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## The solution to another problem of Putnam Mathematical Competition 2019

Here is two solutions with Maple of the problem A2 of  Putnam Mathematical Competition 2019 . The first solution is entirely based on the use of the  geometry  package; the second solution does not use this package. Since the triangle is defined up to similarity, without loss of generality, we can set its vertices  A(0,0) , B(1,0) , C(x0,y0)  and then calculate the parameters  x0, y0  using the conditions of the problem.

The problem

A2: In the triangle ∆ABC, let G be the centroid, and let I be the center of the
inscribed circle. Let α and β be the angles at the vertices A and B, respectively.
Suppose that the segment IG is parallel to AB and that  β = 2*arctan(1/3).  Find α.

 > # Solution 1 with the geometry package restart; # Calculation with(geometry): local I: point(A,0,0): point(B,1,0): point(C,x0,y0): assume(y0>0,-y0*(-1+x0-((1-x0)^2+y0^2)^(1/2))+y0*((x0^2+y0^2)^(1/2)+x0) <> 0): triangle(t,[A,B,C]): incircle(ic,t, 'centername'=I): Cn:=coordinates(I): centroid(G,t): CG:=coordinates(G): a:=-expand(tan(2*arctan(1/3))): solve({Cn[2]=CG[2],y0/(x0-1)=a}, explicit); point(C,eval([x0,y0],%)): answer=FindAngle(line(AB,[A,B]),line(AC,[A,C])); # Visualization (G is the point of medians intersection) triangle(t,[A,B,C]): incircle(ic,t, 'centername'=I): centroid(G,t): segment(s,[I,G]): median(mB,B,t): median(mC,C,t): draw([A(symbol=solidcircle,color=black),B(symbol=solidcircle,color=black),C(symbol=solidcircle,color=black),I(symbol=solidcircle,color=green),G(symbol=solidcircle,color=blue),t(color=black),s(color=red,thickness=2),ic(color=green),mB(color=blue,thickness=0),mC(color=blue,thickness=0)], axes=none, size=[800,500], printtext=true,font=[times,20]);
 > # Solution 2 by a direct calculation # Calculation restart; local I; sinB:=y0/sqrt(x0^2+y0^2): cosB:=x0/sqrt(x0^2+y0^2): Sol1:=eval([x,y],solve({y=-(x-1)/3,y=(sinB/(1+cosB))*x}, {x,y})): tanB:=expand(tan(2*arctan(1/3))): Sol2:=solve({y0/3=Sol1[2],y0=-tanB*(x0-1)},explicit); A:=[0,0]: B:=[1,0]: C:=eval([x0,y0],Sol2[2]): AB:=<(B-A)[]>: AC:=<(C-A)[]>: answer=arccos(AB.AC/sqrt(AB.AB)/sqrt(AC.AC)); # Visualization with(plottools): with(plots): ABC:=curve([A,B,C,A]): I:=simplify(eval(Sol1,Sol2[2])); c:=circle(I,eval(Sol1[2],Sol2[2]),color=green): G:=(A+B+C)/~3; IG:=line(I,G,color=red,thickness=2): P:=pointplot([A,B,C,I,G], color=[black\$3,green,blue], symbol=solidcircle): T:=textplot([[A[],"A"],[B[],"B"],[C[],"C"],[I[],"I"],[G[],"G"]], font=[times,20], align=[left,below]): M:=plot([[(C+t*~((A+B)/2-C))[],t=0..1],[(B+t*~((A+C)/2-B))[],t=0..1]], color=blue, thickness=0): display(ABC,c,IG,P,T,M, scaling=constrained, axes=none,size=[800,500]);
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