Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$. Show that $A,X,Y$ are collinear.

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.

Let $ABC$ be an acute, non isosceles triangle, $AX, BY, CZ$ are the altitudes with $X, Y, Z$ belong to $BC, CA,AB$ respectively. Respectively denote $(O_1), (O_2), (O_3)$ as the circumcircles of triangles $AY Z, BZX, CX Y$ . Suppose that $(K)$ is a circle that internal tangent to $(O_1), (O_2), (O_3)$. Prove that $(K)$ is tangent to circumcircle of triangle $ABC$.

Consider the isosceles triangle $ABC$ inscribed in the semicircle of radius $ r$. If the $\vartriangle BCD$ and $\vartriangle CAE$ are equilateral, determine the altitude of $\vartriangle DEC$ on the side $DE$ in terms of $ r$. [img]https://cdn.artofproblemsolving.com/attachments/6/3/772ff9a1fd91e9fa7a7e45ef788eec7a1ba48e.png[/img]

Let $ABC$ be an acute triangle with circumcenter $O$ and circumcircle $\Omega$. Choose points $D, E$ from sides $AB, AC$, respectively, and let $\ell$ be the line passing through $A$ and perpendicular to $DE$. Let $\ell$ intersect the circumcircle of triangle $ADE$ and $\Omega$ again at points $P, Q$, respectively. Let $N$ be the intersection of $OQ$ and $BC$, $S$ be the intersection of $OP$ and $DE$, and $W$ be the orthocenter of triangle $SAO$. Prove that the points $S$, $N$, $O$, $W$ are concyclic. [i]Proposed by Li4 and me.[/i]

Triangle $ABC$ is right angled such that $\angle ACB=90^{\circ}$ and $\frac {AC}{BC} = 2$. Let the line parallel to side $AC$ intersects line segments $AB$ and $BC$ in $M$ and $N$ such that $\frac {CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$ lies point $K$ such that $OM+OK=KN$. Let $T$ be the intersection point of angle bisector of $\angle ABC$ and line from $K$ perpendicular to $AN$. Determine value of $\angle MTB$.

Let $ABCD$ be a parallelogram which is not a rectangle and $E$ be the point in its plane such that $AE \perp AB$ and $CE \perp CB$. Prove that $\angle DEA = \angle CEB$.

Two straight lines $a$ and $b$ are given and also points $A$ and $B$. Point $X$ slides along the line $a$, and point $Y$ slides along the line $b$, so that $AX \parallel BY$. Find the locus of the intersection point of $AY$ with $XB$.

We´re given an inscriptible quadrilateral $ DEFG $ having some vertices on the sides of a triangle $ ABC, $ and some vertices (at least one of them) coinciding with the vertices of the same triangle. Knowing that the lines $ DF $ and $ EG $ aren´t parallel, find the locus of their intersection. [i]Dan Brânzei[/i]

Let $ABC$ be a triangle without obtuse angles, $M$ the midpoint of $BC$, $K$ the midpoint of $BM$. What is the largest value of the angle $\angle KAM$?

Given a pentagon with all equal sides. a) Prove that there exist such a point on the maximal diagonal, that every side is seen from it inside a right angle. (side $AB$ is seen from the point $C$ inside an arbitrary angle that is greater or equal than $\angle ACB$) b) Prove that the circles constructed on its sides as on the diameters cannot cover the pentagon entirely.

$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \] where $R$ is the radius of the circumscribing circle.

[b]p1.[/b] Insert "$+$" signs between some of the digits in the following sequence to obtain correct equality: $$1\,\,\,\, 2\,\,\,\, 3\,\,\,\, 4\,\,\,\,5\,\,\,\, 6\,\,\,\, 7 = 100$$ [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. $\frac25$ of his drink is orange juice and the rest is apple juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $\frac35$ of orange juice? [b]p4.[/b] A train moving at $55$ miles per hour meets and is passed by a train moving moving in the opposite direction at $35$ miles per hour. A passenger in the first train sees that the second train takes $8$ seconds to pass him. How long is the second train? [b]p5.[/b] It is easy to arrange $16$ checkers in $10$ rows of $4$ checkers each, but harder to arrange $9$ checkers in $10$ rows of $3$ checkers each. Do both. [b]p6.[/b] Every human that lived on Earth exchanged some number of handshakes with other humans. Show that the number of people that made an odd number of handshakes is even. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three parallel lines $L_1, L_2, L_2$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$ and the perpendicular distance between lines $L_2$ and $L_3$ is also $3$. A square $ABCD$ is constructed such that $A$ lies on $L_1$, $B$ lies on $L_3$ and $C$ lies on $L_2$. Find the area of the square.

Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example. $(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3. $(q)$ In $\triangle{ABC},\ \triangle{ABD}$, if $AC<AD$ and $BC<BD$, then $\angle{C}>\angle{D}$. 35 points

Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$. [i]Proposed by Kevin Sun[/i]

Let $ABC$ be a triangle with ${AB\ne BC}$; and let ${BD}$ be the internal bisector of $\angle ABC,\ $, $\left( D\in AC \right)$. Denote by ${M}$ the midpoint of the arc ${AC}$ which contains point ${B}$. The circumscribed circle of the triangle ${\vartriangle BDM}$ intersects the segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of ${A}$ with respect to ${K}$. If ${DJ\cap AM=\left\{O\right\}}$, prove that the points ${J, B, M, O}$ belong to the same circle.

$ABCD$ is a parallelogram. $K$ is the circumcircle of $ABD$. The lines $BC$ and $CD$ meet $K$ again at $E$ and $F$. Show that the circumcenter of $CEF$ lies on $K$.

In an isosceles triangle $ABC$ in which $AC = BC$ and $\angle ABC < 60^o$, $I$ and $O$ are the centers of the inscribed and circumscribed circles, respectively. For the triangle $BIO$, the circumscribed circle intersects the side $BC$ again at $D$. Prove that: i) lines $AC$ and $DI$ are parallel, ii) lines $OD$ and $IB$ are perpendicular.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and $AB\neq AC$. Let $D$ and $E$ lie on the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle BAE=\angle DAC$. Let the incenters of $BAE$ and $CAD$ be $X$ and $Y$, respectively, and let the external tangents of the incircles of $BAE$ and $CAD$ intersect at $Z$. Prove that $Z$ lies on the common chord of $\Gamma$ and the circumcircle of $AXY$.

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

Let $A_1,B_1,C_1$ are the midpoints of the sides of the triangle $ABC, I$ is the center of the circle inscribed in it. Let $C_2$ be the intersection point of lines $C_1 I$ and $A_1B_1$. Let $C_3$ be the intersection point of lines $CC_2$ and $AB$. Prove that line $IC_3$ is perpendicular to line $AB$. (A. Zaslavsky)

A chord $PQ$ of the circumcircle of a triangle $ABC$ meets the sides $BC, AC$ at points $A', B'$ respectively. The tangents to the circumcircle at $A$ and $B$ meet at a point $X$, and the tangents at points $P$ and $Q$ meet at point $Y$. The line $XY$ meets $AB$ at a point $C'$. Prove that the lines $AA', BB'$ and $CC'$ concur.

Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.

Let a triangle $ABC$ be given. On a ruler three segment congruent to the sides of this triangle are marked. Using this ruler construct the orthocenter of the triangle formed by the tangency points of the sides of $ABC$ with its incircle.