$, are $4n^2$ real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than $1$. Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed $n$.