MathDB

19th Cabri Clubs 2007

Part of Cabri Clubs

Subcontests

(1)
2

19th Cabri Clubs 2007, round 1, level 1, 3 problems, Argentinian geo contest

level 1
p1. Given an acute triangle ABCABC, point DD in BCBC and point EE in ACAC are marked, so that ∠AEB=∠ADB=90o\angle AEB = \angle ADB = 90^o. The bisectors of the angles ∠CAD\angle CAD and ∠CBE\angle CBE intersect at FF. Find the angle ∠AFB\angle AFB.
p2. [color=#f00](figure missing) Let ABCABC be an equilateral triangle with side 22 and let M,NM, N and PP be the midpoints of ABAB, BCBC and CACA respectively. Draw the three circles of radius 11 centered at A,BA, B, and CC. a) Calculate the area of ​​the shaded region in figure AA. b) The circle that passes through M,NM, N and PP. is drawn. Calculate the area of ​​the shaded region in figure BB.
p3. Let ABCABC be a triangle. Let DD be the midpoint of ABAB and let EE be a point on segment BCBC such that BE=2ECBE = 2 EC. Knowing that ∠BAE=∠ADC\angle BAE = \angle ADC , find the angle ∠BAC\angle BAC.

19th Cabri Clubs 2007, round 1, level 2, 3 problems, Argentinian geo contest

level 2
p4. Let ABCABC be a triangle with side ABAB less than side ACAC. DD is marked on the side ACAC, such that AD=ABAD = AB. Knowing that ∠ABC−∠ACB=48o\angle ABC - \angle ACB = 48^o, calculate the measure of the angle ∠DBC\angle DBC.
p5. Let ABCABC be a right triangle at AA. D,ED, E, and FF are marked on side BCBC so that ADAD is perpendicular to BCBC, AEAE is the bisector of ∠BAC\angle BAC, and FF is the midpoint of BCBC. Prove that AEAE is a bisector of ∠DAF\angle DAF.
p6. Let ABCDABCD be a parallelogram and EE a point on side BCBC. The line AEAE, which intersects the extension of the side DCDC at FF, and the diagonal BDBD, which intersects the segment AEAE at GG. Knowing that AG=6AG = 6 and GE=4GE = 4, find the length of the drawn segment EFEF.