Given a triangle $ABC$. Its incircle is tangent to $BC, CA, AB$ at $D, E, F$ respectively. Let $K, L$ be points on $CA, AB$ respectively such that $K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF$. Prove that the circumcircle of $AKL$ is tangent to the incircle of $ABC$.

Let $ABC$ be a triangle inscribed in the circle $(O)$ and $P$ is a point inside the triangle $ABC$. Let $D$ be a point on $(O)$ such that $AD \perp AP$. The line $CD$ cuts the perpendicular bisector of $BC$ at $M$. The line $AD$ cuts the line passing through $B$ and is perpendicular to $BP$ at $Q$. Let $N$ be the reflection of $Q$ through $M$. Prove that $CN \perp CP$.

Let $ABC$ be an acute scalene triangle, with the feet of $A,B,C$ onto $BC,CA,AB$ being $D,E,F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC,CA,AB$ are $W_a,W_b,W_c$ respectively. Finally, let $N$ and $I$ be the circumcenter and the incenter of $W_aW_bW_c$ respectively. Prove that, if $N$ coincides with the nine-point center of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$. [i]Proposed by Navneel Singhal, India and Massimiliano Foschi, Italy[/i]

A convex quadrilateral $ABCD$ is circumscribed about circle $\omega$. A tangent to $\omega$ parallel to $AC$ intersects $BD$ at a point $P$ outside of $\omega$. The second tangent from $P$ to $\omega$ touches $\omega$ at a point $T$. Prove that $\omega$ and circumcircle of $ATC$ are tangent. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

The incircle of triangle $ABC$, is tangent to sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. The reflection of $F$ with respect to $B$ and the reflection of $E$ with respect to $C$ are $T$ and $S$ respectively. Prove that the incenter of triangle $AST$ is inside or on the incircle of triangle $ABC$. [i]Proposed by Mehdi E'tesami Fard[/i]

Let $A,B$ be two distinct points on a given circle $O$ and let $P$ be the midpoint of the line segment AB. Let $O_1$ be the circle tangent to the line $AB$ at $P$ and tangent to the circle $O$. Let $l$ be the tangent line, different from the line $AB$, to $O_1$ passing through $A$. Let $C$ be the intersection point, different from $A$, of $l$ and $O$. Let $Q$ be the midpoint of the line segment $BC$ and $O_2$ be the circle tangent to the line $BC$ at $Q$ and tangent to the line segment $AC$. Prove that the circle $O_2$ is tangent to the circle $O$.

In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$. [i]Author: Ray Li[/i]

Let $ABC$ be triangle with circumcircle $(O)$ of fixed $BC$, $AB \ne AC$ and $BC$ not a diameter. Let $I$ be the incenter of the triangle $ABC$ and $D = AI \cap BC, E = BI \cap CA, F = CI \cap AB$. The circle passing through $D$ and tangent to $OA$ cuts for second time $(O)$ at $G$ ($G \ne A$). $GE, GF$ cut $(O)$ also at $M, N$ respectively. a) Let $H = BM \cap CN$. Prove that $AH$ goes through a fixed point. b) Suppose $BE, CF$ cut $(O)$ also at $L, K$ respectively and $AH \cap KL = P$. On $EF$ take $Q$ for $QP = QI$. Let $J$ be a point of the circimcircle of triangle $IBC$ so that $IJ \perp IQ$. Prove that the midpoint of $IJ$ belongs to a fixed circle.

Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$ at points $G, H,$ and $J,$ respectively. The points $G, D, E, H, J,$ and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\overarc{DE}+2\cdot \overarc{HJ} + 3\cdot \overarc{FG},$ where the arcs are measured in degrees. [asy] import olympiad; size(6cm); defaultpen(fontsize(10pt)); pair B = (0, 0), A = (Cos(60), Sin(60)), C = (Cos(60)+Sin(60)/Tan(36), 0), D = midpoint(B--C), E = midpoint(A--C), F = midpoint(A--B); guide circ = circumcircle(D, E, F); pair G = intersectionpoint(B--D, circ), J = intersectionpoints(A--F, circ)[0], H = intersectionpoints(A--E, circ)[0]; draw(B--A--C--cycle); draw(D--E--F--cycle); draw(circ); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); dot(J); label("$A$", A, (0, .8)); label("$B$", B, (-.8, -.8)); label("$C$", C, (.8, -.8)); label("$D$", D, (0, -.8)); label("$E$", E, (.8, .2)); label("$F$", F, (-.8, .2)); label("$G$", G, (0, .8)); label("$H$", H, (-.2, -1)); label("$J$", J, (.2, -.8)); [/asy]

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.

Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that \[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]\[, IA'+IB'+IC' \geq IA+IB+IC\]

An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that \[ \angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.\] The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

In triangle $\triangle ABC$, $AB<AC$, $M,D,E$ are the midpoints of $BC$, the arcs $BAC$ and $BC$ of the circumcircle of $\triangle ABC$ respectively. The incircle of $\triangle ABC$ touches $AB$ at $F$, $AE$ meets $BC$ at $G$, and the perpendicular to $AB$ at $B$ meets segment $EF$ at $N$. If $BN=EM$, prove that $DF$ is perpendicular to $FG$.

Let the points $D$ and $E$ lie on the sides $BC$ and $AC$, respectively, of the triangle $ABC$, satisfying $BD=AE$. The line joining the circumcentres of the triangles $ADC$ and $BEC$ meets the lines $AC$ and $BC$ at $K$ and $L$, respectively. Prove that $KC=LC$.

Let $\triangle{ABC}$ be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ and incenter $I$. Suppose the orthocenter $H$ of $BIC$ lies inside $\omega$. Let $M$ be the midpoint of the longer arc $BC$ of $\omega$. Let $N$ be the midpoint of the shorter arc $AM$ of $\omega$. Prove that there exists a circle tangent to $\omega$ at $N$ and tangent to the circumcircles of $BHI$ and $CHI$.

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that \[ R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}\] where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$

The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid. [asy] pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6); draw(B--C--P--D--C^^D--A); filldraw(A--P--B--cycle, gray, black); label("$A$", A, SE); label("$B$", B, NE); label("$C$", C, NW); label("$D$", D, SW); label("$P$", P, N); label("$13\sqrt{3}$", A--D, S); label("$12\sqrt{3}$", A--B, E);[/asy]

The quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$ and $BD$ do not pass through $O$. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.

An acute $\triangle ABC$ with circumcircle $\Gamma$ is given. $I$ and $I_a$ are the incenter and $A-$excenter of $\triangle ABC$ respectively. The line $AI$ intersects $\Gamma$ again at $D$ and $A'$ is the antipode of $A$ with respect to $\Gamma$. $X$ and $Y$ are point on $\Gamma$ such that $\angle IXD = \angle I_aYD = 90^\circ$. The tangents to $\Gamma$ at $X$ and $Y$ intersect in point $Z$. Prove that $A', D$ and $Z$ are collinear. [i]Proposed by Nikola Velov[/i]

Let $ABC$ be a triangle with right angle $ACB$. Denote by $F$ the projection of $C$ on $AB$. A circle $\omega$ touches $FB$ at point $P$, touches $CF$ at point $Q$, and the circumcircle of $ABC$ at point $R$. Prove that the points $A, Q, R$ all lie on the same line and $AP=AC$.

If the circumradius of a regular $n$-gon is $1$ and the ratio of its perimeter over its area is $\dfrac{4\sqrt 3}{3}$, what is $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

Let $ABC$ be an acute and scalene of circumcircle $\Gamma$ and orthocenter $H$. Let $A_1,B_1,C_1$ be the second intersection points of the lines $AH, BH, CH$ with $\Gamma$, respectively. The lines that pass through $A_1,B_1,C_1$ and are parallel to $BC,CA, AB$ intersect again to $\Gamma$ at $A_2,B_2,C_2$, respectively. Let $M$ be the intersection point of $AC_2$ and $BC_1, N$ the intersection point of $BA_2$ and $CA_1$, and $P$ the intersection point of $CB_2$ and $AB_1$. Prove that $\angle MNB = \angle AMP$ .