MathDB

17th Cabri Clubs 2002

Part of Cabri Clubs

Subcontests

(1)
3

17th Cabri Clubs 2002, round 1, level A, 4 problems, Argentinian geo contest

level A
p1. Construct the following figure where C1C_1, C2C_2 and C3C_3 are three circles of radii 33, 33 and 22 respectively, tangent to each other. https://cdn.artofproblemsolving.com/attachments/9/7/841f4d087876cdf5cadb3211a1b3409bce42a4.gif
p2. In the figure of the previous problem, find the lengths of O3AO_3A and O1BO_1B.
p3. Let ABCDABCD be a trapezoid with AB∥CDAB \parallel CD such that AB=BCAB = BC . CD=ACCD = AC and ∠BCD=60o\angle BCD = 60^o. Find the measure of the angles of the trapezoid ABCDABCD.
p4. Construct the figure of the previous problem.

17th Cabri Clubs 2002, round 1, level B, 4 problems, Argentinian geo contest

level B
p5. Construct the following figure formed by a square and two equilateral triangles and calculate the length of the segment PQPQ knowing that AB=2AB = 2. https://cdn.artofproblemsolving.com/attachments/3/e/99cd1f1c5145e5f736f3cefcf41dae04f8b44e.gif
p6. Construct two equilateral triangles, so that the smaller one has one sixth of the area that the larger one has.
p7. Let ABCABC be any triangle and MM the midpoint of BCBC. Construct a circle that passes through AA and is tangent to side BCBC at point MM.
p8. If in the previous problem, the radius of the circle has length 1 1 and the altitude from AA has length 1.51.5, find the length of AMAM and the possible measures of the angle ∠APM\angle APM where PP is the intersection of the circle with ABAB.

17th Cabri Clubs 2002, round 1, level C, 4 problems, Argentinian geo contest

level C
p9. For the pentagon ABCDEABCDE, we know that: ∙\bullet AB = 10cm ∙\bullet BC = 12cm ∙\bullet CD = 15cm ∙\bullet EA = 12cm. ∙\bullet Also, the angles B,CB, C, and EE of the pentagon are right. Find the area of ​​the pentagon.
p10. Given three points MM, DD and EE, construct a triangle ABCABC such that: Let MM be the midpoint of BCBC. Let DD be the foot of the altitude from AA Let EE be the foot of the altitude from BB
p11. Construct the following figure where: https://cdn.artofproblemsolving.com/attachments/a/1/6155e642acd25e339fba51ad02c2d5e543199f.gif ∙\bullet AD=CDAD = CD and AE=EBAE = EB ∙\bullet ∠CDB=45o\angle CDB = 45^o and ∠EBC=60o\angle EBC = 60^o
p12. Let ABCABC be an equilateral triangle with side kk and let S1S_1, S2S_2 and S3S_3 be the circles of radius kk and centers A,BA, B and CC respectively. Let XX be the point of intersection between S1S_1 and S2S_2 (other than CC). Let YY be the point of intersection of S2S_2 and S3S_3 (other than AA). Let ZZ be the point of intersection of S3S_3 and S1S_1 (other than BB). Let PP be a point in S1S_1. Draw the line PXPX that intersects S2S_2 at QQ. Draw the line QYQY that intersect S3S_3 at RR and draw the line RZRZ that intersect S1S_1 at TT. Prove that PTPT is a diameter of S1S_1.