In a triangle $ABC$, the bisector of angle $BAC$ intersects $BC$ at $D$. The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$. Line $BM$ meets $\Gamma$ again at $P$. Prove that line $AP$ is a median of $\triangle ABD.$

Let $ABC$ be a triangle and let $X$ and $Y$ be points on the $A$-symmedian such that $AX = XB$ and $AY = YC$. Let $BX$ and $CY$ meet at $Z$. Let the $Z$-excircle of triangle $XYZ$ touch $ZX$ and $ZY$ at $E$ and $F$. Show that $A$, $E$, $F$ are collinear.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Let $M$ be the midpoint of cathetus $AB$ of triangle $ABC$ with right angle $A$. Point $D$ lies on the median $AN$ of triangle $AMC$ in such a way that the angles $ACD$ and $BCM$ are equal. Prove that the angle $DBC$ is also equal to these angles.

In a triangle $\triangle ABC$, $D$ and $E$ are respectively on $AB$ and $AC$ such that $DE\parallel BC$. $P$ is the intersection of $BE$ and $CD$. $M$ is the second intersection of $(APD)$ and $(BCD)$ , $N$ is the second intersection of $(APE)$ and $(BCE)$. $w$ is the circle passing through $M$ and $N$ and tangent to $BC$. Prove that the lines tangent to $w$ at $M$ and $N$ intersect on $AP$.

Squares $CAKL$ and $CBMN$ are constructed on the sides of acute-angled triangle $ABC$, outside of the triangle. Line $CN$ intersects line segment $AK$ at $X$, while line $CL$ intersects line segment $BM$ at $Y$. Point $P$, lying inside triangle $ABC$, is an intersection of the circumcircles of triangles $KXN$ and $LYM$. Point $S$ is the midpoint of $AB$. Prove that angle $\angle ACS=\angle BCP$.

Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.

Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$. [i](United Kingdom) David Monk[/i]

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$. [i]Proposed by Evan Chen[/i]

The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $ABC$ meet at $X$. Let $M$ be the midpoint of $BC$. Prove that [i](a)[/i] $\angle BAM = \angle CAX$, and [i](b)[/i] $\frac{AM}{AX} = \cos\angle BAC.$

On sides $AB$ and $BC$ of a non-isosceles triangle $ABC$ are selected points $C_1$ and $A_1$ such that the quadrilateral $AC_1A_1C$ is cyclic. Lines $CC_1$ and $AA_1$ intersect at point $P$. Line $BP$ intersects the circumscribed circle of triangle $ABC$ at the point $Q$. Prove that the lines $QC_1$ and $CM$, where $M$ is the midpoint of $A_1C_1$, intersect at the circumscribed circles of triangle $ABC$.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Let $ABC$ be a triangle with $AB = 130$, $BC = 140$, $CA = 150$. Let $G$, $H$, $I$, $O$, $N$, $K$, $L$ be the centroid, orthocenter, incenter, circumenter, nine-point center, the symmedian point, and the de Longchamps point. Let $D$, $E$, $F$ be the feet of the altitudes of $A$, $B$, $C$ on the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$. Let $X$, $Y$, $Z$ be the $A$, $B$, $C$ excenters and let $U$, $V$, $W$ denote the midpoints of $\overline{IX}$, $\overline{IY}$, $\overline{IZ}$ (i.e. the midpoints of the arcs of $(ABC)$.) Let $R$, $S$, $T$ denote the isogonal conjugates of the midpoints of $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Let $P$ and $Q$ denote the images of $G$ and $H$ under an inversion around the circumcircle of $ABC$ followed by a dilation at $O$ with factor $\frac 12$, and denote by $M$ the midpoint of $\overline{PQ}$. Then let $J$ be a point such that $JKLM$ is a parallelogram. Find the perimeter of the convex hull of the self-intersecting $17$-gon $LETSTRADEBITCOINS$ to the nearest integer. A diagram has been included but may not be to scale. [asy] size(6cm); import olympiad; import cse5; pair A = dir(110); pair B = dir(210); pair C = dir(330); pair D = foot(A,B,C); pair E = foot(B,C,A); pair F = foot(C,A,B); pair G = centroid(A,B,C); pair H = orthocenter(A,B,C); pair I = incenter(A,B,C); pair isocon(pair targ) { return extension(A,2*foot(targ,I,A)-targ, C,2*foot(targ,I,C)-targ); } pair O = circumcenter(A,B,C); pair K = isocon(G); pair N = midpoint(O--H); pair U = extension(O,midpoint(B--C),A,I); pair V = extension(O,midpoint(C--A),B,I); pair W = extension(O,midpoint(A--B),C,I); pair X = -I + 2*U; pair Y = -I + 2*V; pair Z = -I + 2*W; pair R = isocon(midpoint(A--D)); pair S = isocon(midpoint(B--E)); pair T = isocon(midpoint(C--F)); pair L = 2*H-O; pair P = 0.5/conj(G); pair Q = 0.5/conj(H); pair M = midpoint(P--Q); pair J = K+M-L; draw(A--B--C--cycle); void draw_cevians(pair target) { draw(A--extension(A,target,B,C)); draw(B--extension(B,target,C,A)); draw(C--extension(C,target,A,B)); } draw_cevians(H); draw_cevians(G); draw_cevians(I); draw(unitcircle); draw(circumcircle(D,E,F)); draw(O--P); draw(O--Q); draw(P--Q); draw(CP(X,foot(X,B,C))); draw(CP(Y,foot(Y,C,A))); draw(CP(Z,foot(Z,A,B))); draw(J--K--L--M); draw(X--Y--Z--cycle); draw(A--X); draw(B--Y); draw(C--Z); draw(A--foot(X,A,B)); draw(A--foot(X,A,C)); draw(B--foot(Y,B,C)); draw(B--foot(Y,B,A)); draw(C--foot(Z,C,A)); draw(C--foot(Z,C,B)); pen p = black; dot(A, p); dot(B, p); dot(C, p); dot(D, p); dot(E, p); dot(F, p); dot(G, p); dot(H, p); dot(I, p); dot(J, p); dot(K, p); dot(L, p); dot(M, p); dot(N, p); dot(O, p); dot(P, p); dot(Q, p); dot(R, p); dot(S, p); dot(T, p); dot(U, p); dot(V, p); dot(W, p); dot(X, p); dot(Y, p); dot(Z, p); [/asy]

Cut the non-equilateral triangle into four similar triangles, among which not all are the same.