Prove that inside any acute-angled triangle, there exists a point $P$ such that the feet of the perpendiculars dropped from $P$ to the sides of the triangle are the vertices of an equilateral triangle. (NB Vassiliev)

Given is a triangle $\triangle{ABC}$,with $AB<AC<BC$,inscribed in circle $c(O,R)$.Let $D,E,Z$ be the midpoints of $BC,CA,AB$ respectively,and $K$ the foot of the altitude from $A$.At the exterior of $\triangle{ABC}$ and with the sides $AB,AC$ as diameters,we construct the semicircles $c_1,c_2$ respectively.Suppose that $P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1$ and $R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2$.Finally,let $M$ be the intersection of the lines $PS,RT$. [b]i.[/b] Prove that the lines $PR,ST$ intersect at $A$. [b]ii.[/b] Prove that the lines $PR\cap MD$ intersect on $c$. [asy]import graph; size(8cm); 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 = -4.8592569519241255, xmax = 12.331775417316715, ymin = -3.1864435704043403, ymax = 6.540061585876658; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((0.6699432366054657,3.2576036755978928)--(0.,0.)--(5.,0.)--cycle, aqaqaq); /* draw figures */ draw((0.6699432366054657,3.2576036755978928)--(0.,0.), uququq); draw((0.,0.)--(5.,0.), uququq); draw((5.,0.)--(0.6699432366054657,3.2576036755978928), uququq); draw(shift((0.33497161830273287,1.6288018377989464))*xscale(1.662889476749906)*yscale(1.662889476749906)*arc((0,0),1,78.3788505217281,258.3788505217281)); draw(shift((2.834971618302733,1.6288018377989464))*xscale(2.7093067970187343)*yscale(2.7093067970187343)*arc((0,0),1,-36.95500560847834,143.0449943915217)); draw((0.6699432366054657,3.2576036755978928)--(0.6699432366054657,0.)); draw((-0.9938564482532047,2.628510486065423)--(2.5,0.)); draw((0.6699432366054657,0.)--(0.,3.2576036755978923)); draw((0.6699432366054657,0.)--(5.,3.257603675597893)); draw((2.5,0.)--(3.3807330143335355,4.282570444700163)); draw((-0.9938564482532047,2.628510486065423)--(2.5,4.8400585427926455)); draw((2.5,4.8400585427926455)--(5.,3.257603675597893)); draw((-0.9938564482532047,2.628510486065423)--(3.3807330143335355,4.282570444700163), linewidth(1.2) + linetype("2 2")); draw((0.,3.2576036755978923)--(5.,3.257603675597893), linewidth(1.2) + linetype("2 2")); draw(circle((2.5,1.18355242571055), 2.766007292905304), linewidth(0.4) + linetype("2 2")); draw((2.5,4.8400585427926455)--(2.5,0.), linewidth(1.2) + linetype("2 2")); /* dots and labels */ dot((0.6699432366054657,3.2576036755978928),linewidth(3.pt) + dotstyle); label("$A$", (0.7472169504504719,2.65), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.2,-0.4), NE * labelscalefactor); dot((5.,0.),linewidth(3.pt) + dotstyle); label("$C$", (5.028818057451246,-0.34281415594345044), NE * labelscalefactor); dot((2.5,0.),linewidth(3.pt) + dotstyle); label("$D$", (2.4275434226319077,-0.32665717063401356), NE * labelscalefactor); dot((2.834971618302733,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$E$", (3.073822835009383,1.5637101105701008), NE * labelscalefactor); dot((0.33497161830273287,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$Z$", (0.003995626216375389,1.402140257475732), NE * labelscalefactor); dot((0.6699432366054657,0.),linewidth(3.pt) + dotstyle); label("$K$", (0.6179610679749769,-0.3105001853245767), NE * labelscalefactor); dot((-0.9938564482532047,2.628510486065423),linewidth(3.pt) + dotstyle); label("$P$", (-1.0785223895158957,2.7916409940873033), NE * labelscalefactor); dot((0.,3.2576036755978923),linewidth(3.pt) + dotstyle); label("$S$", (-0.14141724156855653,3.454077391774215), NE * labelscalefactor); dot((5.,3.257603675597893),linewidth(3.pt) + dotstyle); label("$T$", (5.061132028070119,3.3571354799175936), NE * labelscalefactor); dot((3.3807330143335355,4.282570444700163),linewidth(3.pt) + dotstyle); label("$R$", (3.445433497126431,4.375025554412117), NE * labelscalefactor); dot((2.5,4.8400585427926455),linewidth(3.pt) + dotstyle); label("$M$", (2.5567993051074027,4.940520040242407), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

Let $ ABC$ be a triangle with $ G$ as its centroid. Let $ P$ be a variable point on segment $ BC$. Points $ Q$ and $ R$ lie on sides $ AC$ and $ AB$ respectively, such that $ PQ \parallel AB$ and $ PR \parallel AC$. Prove that, as $ P$ varies along segment $ BC$, the circumcircle of triangle $ AQR$ passes through a fixed point $ X$ such that $ \angle BAG = \angle CAX$.

In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$

A circle $\omega$ with center $I$ is located inside the circle $\Omega$ with center $O$. Ray $IO$ intersects $\omega$ and $\Omega$ at $P_1$ and $P_2$ respectively. On $\Omega$ an arbitrary point $A \neq P_2$ is chosen. The circumcircle of the triangle $P_1P_2A$ intersects $\omega$ for the second time at $X$. Line $AX$ intersects $\Omega$ for the second time at $Y$. Prove that lines $XP_1$ and $YP_2$ are perpendicular to each other

$P$ 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 $O$ is circumcenter of triangle $ABC$. Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$. Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is \[\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}\]

In $\vartriangle ABC$, $AB = AC$ and $\angle BAC = 20^o$. A point $D$ lies on the side $AB$ and $AD = BC$. Find $\angle BCD$. (LF. Sharygin, Moscow)

Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$.

The circle is inscribed in $\vartriangle ABC$. Let $L, M, N$ be the tangent points of the circle with sides $AB, AC, BC$, respectively. Prove that $\angle MLN$ is always an acute angle.

Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.

A teacher gives each student in the class the following task in their exercise book . "Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$." Prove that each student would obtain the same result . [img]https://1.bp.blogspot.com/-K3kHWWWgxgU/XWHRQ8WqqPI/AAAAAAAAKjE/0iO4-Yz6p9AcM2mklprX_M18xTyg9O81gCK4BGAYYCw/s200/TOT%2B1985%2BAutumn%2BJ3.png[/img] ( A . K . Tolpygo, Kiev)

Can the number obtained by writing the numbers from 1 to $n$ in order ($n > 1$) be the same when read left-to-right and right-to-left? [i]N. Agakhanov[/i]

Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$. $A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started. If $A$ starts the game, determine which player has a winning strategy.

Given a circle of radius 2, there are many line segments of length 2 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. $\text{(A)}\ \frac \pi 4 \qquad \text{(B)}\ 4 - \pi \qquad \text{(C)}\ \frac \pi 2 \qquad \text{(D)}\ \pi \qquad \text{(E)}\ 2\pi$

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$

In a right-angled triangle $ABC$ ($\angle A = 90^o$), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of angle $C$, find all possible values for angles $B$ and $C$. by Mahdi Etesami Fard

Let $A,B,C$ be points on $[OX$ and $D,E,F$ be points on $[OY$ such that $|OA|=|AB|=|BC|$ and $|OD|=|DE|=|EF|$. If $|OA|>|OD|$, which one below is true? $\textbf{(A)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)>\text{Area}(DBF)$ $\textbf{(B)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)=\text{Area}(DBF)$ $\textbf{(C)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)<\text{Area}(DBF)$ $\textbf{(D)}$ If $m(\widehat{XOY})<45^\circ$ then $\text{Area}(AEC)<\text{Area}(DBF)$, and if $45^\circ < m(\widehat{XOY})<90^\circ$ then $\text{Area}(AEC)>\text{Area}(DBF)$ $\textbf{(E)}$ None of above

Prove that the bases of the altitudes and medians of an acute-angled triangle lie on the same circle.

Let $ABC$ be a triangle, and let $a$, $b$, and $c$ be the lengths of the sides opposite vertices $A$, $B$, and $C$, respectively. Let $R$ be its circumradius and $r$ its inradius. Suppose that $b + c = 2a$ and $R = 3r$. The excircle relative to vertex $A$ intersects the circumcircle of $ABC$ at points $P$ and $Q$. Let $U$ be the midpoint of side $BC$, and let $I$ be the incenter of $ABC$. Prove that $U$ is the centroid of triangle $QIP$.

The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \frac{xy}{2}$, show that $\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right)$ or $\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right)$.

Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?

Find the smallest positive real number $r$ with the following property: For every choice of $2023$ unit vectors $v_1,v_2, \dots ,v_{2023} \in \mathbb{R}^2$, a point $p$ can be found in the plane such that for each subset $S$ of $\{1,2, \dots , 2023\}$, the sum $$\sum_{i \in S} v_i$$ lies inside the disc $\{x \in \mathbb{R}^2 : ||x-p|| \leq r\}$.

Two congruent equilateral triangles $\triangle ABC$ and $\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\overline{AB}$, $\overline{AC}$, $\overline{DE}$, and $\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\tfrac{AW}{BW} = \tfrac29$ , $\tfrac{AX}{CX} = \tfrac56$ , and $\tfrac{DY}{EY} = \tfrac92$. The ratio $\tfrac{EZ}{FZ}$ can then be written as $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(200); defaultpen(linewidth(0.6)); real r = 3/11, s = 0.52, l = 33, d=5.5; pair A = (l/2,l*sqrt(3)/2), B = origin, C = (l,0), D = (3*l/2+d,l*sqrt(3)/2), E = (l+d,0), F = (2*l+d,0); pair W = r*B+(1-r)*A, X = s*C+(1-s)*A, Y = extension(W,X,D,E), Z = extension(W,X,E,F); draw(E--D--F--B--A--C^^W--Z); dot("$A$",A,N); dot("$B$",B,S); dot("$C$",C,S); dot("$D$",D,N); dot("$E$",E,S); dot("$F$",F,S); dot("$W$",W,0.6*NW); dot("$X$",X,0.8*NE); dot("$Y$",Y,dir(100)); dot("$Z$",Z,dir(70)); [/asy]

Given an octagon with the equal angles. The lengths of all the sides are integers. Prove that the opposite sides are equal in pairs. [u]alternate wording[/u] Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.