Let $AB$ be an acute scalene triangle. A point \( D \) varies on its side \( BC \). The points \( P \) and \( Q \) are the midpoints of the arcs \( \widehat{AB} \) and \( \widehat{AC} \) (not containing \( D \)) of the circumcircles of triangles \( ABD \) and \( ACD \), respectively. Prove that the circumcircle of triangle \( PQD \) passes through a fixed point, independent of the choice of \( D \) on \( BC \).

Triangle $ABC$ has circumcircle $\Omega$. Chord $XY$ of $\Omega$ intersects segment $AC$ at point $E$ and segment $AB$ at point $F$ such that $E$ lies between $X$ and $F$. Suppose that $A$ bisects arc $\widehat{XY}$. Given that $EC = 7, FB = 10, AF = 8$, and $YF - XE = 2$, find the perimeter of triangle $ABC$.

Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$, and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at $F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals $AEFA', BDFB', CDEC'$ are inscribable. (1) Prove that $DEA'B'$ is inscribable. (2) Prove that $DA', EB', FC'$ are concurrent.

Prove that the sides of any equilateral triangle you can either increase everything or decrease everything by the same amount so that you get a right triangle.

What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$?

Construct a right triangle, given two medians drawn to its legs.

Can one paint four vertices of a cube red and the other four points black so that any plane passing through three points of the same colour contains a vertex of the other colour? (Mebius, Sharygin)

A semicircle is joined to the side of a triangle, with the common edge removed. Sixteen points are arranged on the figure, as shown below. How many non-degenerate triangles can be drawn from the given points? [asy] draw((0,-2)--arc((0,0),1,0,180)--cycle); dot((-0.8775,-0.245)); dot((-0.735,-0.53)); dot((-0.5305,-0.939)); dot((-0.3875,-1.225)); dot((-0.2365,-1.527)); dot((0.155,-1.69)); dot((0.306,-1.388)); dot((0.4,-1.2)); dot((0.551,-0.898)); dot((0.837,-0.326)); dot(dir(25)); dot(dir(50)); dot(dir(65)); dot(dir(100)); dot(dir(115)); dot(dir(140)); [/asy]

Triangle $\triangle{ABC}$,O=circumcenter of (ABC),OA=R,the A-excircle intersect (AB),(BC),(CA) at points F,D,E. If the A-excircle has radius R prove that $OD\perp EF$

Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$. (a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$. (b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$. [i]James Lin[/i]

Let $\triangle ABC $ be an acute triangle.$O$ denote its circumcenter.Points $D$,$E$,$F$ are the midpoints of the sides $BC$,$CA$,and $AB$.Let $M$ be a point on the side $BC$ . $ AM \cap EF = \big\{ N \big\} $ . $ON \cap \big( ODM \big) = \big\{ P \big\} $ Prove that $M'$ lie on $\big(DEF\big)$ where $M'$ is the symmetrical point of $M$ thought the midpoint of $DP$.

The tetrahedron $OABC$ has all angles at vertex $O$ equal to $60^o$. Prove that $$AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2$$

Let $SABCD$ be a pyramid with the vertex $S$ whose all edges are equal. Points $M$ and $N$ on the edges $SA$ and $BC$ respectively are such that $MN$ is perpendicular to both $SA$ and $BC$. Find the ratios $SM:MA$ and $BN:NC$.

Let $ABC$ be an acute angled triangle. Points $E$ and $F$ are chosen on the sides $AC$ and $AB$, respectively, such that \[BC^2=BA\times BF+CE\times CA.\] Prove that for all such $E$ and $F$, circumcircle of the triangle $AEF$ passes through a fixed point different from $A$.

$A, B, C$ are collinear with $B$ betweeen $A$ and $C$. $K_{1}$ is the circle with diameter $AB$, and $K_{2}$ is the circle with diameter $BC$. Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$. The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$. Show that the lines $AR$ and $CS$ meet on the perpendicular to $AC$ at $B$.

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

Let $ABCD$ be a rhombus and $P$ be a point on its side $BC$. The circle passing through $A, B$, and $P$ intersects $BD$ once more at the point $Q$ and the circle passing through $C,P$ and $Q$ intersects $BD$ once more at the point $R$. Prove that $A, R$ and $P$ lie on the one straight line. (D. Fomin, Leningrad)

Let the altitude from $A$ and $B$ of triangle $ABC$ meet the circumcircle of $ABC$ again at $D$ and $E$ respectively. Let $DE$ meet $AC$ and $BC$ at $P$ and $Q$ respectively. Show that $ABQP$ is cyclic

The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a$\sqrt{17}$. Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.

$A_1A_2A_3A_4$ is cyclic quadrilateral of $\odot O$. $H_1,H_2,H_3,H_4$ are orthocentres of $\triangle A_2A_3A_4,\triangle A_3A_4A_1,\triangle A_4A_1A_2,\triangle A_1A_2A_3$. Prove that $H_1,H_2,H_3,H_4$ are concyclic, and determine its center.

A rubber band of negligible thickness encloses three pegs that lie in a perfect line, as shown. Each peg has a diameter of $4$ cm, as shown. What is the length of the rubber band used, in centimeters? All pegs shown are congruent circles. [asy] size(120); draw(circle((0,0),1));draw(circle((0,2),1));draw(circle((0,4),1)); dot((0,0)^^(0,2)^^(0,4)); draw((-1,0)--(-1,4)--arc((0,4),1,180,0)--(1,4)--(1,0)--arc((0,0),1,360,180),linewidth(2)); draw((-1,0)--(1,0),dotted); label("$4$ cm", (-0.38,0)--(1,0), N); [/asy] $\textbf{(A) }8\qquad\textbf{(B) }8+4\pi\qquad\textbf{(C) }16+4\pi\qquad\textbf{(D) }16+8\pi\qquad\textbf{(E) }16\pi$

A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.

Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.

[b]p1.[/b] Solve for $x$ and $y$ if $\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}$ and $\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}$ [b]p2.[/b] Find positive integers $p$ and $q$, with $q$ as small as possible, such that $\frac{7}{10} <\frac{p}{q} <\frac{11}{15}$. [b]p3.[/b] Define $a_1 = 2$ and $a_{n+1} = a^2_n -a_n + 1$ for all positive integers $n$. If $i > j$, prove that $a_i$ and $a_j$ have no common prime factor. [b]p4.[/b] A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it. Prove that the number of these smaller triangular regions is always odd. [b]p5.[/b] In triangle $ABC$, let $\angle ABC=\angle ACB=40^o$ is extended to $D$ such that $AD=BC$. Prove that $\angle BCD=10^o$. [img]https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1 [/u] [b]p1.[/b] In the convex quadrilateral $ABCD$ with diagonals $AC$ and $BD$, you know that angle $BAC$ is congruent to angle $CBD$, and that angle $ACD$ is congruent to angle $ADB$. Show that angle $ABC$ is congruent to angle $ADC$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/41cd120813d5541dc73c5d4a6c86cc82747fcc.png[/img] [b]p2.[/b] In how many different ways can you place $12$ chips in the squares of a $4 \times 4$ chessboard so that (a) there is at most one chip in each square, and (b) every row and every column contains exactly three chips. [b]p3.[/b] Students from Hufflepuff and Ravenclaw were split into pairs consisting of one student from each house. The pairs of students were sent to Honeydukes to get candy for Father's Day. For each pair of students, either the Hufflepuff student brought back twice as many pieces of candy as the Ravenclaw student or the Ravenclaw student brought back twice as many pieces of candy as the Hufflepuff student. When they returned, Professor Trelawney determined that the students had brought back a total of $1000$ pieces of candy. Could she have possibly been right? Why or why not? Assume that candy only comes in whole pieces (cannot be divided into parts). [b]p4.[/b] While you are on a hike across Deception Pass, you encounter an evil troll, who will not let you across the bridge until you solve the following puzzle. There are six stones, two colored red, two colored yellow, and two colored green. Aside from their colors, all six stones look and feel exactly the same. Unfortunately, in each colored pair, one stone is slightly heavier than the other. Each of the lighter stones has the same weight, and each of the heavier stones has the same weight. Using a balance scale to make TWO measurements, decide which stone of each color is the lighter one. [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2 [/u] [b]p6.[/b] Consider a set of finitely many points on the plane such that if we choose any three points $A,B,C$ from the set, then the area of the triangle $ABC$ is less than $1$. Show that all of these points can be covered by a triangle whose area is less than $4$. [b]p7.[/b] A palindrome is a number that is the same when read forward and backward. For example, $1771$ and $23903030932$ are palindromes. Can the number obtained by writing the numbers from $1$ to $n$ in order be a palindrome for some $n > 1$ ? (For example, if $n = 11$, the number obtained is $1234567891011$, which is not a palindrome.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].