Subcontests
(1)1996 Cabri Clubs 1st , Round 2, 6 problems, Argentinian geo contest
level 1p1. Given a circle ω and two points A and M on ω. Take a point B on ω and construct a point C such that the segment AM is the median of ABC. Find B so that the area of triangle ABC is maximum.
p2. A triangle ABC and a point P are given. Let S1 be the symmetric point of P wrt A, S2 the symmetric point of S1 wrt B and S3 the symmetric point of S2 wrt C. Find the locus of S3 as P moves.
p3. Given a segment AB and a point X on AB. The line r, perpendicular to AB, is drawn at X. Let O be a point on the line r and ω the circle with center O passing through A. The tangents to ω are traced through B, touching ω at T1 and T2. Find the locus of T1 and T2 as O moves on r.level 2
p4. Given a circle ω and a chord DA on the circle. Take another point B on ω (different from A and D) and draw the line s passing through A perpendicular on DB. The intersection point of line s and line DB is C. Find point B so that the area of triangle ABC is maximum.
p5. Given a triangle ABC, let P be any point in the plane. The perpendicular bisectors of AP, BP and CP that intersect at three points, A′, B′ and C′, are drawn. Find a point P so that the triangle ABC and A′B′C′ are similar.
p6. Let C1 and C2 be two intersecting circles at A and B where AB is the diameter of C1. Let P be a point on C2, in the interior of C1. Construct two points C and D on C1 such that CD is perpendicular to AB and ∠CPD is right, using only the constructions of:
∙ a line given by two points
∙ intersection of 2 objects
∙ line perpendicular on a line. 1996 Cabri Clubs 1st, finals, 6 problems, Argentinian geo contest
level 1
p1. i. Given three not collinear points A, B and C, find points A′, B′ and C′ on BC, AC and AB, respectively, such that the triangles ABC and A′B′C′ have the same centroid.
ii. Given three not collinear points A, B and C, find points A′, B′ and C′ on BC, AC and AB, respectively, such that the triangles ABC and A′B′C′ have the same circumcenter.
p2. Given a parallelogram ABCD, find four points A′, B′, C′ and D′ all interior to the parallelogram such that the triples of points AA′D′, BB′A′, CC′B′ and DD′C′ are collinear and the polygons ADD′, DC′C, CB′B, AA′B and A′B′C′D′ have equal areas.
p3. Construct a circle with center O. Let AB and SJ be two perpendicular diameters, in the minor arc BS take a variable point H . For each H we determine F, the intersection point between AH and SB and T the intersection point between SA and BH.
i. Show that TF and AB are orthogonal.
ii. Find the locus of the circumcenter of FSH.
level 2
p4. Given a triangle ABC, find the locus of the points P inside the triangle such that d(P,BC)=d(P,AC)+d(P,AB).Note: d(P,BC) is the distance from P from line BC.
p5. Construct a circle and take a fixed point A on it. Draw a variable line r through A that is secant to the circle at A and B. Construct an isosceles trapezoid ABCD of bases AB and CD such that AD=DC=1/2AB.
i. Show that line BC passes through a fixed point.
ii. Find the locus of C.
iii. A line t perpendicular to AC is drawn through D. Let P be the intersection point between t and AB. Take J on the ray AB and X on the ray PD such that AJ=PX. Show that the perpendicular bisector of JX passes through the incenter of triangle APD.
p6. Construct a triangle ABC, and then find points A′, B′ and C′ on BC, AC and AB, respectively, such that A′B′ is perpendicular to BC, B′C′ is perpendicular to AC and C′A′ is perpendicular to AB.