Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if $\angle BAD=\angle CAD$. [i]Dan Branzei[/i]

Let $n$ be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves: (1) select a row and multiply each entry in this row by $n$; (2) select a column and subtract $n$ from each entry in this column. Find all possible values of $n$ for which the following statement is true: Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is $0$.

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

Let $k$ be a circle centered at $O$. Let $\overline{AB}$ be a chord of that circle and $M$ its midpoint. Tangent on $k$ at points $A$ and $B$ intersect at $T$. The line $\ell$ goes through $T$, intersect the shorter arc $AB$ at the point $C$ and the longer arc $AB$ at the point $D$, so that $|BC| = |BM|$. Prove that the circumcenter of the triangle $ADM$ is the reflection of $O$ across the line $AD$

In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$

Let $ABC$ be a triangle with $AB \ne AC$. Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$

Let $ABC$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_1, B_1, C_1$ denote the reflections of $P$ in the sides $BC,CA,AB$ respectively. Prove that (i) If $P$ is the incenter or an excentre of $ABC$, then $P$ is the circumenter of $A_1B_1C_1$; (ii) If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_1B_1C_1$; (iii) If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_1B_1C_1$.

Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle. Prove that \[AX+AY+BC>AB+AC\]

Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].

Points $A, B$, and $C$ lie on a circle with centre $M$. The reflection of point $M$ in the line $AB$ lies inside triangle $ABC$ and is the intersection of the angle bisectors of angles $A$ and $B$. Line $AM$ intersects the circle again in point $D$. Show that $|CA| \cdot |CD| = |AB| \cdot |AM|$.

Consider a convex $100$-gon $A_1A_2...A_{100}$. Draw the diagonal $A_{43}A_{81}$ which divides it into two convex polygons $P_1,P_2$. How many vertices and how diagonals, has each of the polygons $P_1,P_2$?

Let $\omega_1$ and $\omega_2$ be two different circles that intersect at two different points, $X$ and $Y$. Let lines $l_1$ and $l_2$ be common tangent lines of these circles such that $l_1$ is tangent $\omega_1$ at $A$ and $\omega_2$ at $C$ and $l_2$ is tangent $\omega_1$ at $B$ and $\omega_2$ at $D$. Let $Z$ be the reflection of $Y$ respect to $l_1$ and let $BC$ and $\omega_1$ meet at $K$ for the second time. Let $AD$ and $\omega_2$ meet at $L$ for the second time. Prove that the line tangent to $\omega_1$ and passes through $K$ and the line tangent to $\omega_2$ and passes through $L$ meet on the line $XZ$.

Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

A string having a small loop in one end is set over a horizontal pipe so that the ends hang loosely. After that, the other end is put through the loop, pulled as far as possible from the pipe and fixed in that position whereby this end of the string is farther from the pipe than the loop. Let $\alpha$ be the angle by which the string turns at the point where it passes through the loop (see picture). Find $\alpha$. [img]https://cdn.artofproblemsolving.com/attachments/2/1/018bb16d80956699e11c641bad9bb3d0083770.png[/img]

A circle $\Gamma$ is tangent to the side $BC$ of a triangle $ABC$ at $X$ and tangent to the side $AC$ at $Y$. A point $P$ is taken on the side $AB$. Let $XP$ and $YP$ intersect $\Gamma$ at $K$ and $L$ for the second time, $AK$ and $BL$ intersect $\Gamma$ at $R$ and $S$ for the second time. Prove that $XR$ and $YS$ intersect on $AB$.

In the figure below $\triangle ABC$, $\triangle DEF$, and $\triangle GHI$ are overlapping equilateral triangles, $C$ and $F$ lie on $\overline{BD}$, $F$ and $I$ lie on $\overline{EG}$, and $C$ and $I$ lie on $\overline{AH}$. Length $AB = 2FC$, $DE = 3FC$, and $GH = 4FC$. Given that the area of $\triangle FCI$ is $3$, find the area of the hexagon $ABGHDE$. [asy] size(5cm); pen dps = fontsize(10); defaultpen(dps); pair A,B,C,D,E,F,G,H,I; G=origin; H=(4,0); I=(2,2*sqrt(3)); F=(1.5,3*sqrt(3)/2); C=F+(1,0); B=F-(1,0); D=C+(2,0); A=F+(0,sqrt(3)); E=C+(0.5,3*sqrt(3)/2); draw(A--H--G--E--D--B--cycle); label("$A$",A,N*.5); label("$B$",B,S*.5); label("$C$",C,SW*.5); label("$D$",D,S*.5); label("$E$",E,N*.5); label("$F$",F,SE*.5); label("$G$",G,S*.5); label("$H$",H,S*.5); label("$I$",I,N*2); [/asy]

In triangle $ABC$, let $D$ be the midpoint of $BC$, and $BE$, $CF$ are the altitudes. Prove that $DE$ and $DF$ are both tangents to the circumcircle of triangle $AEF$

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

$ABCD$ is a square side $1$. $P$ and $Q$ lie on the side $AB$ and $R$ lies on the side $CD$. What are the possible values for the circumradius of $PQR$?

In trapezoid $ABCD$ the sides $BC$ and $AD$ are parallel, $AC = BC + AD$, and the angle between the diagonals is equal to $ 60^o$. Prove that $AB = CD$. (Stanislav Smirnov, St Petersburg)

Given a triangle in which the sides $ a $, $ b $, $ c $ form an arithmetic progression and the angles also form an arithmetic progression. Find the ratios of the sides of this triangle.

Let $P$ be an interior point of triangle $ABC$. The lines $AP$, $BP$ and $CP$ divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of $ABC$?

Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.

Suppose we climb a mountain that is a cone with radius $100$ and height $4$. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height $z = 2$). Our climbing speed starts at $v_0=2$ but gets slower at a rate inversely proportional to the distance to the mountain top (so at height $z$ the speed $v$ is $(h-z)v_0/h$). Find the minimum time needed to get to the destination.

In a trapezoid, the sum of the lengths of the side and the diagonal is equal to the sum of the lengths of the other side and the other diagonal. Prove that the trapezoid is isosceles.