MathDB

2nd Cabri Clubs 1996

Part of Cabri Clubs

Subcontests

(1)
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1996 Cabri Clubs 2nd, finals, level 1, 4 problems, Argentinian geo contest

level 1
p1. Three points are given O,GO, G and MM. Construct a triangle in such a way that OO is its circumcenter, GG is its centroid, and MM is the midpoint of one side.
p2. Let ABCABC be a triangle and HH its orthocenter. The height is drawn from AA, which intersects BCBC at DD. On the extension of the altitude ADAD the point EE is taken in such a way that the angles ∠CAD\angle CAD and ∠CBE\angle CBE are equal. Prove that BE=BHBE = BH.
p3. Let ω\omega be a circle, and MM be a variable point on its exterior. From MM the tangents to ω\omega . Let AA and BB be the touchpoints. Find the locus of the incenter of the triangle MABMAB as MM varies.
p4. i) Find a point DD in the interior of a triangle ABCABC such that the areas of the triangles ABDABD, BCDBCD and CADCAD are equal.
ii) The same as i) but with DD outside ABCABC.

1996 Cabri Clubs 2nd Round 1, 6 problems, Argentinian geo contest

level 1
p1. Construct the given figure, where ABCDABCD is a square and AEFAEF is an equilateral triangle. https://cdn.artofproblemsolving.com/attachments/f/c/1b4043aeed5992ddb8739eec5a8e72ebf4cf91.gif
p2. Let ABCABC be an isosceles triangle (AB=ACAB = AC). We draw the perpendicular bisector mm of ACAC and the bisector nn of angle ∠C\angle C. If m,nm, n and ABAB intersect at a single point, how much is angle ∠A\angle A?
p3. Let A A, B B, and CC be points on a circle. Let us call the orthocenter of the triangle HH. Find the locus of HH as AA moves around the circle.

level 2
p4. Given 33 points A A, B B and CC, construct the isosceles trapezoid ABCDABCD where AB=CDAB = CD and BCBC is parallel to ADAD (BCBC different from ADAD).
p5. Let A A, B B and CC be points on a circle. Let's call the centroid of the triangle GG. Find the locus of GG as AA moves along the circle.
p6. Given a triangle ABCABC, let DD, EE, and FF be the midpoints of the sides BCBC, CACA, and ABAB, respectively. From DD the lines M1M_1 and M2M_2 are drawn, perpendicular on ABAB and ACAC respectively. From EE the lines M3M_3 and M4M_4 are drawn, perpendicular on BCBC and ABAB respectively. From FF the lines M5M_5 and M6M_6 are drawn perpendicular on ACAC and BCBC respectively. Let A′A' be the intersection between M4M_4 and M5M_5. Let B′B' be the intersection between M6M_6 and M1M_1. Let C′C' be the intersection between M2M_2 and M3M_3. Show that the triangles ABCABC and A′B′C′A'B'C' are similar and find the ratio of similarity.

1996 Cabri Clubs 2nd, finals, level 2, 4 problems, Argentinian geo contest

level 2
p5. Let ABCABC be a triangle and MM be a variable point on ABAB. NN is the point on the prolongation of ACAC such that CN=BMCN = BM and that it does not belong to the ray CACA. The parallelogram BMNPBMNP is constructed (in that order). Find the locus of PP as MM varies.
p6. Let ABCDABCD be a quadrilateral. Let C1,C2,C3,C4C_1, C_2, C_3, C_4 be the circles of diameters AB,BC,CDAB, BC, CD and DADA respectively. Let P,Q,RP, Q, R and SS be the points of intersection (which are not vertices of ABCDABCD) of C1C_1 and C2C_2, C2C_2 and C3C_3, C3C_3 and C4C_4, C4C_4 and C1C_1 respectively. Show that the quadrilaterals ABCDABCD and PQRSPQRS are similar.
p7. M,NM, N, and P are three collinear points, with NN between MM and PP. Let r r be the perpendicualr bisector of NPNP. A point OO is taken over r r. ω\omega is the circle with center OO passing through NN. The tangents through MM to ω\omega intersect ω\omega at TT and T′T'. Find the locus of the centroid of the triangle MTT′MTT' a OO varies over rr.
p8. P,QP, Q and RR are the centers of three circles that pass through the same point OO. Let AA, BB and CC be the points (other than OO) of intersection of the circles. Prove that A,BA, B and CC are collinear if and only if O,Q,PO, Q, P and RR are in the same circle.
[hide=PS.] p8 might have a typo, as [url=https://www.oma.org.ar/enunciados/2da2da.htm]here in my source it was incorrect, and I tried correcting it.