Školjka

%V0 Let $X,Y,Z$ be the midpoints of the small arcs $BC,CA,AB$ respectively (arcs of the circumcircle of $ABC$). $M$ is an arbitrary point on $BC$, and the parallels through $M$ to the internal bisectors of $\angle B,\angle C$ cut the external bisectors of $\angle C,\angle B$ in $N,P$ respectively. Show that $XM,YN,ZP$ concur.

%V0 Let $ABC$ be a triangle. $D$ is a point on the side $(BC)$. The line $AD$ meets the circumcircle again at $X$. $P$ is the foot of the perpendicular from $X$ to $AB$, and $Q$ is the foot of the perpendicular from $X$ to $AC$. Show that the line $PQ$ is a tangent to the circle on diameter $XD$ if and only if $AB = AC$.

%V0 The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet the opposite sides at $D,E, F,$ respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R,$ respectively. The line $EF$ meets $BC$ at $P.$ Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC.$

%V0 In an acute-angled triangle $ABC,$ let $AD,BE$ be altitudes and $AP,BQ$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $D, E,$ and $I$ are collinear if and only if the points $P, Q,$ and $O$ are collinear.

%V0 Let $A_1A_2A_3$ be a non-isosceles triangle with incenter $I.$ Let $C_i,$ $i = 1, 2, 3,$ be the smaller circle through $I$ tangent to $A_iA_{i+1}$ and $A_iA_{i+2}$ (the addition of indices being mod 3). Let $B_i, i = 1, 2, 3,$ be the second point of intersection of $C_{i+1}$ and $C_{i+2}.$ Prove that the circumcentres of the triangles $A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

%V0 Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality $$(n + 1)h_n+1 - nh_n > r.$$ Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$