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15th Cabri Clubs 2001, round 1, level C, 4 problems, Argentinian geo contest

Source:

December 4, 2021
geometrygeometric constructionconstructioncabri clubsLocus

Problem Statement

level C
p10. Construct the following figure, where ABCDABCD is a rectangle, PQCPQC is equilateral and 2PD=PA2 PD = PA. https://cdn.artofproblemsolving.com/attachments/3/d/6fd25309c373e91b8d837df7378725b6b6fd7d.gif
p11. Let ABCABC be an isosceles triangle, with base AB=10AB = 10 cm. Let MM and NN be the midpoints of the sides ACAC and BCBC respectively. Let GG be the point of intersection of BMBM and NANA. If the angle AGBAGB is right, find the area of ​​ABCABC.
p12. Let A,B,CA, B, C, and DD be four collinear points such that AB=BC=CDAB = BC = CD. Let PP be a point on the plane such that 2∠APB=∠BPC=2∠PD2 \angle APB = \angle BPC = 2 \angle PD. Find the measure of angle ∠APB\angle APB.
p13. Let S and RR be two circles with centers O1O_1 and O2O_2 respectively. Let PP be a point on SS. A parallel to O1O2O_1O_2 is drawn by PP, which intersects RR at AA and BB. Lines PO1PO_1 and AO2AO_2 intersect at point EE. Find the locus of EE as PP moves on SS.
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