Let $ABC$ a triangle inscribed in a circle $(O)$ with orthocenter $H$. Two lines $d_1$ and $d_2$ are mutually perpendicular at $H$. Let $d_1$ meet $BC,CA,AB$ at $X_1, Y_1,Z_1$ respectively. Let $A_1B_1C_1$ be a triangle formed by the line through $X_1$ perpendicular to $BC$, the line through $Y_1$ perpendicular to CA, the line through $Z_1$ perpendicular perpendicular to $AB$. Triangle $A_2B_2C_2$ is defined in the same manner. Prove that the circumcircles of triangles $A_1B_1C_1$ and $A_2B_2C_2$ touch each other at a point on $(O)$. Nguyễn Văn Linh

Let $P$ be an arbitrary point in the plane of triangle $ABC$. Lines $PA, PB, PC$ meets the perpendicular bisectors of $BC,CA,AB$ at $O_a,O_b,O_c$ respectively. Let $(O_a)$ be the circle with center $O_a$ passing through two points $B,C$, two circles $(O_b), (O_c)$ are defined in the same manner. Two circles $(O_b), (O_c)$ meets at $A_1$, distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Let $Q$ be an arbitrary point in the plane of $ABC$ and $QB,QC$ meets $(O_c)$ and $(O_b)$ at $A_2,A_3$ distinct from $B,C$. Similarly, we have points $B_2,B_3,C_2,C_3$. Let $(K_a), (K_b), (K_c)$ be the circumcircles of triangles $A_1A_2A_3, B_1B_2B_3, C_1C_2C_3$. Prove that (a) three circles $(K_a), (K_b), (K_c)$ have a common point. (b) two triangles $K_aK_bK_c, ABC$ are similar. Trần Quang Hùng

Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency. Nguyễn Văn Linh

Let $ABC$ be a triangle inscribed in a circle $(O)$. Let $P$ be an arbitrary point in the plane of triangle $ABC$. Points $A',B',C'$ are the reflections of $P$ about the lines $BC,CA,AB$ respectively. $X$ is the intersection, distinct from $A$, of the circle with diameter $AP$ and the circumcircle of triangle $AB'C'$. Points $Y,Z$ are defined in the same way. Prove that five circles $(O), (AB'C')$, $(BC'A'), (CA'B'), (XY Z)$ have a point in common. Nguyễn Văn Linh

Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ of the triangle touches sides $BC,CA,AB$ at $A_0,B_0$, and $C_0$. Points $A_1,B_1$, and $C_1$ are on $BC,CA,AB$ such that $BA1 = CA_0, CB_1 = AB_0, AC_1 = BC_0$. Prove that the circumcircles $(IAA1), (IBB_1), (ICC_1)$ pass all through a common point, distinct from $I$. Nguyễn Minh Hà