MathDB

15th Cabri Clubs 2001

Part of Cabri Clubs

Subcontests

(1)
3

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

level B
p6. Let ABCDABCD be a parallelogram with ∠DAB\angle DAB acute. The bisector of ∠DBC\angle DBC intersects side CDCD at QQ and line ADAD at P. If AD=DPAD = DP, find the measure of angle ∠DQB\angle DQB.
p7. Construct the following figure formed by 44 circles of equal radius, tangent to each other and tangent to the sides of the parallelogram. https://cdn.artofproblemsolving.com/attachments/6/9/da7261850412b1277f331aa2e3c8bb6ceb5cf2.gif
p8. If in the previous problem the radius of the circles is 1cm, find the area of ​​the parallelogram.
p9. Given a square ABCDABCD, construct an isosceles triangle PQRPQR with the same perimeter as ABCDABCD and such that the base of PQRPQR measures half of the other two sides.

15th Cabri Clubs 2001, round 1, level A, 5 problems, Argentinian geo contest

level A
p1. Construct the following figure where ABCABC, CDECDE and FGHFGH are equilateral triangles, AEFHAEFH is a rectangle and AC=CE=EFAC = CE = EF.https://cdn.artofproblemsolving.com/attachments/f/b/5f39b1462b52f0404a6d73ef9343cfb8bb39f5.gif
p2. In the figure in problem 1, find the measure of segment BGBG.
p3. In the figure in problem 1, find the angle formed by lines BHBH and DFDF.
p4. Let ABCDABCD be a square, of area 3030. Let PP be a point on side ABAB such that 2AP=PB2 AP = PB. Let QQ be a point on side BCBC such that BQ=QCBQ = QC. Let RR be a point on side ADAD such that 2DR=RA2 DR = RA. Line PQPQ intersects line CDCD at MM and line PRPR intersects line CDCD at NN. Find the area of ​​the triangle MNPMNP.
p5. Given a square ABCDABCD, construct an isosceles triangle of equal area and with one side equal to the side of the square.

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

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.