MathDB

2

Part of 2008 Greece Team Selection Test

Problems(2)

Tourism In Five Villages

Source: 2008 Greek TST,Pr.2

5/25/2016
In a village X0X_0 there are 8080 tourists who are about to visit 55 nearby villages X1,X2,X3,X4,X5X_1,X_2,X_3,X_4,X_5.Each of them has chosen to visit only one of them.However,there are cases when the visit in a village forces the visitor to visit other villages among X1,X2,X3,X4,X5X_1,X_2,X_3,X_4,X_5.Each tourist visits only the village he has chosen and the villages he is forced to.If X1,X2,X3,X4,X5X_1,X_2,X_3,X_4,X_5 are totally visited by 40,60,65,70,7540,60,65,70,75 tourists respectively,then find how many tourists had chosen each one of them and determine all the ordered pairs (Xi,Xj):i,j∈{1,2,3,4,5}(X_i,X_j):i,j\in \{1,2,3,4,5\} which are such that,the visit in XiX_i forces the visitor to visit XjX_j as well.
combinatorics
Six Lines Are Concurrent

Source: 2008 Greek TST,Pr.3

5/25/2016
The bisectors of the angles ∠A,∠B,∠C\angle{A},\angle{B},\angle{C} of a triangle △ABC\triangle{ABC} intersect with the circumcircle c1(O,R)c_1(O,R) of △ABC\triangle{ABC} at A2,B2,C2A_2,B_2,C_2 respectively.The tangents of c1c_1 at A2,B2,C2A_2,B_2,C_2 intersect each other at A3,B3,C3A_3,B_3,C_3 (the points A3,AA_3,A lie on the same side of BCBC,the points B3,BB_3,B on the same side of CACA,and C3,CC_3,C on the same side of ABAB).The incircle c2(I,r)c_2(I,r) of △ABC\triangle{ABC} is tangent to BC,CA,ABBC,CA,AB at A1,B1,C1A_1,B_1,C_1 respectively.Prove that A1A2,B1B2,C1C2,AA3,BB3,CC3A_1A_2,B_1B_2,C_1C_2,AA_3,BB_3,CC_3 are concurrent.
[asy]import graph; size(11cm); 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 = -9.26871978147865, xmax = 19.467150423463277, ymin = -6.150626456647122, ymax = 10.10782642246474; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274);
draw((1.0409487561836381,4.30054785243355)--(0.,0.)--(6.,0.)--cycle, aqaqaq); /* draw figures */ draw((1.0409487561836381,4.30054785243355)--(0.,0.), uququq); draw((0.,0.)--(6.,0.), uququq); draw((6.,0.)--(1.0409487561836381,4.30054785243355), uququq); draw(circle((3.,1.550104087253063), 3.376806580383107)); draw(circle((1.9303371951242874,1.5188413314630436), 1.5188413314630436)); draw((1.0226422135625703,7.734611112525813)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((-1.2916762981259242,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((-0.2820306621765219,2.344520485530311)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((1.0559139088339535,1.4932847901569466)--(5.212367857300808,4.101231513568902), linetype("2 2")); draw((1.0559139088339535,1.4932847901569466)--(3.,-1.8267024931300442), linetype("2 2")); draw((12.047991949367804,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((1.0226422135625703,7.734611112525813)--(-1.2916762981259242,-1.8267024931300444)); draw((-1.2916762981259242,-1.8267024931300444)--(12.047991949367804,-1.8267024931300444)); draw((12.047991949367804,-1.8267024931300444)--(1.0226422135625703,7.734611112525813)); /* dots and labels */ dot((1.0409487561836381,4.30054785243355),linewidth(3.pt) + dotstyle); label("AA", (0.5889800538632699,4.463280489351154), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("BB", (-0.5723380089304358,-0.10096957139619551), NE * labelscalefactor); dot((6.,0.),linewidth(3.pt) + dotstyle); label("CC", (6.233525986976863,0.06107480945873997), NE * labelscalefactor); label("c1c_1", (1.9663572911302232,5.111458012770896), NE * labelscalefactor); dot((3.,-1.8267024931300442),linewidth(3.pt) + dotstyle); label("A2A_2", (2.9386235762598374,-2.3155761097469805), NE * labelscalefactor); dot((5.212367857300808,4.101231513568902),linewidth(3.pt) + dotstyle); label("B2B_2", (5.315274495465561,4.274228711687063), NE * labelscalefactor); dot((-0.2820306621765219,2.344520485530311),linewidth(3.pt) + dotstyle); label("C2C_2", (-0.9234341674494632,2.6807922999468636), NE * labelscalefactor); dot((1.0226422135625703,7.734611112525813),linewidth(3.pt) + dotstyle); label("A3A_3", (1.1291279900463889,7.893219884113956), NE * labelscalefactor); dot((-1.2916762981259242,-1.8267024931300444),linewidth(3.pt) + dotstyle); label("B3B_3", (-1.8146782621516093,-1.4783468086631473), NE * labelscalefactor); dot((12.047991949367804,-1.8267024931300444),linewidth(3.pt) + dotstyle); label("C3C_3", (12.148145888182015,-1.6673985863272387), NE * labelscalefactor); dot((1.9303371951242874,1.5188413314630436),linewidth(3.pt) + dotstyle); label("II", (2.047379481557691,1.681518618008095), NE * labelscalefactor); dot((1.9303371951242878,0.),linewidth(3.pt) + dotstyle); label("A1A_1", (1.4532167517562602,-0.5600953171518461), NE * labelscalefactor); label("c2c_2", (1.5072315453745722,3.247947632939138), NE * labelscalefactor); dot((2.9254299438737803,2.666303492733126),linewidth(3.pt) + dotstyle); label("B1B_1", (2.8576013858323694,3.1129106488933584), NE * labelscalefactor); dot((0.45412477306806903,1.8761589424582812),linewidth(3.pt) + dotstyle); label("C1C_1", (0,2.3296961414278368), NE * labelscalefactor); dot((1.0559139088339535,1.4932847901569466),linewidth(3.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]
geometrycircumcircle