suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

A finite set $S{}$ of $n{}$ points is given in the plane. No three points lie on the same line. The number of non-self-intersecting closed $n{}$-link polylines with vertices at these points will be denoted by $f(S).$ Prove that [list=a] [*]$f(S)>0$ for all sets $S{};$ [*]$f(S)=1$ if and only if all the points of $S{}$ lie on the convex hull of $S{};$ [*]if $f(S)>1$ then $f(S)\geqslant n-1$, with equality if and only if one point of $S$ lies inside the convex hull; [*]if exactly two points of $S{}$ lie inside the convex hull, then\[f(S)\geqslant\frac{(n-2)(n-3)}{2}.\] [/list]Let $n\geqslant 3.$ Denote by $F(n)$ the largest possible value of the function $f(S)$ over all admissible sets $S{}$ of $n{}$ points. Prove that \[F(n)\geqslant3\cdot 2^{(n-8)/3}.\][i]Proposed by E. Bakaev and D. Magzhanov[/i]

Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.

Let $ABCDE$ be pentagon such that $AE = ED$ and $BC = CD$. It is known that $\angle BAE + \angle EDC + \angle CB A = 360^o$ and that $P$ is the midpoint of $AB$. Show that the triangle $ECP$ is right.

Let $ABC$ be a triangle and let $\Gamma$ be its circumcircle. Let $D$ be a point on $AB$ such that $CD$ is parallel to the line tangent to $\Gamma$ at $A$. Let $E$ be the intersection of $CD$ with $\Gamma$ distinct from $C$, and $F$ the intersection of $BC$ with the circumcircle of $\bigtriangleup ADC$ distinct from $C$. Finally, let $G$ be the intersection of the line $AB$ and the internal bisector of $\angle DCF$. Show that $E,\ G,\ F$ and $C$ lie on the same circle.

Given a segment $AB$ in the plane. Let $C$ be another point in the same plane,$H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$. Find the locus of $M$ for which $A,B,H,I$ are concyclic.

A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\tfrac{AM}{MP} = \tfrac{m}{n}$ for relatively prime positive integers $m, n$, compute $100m+n$. [i]Proposed by Michael Tang[/i]

Square $ABCD$ has side length $36$. Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$. Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$.

In a triangle $ABC$ with $AC = b \ne BC = a$, points $E,F$ are taken on the sides $AC,BC$ respectively such that $AE = BF =\frac{ab}{a+b}$. Let $M$ and $N$ be the midpoints of $AB$ and $EF$ respectively, and $P$ be the intersection point of the segment $EF$ with the bisector of $\angle ACB$. Find the ratio of the area of $CPMN$ to that of $ABC$.

In an acute triangle $ABC$ let $K,L,$ and $M$ be the midpoints of sides $AB,BC,$ and $CA,$ respectively. From each of $K,L,$ and $M$ drop two perpendiculars to the other two sides of the triangle; e.g., drop perpendiculars from $K$ to sides $BC$ and $CA,$ etc. The resulting $6$ perpendiculars intersect at points $Q,S,$ and $T$ as in the figure to form a hexagon $KQLSMT$ inside triangle $ABC.$ Prove that the area of this hexagon $KQLSMT$ is half of the area of the original triangle $ABC.$ [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra; diagram by adihaya*/ import graph; size(10cm); 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 = 11.888712276357234, xmax = 17.841346447833423, ymin = 10.61620970860601, ymax = 15.470685507068502; /* image dimensions */ pen zzttqq = rgb(0.6,0.2,0.); pen qqwuqq = rgb(0.,0.39215686274509803,0.); pair A = (12.488234161849352,12.833838721895551), B = (16.50823416184936,15.093838721895553), C = (16.28823416184936,11.353838721895551), K = (14.498234161849355,13.963838721895552), L = (16.39823416184936,13.223838721895552), M = (14.388234161849356,12.093838721895551), D = (13.615830174638527,13.467760858438725), F = (15.75135711740064,11.562938202365055), G = (15.625830174638523,14.597760858438724), H = (16.435061748056253,13.849907687412797), T = (14.02296781802369,12.74356027153236), Q = (16.032967818023693,13.873560271532357), O = (16.325061748056253,11.979907687412794); draw(A--B--C--cycle, zzttqq); draw((13.426050287639166,13.361068683160477)--(13.532742462917415,13.171288796161116)--(13.722522349916774,13.277980971439364)--D--cycle, qqwuqq); draw((14.054227993863618,12.223925334689998)--(14.133240861538676,12.426796211152979)--(13.930369985075695,12.505809078828037)--(13.851357117400637,12.302938202365056)--cycle, qqwuqq); draw((16.337846386707046,12.19724654447628)--(16.12050752964356,12.210031183127075)--(16.107722890992765,11.992692326063588)--O--cycle, qqwuqq); draw((15.830369985075697,11.765809078828037)--(15.627499108612716,11.844821946503092)--(15.54848624093766,11.641951070040111)--F--cycle, qqwuqq); draw((15.436050287639164,14.491068683160476)--(15.542742462917412,14.301288796161115)--(15.73252234991677,14.407980971439365)--G--cycle, qqwuqq); draw((16.217722890992764,13.86269232606359)--(16.20493825234197,13.645353469000101)--(16.42227710940546,13.63256883034931)--H--cycle, qqwuqq); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true); yaxis(ymin, ymax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw(A--B, zzttqq); draw(B--C, zzttqq); draw(C--A, zzttqq); draw(M--D); draw(K--(13.851357117400637,12.302938202365056)); draw(F--L); draw(L--G); draw(K--H); draw(M--O); /* dots and labels */ dot(A,dotstyle); label("$A$", (12.52502834296331,12.93568440300881), NE * labelscalefactor); dot(B,dotstyle); label("$B$", (16.548187989892043,15.193580123223922), NE * labelscalefactor); dot(C,dotstyle); label("$C$", (16.332661580235147,11.457789022504372), NE * labelscalefactor); dot(K,linewidth(3.pt) + dotstyle); label("$K$", (14.536608166427676,14.02357961365791), NE * labelscalefactor); dot(L,linewidth(3.pt) + dotstyle); label("$L$", (16.43529320388129,13.28463192340569), NE * labelscalefactor); dot(M,linewidth(3.pt) + dotstyle); label("$M$", (14.433976542781535,12.155684063298134), NE * labelscalefactor); dot(D,linewidth(3.pt) + dotstyle); dot((13.851357117400637,12.302938202365056),linewidth(3.pt) + dotstyle); dot(F,linewidth(3.pt) + dotstyle); dot(G,linewidth(3.pt) + dotstyle); dot(H,linewidth(3.pt) + dotstyle); dot((15.922967818023695,12.003560271532354),linewidth(3.pt) + dotstyle); label("$S$", (15.96318773510904,12.063315602016607), NE * labelscalefactor); dot(T,linewidth(3.pt) + dotstyle); label("$T$", (14.064502697655428,12.802263292268826), NE * labelscalefactor); dot(Q,linewidth(3.pt) + dotstyle); label("$Q$", (16.076082521119794,13.931211152376383), NE * labelscalefactor); dot(O,linewidth(3.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

Let $ABC$ be a triangle such that $|AB|=7$, $|BC|=8$, $|AC|=6$. Let $D$ be the midpoint of side $[BC]$. If the circle through $A$, $B$ and $D$ cuts $AC$ at $A$ and $E$, what is $|AE|$? $ \textbf{(A)}\ \dfrac 23 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \dfrac 32 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 3 $

The points $A', B', C', D'$ are selected respectively on the sides $AB, BC, CD, DA$ of the cyclic quadrilateral $ABCD$. It is known that $AA' = BB' = CC' = DD'$ and \[\angle AA'D' =\angle BB'A' =\angle CC'B' =\angle DD'C'.\]Prove that $ABCD$ is a square.

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects the segments $AB$ and $AC$ at the interior points $D$ and $E$, respectively. The lines $BE$ and $CD$ intersects at $F$. Let $G$ be a point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$ and let $H$ be a point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T\neq A$, independent of the choice of $\omega$, such that the circumcircle of triangle $AGH$ passes through $T$.

$ABCD$ is convex quadrilateral. If $W_a$ is product of power of $A$ about circle $BCD$ and area of triangle $BCD$. And define $W_b,W_c,W_d$ similarly.prove $W_a+W_b+W_c+W_d=0$

In a circle $O$, there are six points, $ A$, $ B$, $C$, $D$, $E$, $F$ in a counterclockwise order such that $BD \perp CF$ , and $CF$, $BE$, $AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE \parallel MN$.

Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$.

If the base of a rectangle is increased by $ 10\%$ and the area is unchanged, then the altitude is decreased by: $ \textbf{(A)}\ 9\% \qquad\textbf{(B)}\ 10\% \qquad\textbf{(C)}\ 11\% \qquad\textbf{(D)}\ 11\frac {1}{9}\% \qquad\textbf{(E)}\ 9\frac {1}{11}\%$

It is given a plane with a coordinate system with a beginning at the point $O$. $A(n)$, when $n$ is a natural number is a count of the points with whole coordinates which distances to $O$ are less than or equal to $n$. (a) Find $$\ell=\lim_{n\to\infty}\frac{A(n)}{n^2}.$$ (b) For which $\beta$ $(1<\beta<2)$ does the following limit exist? $$\lim_{n\to\infty}\frac{A(n)-\pi n^2}{n^\beta}$$

Prove $ \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $

[b]7.1[/b] Construct a trapezoid given four sides. [b]7.2[/b] Prove that $$(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .$$ [b]7.3 [/b] In a quadrilateral $ABCD$, $M$ is the midpoint of AB, $N$ is the midpoint of $CD$. Lines $AD$ and BC intersect $MN$ at points $P$ and $Q$, respectively. Prove that if $\angle BQM = \angle APM$ , then $BC=AD$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png[/img] [b]7.4 / 6.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]7.5 / 8.4[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png[/img] [b]7.6 / 6.5 [/b]The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

What fraction of a $5$-dimensional cube is the volume of the inscribed sphere? What fraction is it of a $10$-dimensional cube?

By a [i]fold[/i] of a polygon-shaped paper, we mean drawing a segment on the paper and folding the paper along that. Suppose that a paper with the following figure is given. We cut the paper along the boundary of the shaded region to get a polygon-shaped paper. Start with this shaded polygon and make a rectangle-shaped paper from it with at most 5 number of folds. Describe your solution by introducing the folding lines and drawing the shape after each fold on your solution sheet. (Note that the folding lines do not have to coincide with the grid lines of the shape.) [i]Proposed by Mahdi Etesamifard[/i]

If the perimeter of a rectangle is $ p$ and its diagonal is $ d$, the difference between the length and width of the rectangle is: $ \textbf{(A)}\ \frac {\sqrt {8d^2 \minus{} p^2}}{2} \qquad\textbf{(B)}\ \frac {\sqrt {8d^2 \plus{} p^2}}{2} \qquad\textbf{(C)}\ \frac {\sqrt {6d^2 \minus{} p^2}}{2}$ $ \textbf{(D)}\ \frac {\sqrt {6d^2 \plus{} p^2}}{2} \qquad\textbf{(E)}\ \frac {8d^2 \minus{} p^2}{4}$

A trapezoid $ABCD$ is given where $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the segment $AB$. Point $N$ is located on the segment $CD$ such that $\angle ADN = \frac{1}{2} \angle MNC$ and $\angle BCN = \frac{1}{2} \angle MND$. Prove that $N$ is the midpoint of the segment $CD$. [i]Proposed by Alireza Dadgarnia[/i]

Prove that diagonals of a convex quadrilateral are perpendicular if and only if inside of the quadrilateral there is a point, whose orthogonal projections on sides of the quadrilateral are vertices of a rectangle.