2016 Greece Team Selection Test
Given is a triangle $\triangle{ABC}$,with $AB<AC<BC$,inscribed in circle $c(O,R)$.Let $D,E,Z$ be the midpoints of $BC,CA,AB$ respectively,and $K$ the foot of the altitude from $A$.At the exterior of $\triangle{ABC}$ and with the sides $AB,AC$ as diameters,we construct the semicircles $c_1,c_2$ respectively.Suppose that $P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1$ and $R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2$.Finally,let $M$ be the intersection of the lines $PS,RT$.



i. Prove that the lines $PR,ST$ intersect at $A$.



ii. Prove that the lines $PR\cap MD$ intersect on $c$.



[asy]import graph; size(8cm); 

real labelscalefactor = 0.5; /* changes label-to-point distance */

pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ 

pen dotstyle = black; /* point style */ 

real xmin = -4.8592569519241255, xmax = 12.331775417316715, ymin = -3.1864435704043403, ymax = 6.540061585876658;  /* image dimensions */

pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); 



draw((0.6699432366054657,3.2576036755978928)--(0.,0.)--(5.,0.)--cycle, aqaqaq); 

 /* draw figures */

draw((0.6699432366054657,3.2576036755978928)--(0.,0.), uququq); 

draw((0.,0.)--(5.,0.), uququq); 

draw((5.,0.)--(0.6699432366054657,3.2576036755978928), uququq); 

draw(shift((0.33497161830273287,1.6288018377989464))*xscale(1.662889476749906)*yscale(1.662889476749906)*arc((0,0),1,78.3788505217281,258.3788505217281)); 

draw(shift((2.834971618302733,1.6288018377989464))*xscale(2.7093067970187343)*yscale(2.7093067970187343)*arc((0,0),1,-36.95500560847834,143.0449943915217)); 

draw((0.6699432366054657,3.2576036755978928)--(0.6699432366054657,0.)); 

draw((-0.9938564482532047,2.628510486065423)--(2.5,0.)); 

draw((0.6699432366054657,0.)--(0.,3.2576036755978923)); 

draw((0.6699432366054657,0.)--(5.,3.257603675597893)); 

draw((2.5,0.)--(3.3807330143335355,4.282570444700163)); 

draw((-0.9938564482532047,2.628510486065423)--(2.5,4.8400585427926455)); 

draw((2.5,4.8400585427926455)--(5.,3.257603675597893)); 

draw((-0.9938564482532047,2.628510486065423)--(3.3807330143335355,4.282570444700163), linewidth(1.2) + linetype("2 2")); 

draw((0.,3.2576036755978923)--(5.,3.257603675597893), linewidth(1.2) + linetype("2 2")); 

draw(circle((2.5,1.18355242571055), 2.766007292905304), linewidth(0.4) + linetype("2 2")); 

draw((2.5,4.8400585427926455)--(2.5,0.), linewidth(1.2) + linetype("2 2")); 

 /* dots and labels */

dot((0.6699432366054657,3.2576036755978928),linewidth(3.pt) + dotstyle); 

label("$A$", (0.7472169504504719,2.65), NE * labelscalefactor); 

dot((0.,0.),linewidth(3.pt) + dotstyle); 

label("$B$", (-0.2,-0.4), NE * labelscalefactor); 

dot((5.,0.),linewidth(3.pt) + dotstyle); 

label("$C$", (5.028818057451246,-0.34281415594345044), NE * labelscalefactor); 

dot((2.5,0.),linewidth(3.pt) + dotstyle); 

label("$D$", (2.4275434226319077,-0.32665717063401356), NE * labelscalefactor); 

dot((2.834971618302733,1.6288018377989464),linewidth(3.pt) + dotstyle); 

label("$E$", (3.073822835009383,1.5637101105701008), NE * labelscalefactor); 

dot((0.33497161830273287,1.6288018377989464),linewidth(3.pt) + dotstyle); 

label("$Z$", (0.003995626216375389,1.402140257475732), NE * labelscalefactor); 

dot((0.6699432366054657,0.),linewidth(3.pt) + dotstyle); 

label("$K$", (0.6179610679749769,-0.3105001853245767), NE * labelscalefactor); 

dot((-0.9938564482532047,2.628510486065423),linewidth(3.pt) + dotstyle); 

label("$P$", (-1.0785223895158957,2.7916409940873033), NE * labelscalefactor); 

dot((0.,3.2576036755978923),linewidth(3.pt) + dotstyle); 

label("$S$", (-0.14141724156855653,3.454077391774215), NE * labelscalefactor); 

dot((5.,3.257603675597893),linewidth(3.pt) + dotstyle); 

label("$T$", (5.061132028070119,3.3571354799175936), NE * labelscalefactor); 

dot((3.3807330143335355,4.282570444700163),linewidth(3.pt) + dotstyle); 

label("$R$", (3.445433497126431,4.375025554412117), NE * labelscalefactor); 

dot((2.5,4.8400585427926455),linewidth(3.pt) + dotstyle); 

label("$M$", (2.5567993051074027,4.940520040242407), NE * labelscalefactor); 

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 

 /* end of picture */[/asy]